Theorem binom 7079

Description: The binomial theorem: (A + B)^N is the sum from k = 0 to N of (N C. k) x. ((A^k) x. (B^(N - k)). Theorem 15-2.8 of [Gleason] p. 296. This final piece of the proof combines the 0 < N case of binomlem6 7078 with the N = 0 case.

Hypotheses

Ref	Expression
binomlem.1	|- A e. CC
binomlem.2	|- B e. CC

Assertion

Ref	Expression
binom	|- (N e. NN0 -> ((A + B)^N) = sum_k e. (0...N)((N C. k) x. ((A^(N - k)) x. (B^k))))

Distinct variable groups:   A,k   B,k   k,N

Proof of Theorem binom

Step	Hyp	Ref	Expression
1		elnn0 6110	. 2 |- (N e. NN0 <-> (N e. NN \/ N = 0))
2		binomlem.1	. . . 4 |- A e. CC
3		binomlem.2	. . . 4 |- B e. CC
4	2, 3	binomlem6 7078	. . 3 |- (N e. NN -> ((A + B)^N) = sum_k e. (0...N)((N C. k) x. ((A^(N - k)) x. (B^k))))
5		ax1cn 5288	. . . . . 6 |- 1 e. CC
6		0z 6155	. . . . . 6 |- 0 e. ZZ
7		opreq2 3976	. . . . . . . . . 10 |- (k = 0 -> (0 C. k) = (0 C. 0))
8		0nn0 6122	. . . . . . . . . . 11 |- 0 e. NN0
9		bcn0t 6970	. . . . . . . . . . 11 |- (0 e. NN0 -> (0 C. 0) = 1)
10	8, 9	ax-mp 7	. . . . . . . . . 10 |- (0 C. 0) = 1
11	7, 10	syl6eq 1526	. . . . . . . . 9 |- (k = 0 -> (0 C. k) = 1)
12		opreq2 3976	. . . . . . . . . . . . . 14 |- (k = 0 -> (0 - k) = (0 - 0))
13		0cn 5347	. . . . . . . . . . . . . . 15 |- 0 e. CC
14	13	subid 5410	. . . . . . . . . . . . . 14 |- (0 - 0) = 0
15	12, 14	syl6eq 1526	. . . . . . . . . . . . 13 |- (k = 0 -> (0 - k) = 0)
16	15	opreq2d 3983	. . . . . . . . . . . 12 |- (k = 0 -> (A^(0 - k)) = (A^0))
17		exp0t 6579	. . . . . . . . . . . . 13 |- (A e. CC -> (A^0) = 1)
18	2, 17	ax-mp 7	. . . . . . . . . . . 12 |- (A^0) = 1
19	16, 18	syl6eq 1526	. . . . . . . . . . 11 |- (k = 0 -> (A^(0 - k)) = 1)
20		opreq2 3976	. . . . . . . . . . . 12 |- (k = 0 -> (B^k) = (B^0))
21		exp0t 6579	. . . . . . . . . . . . 13 |- (B e. CC -> (B^0) = 1)
22	3, 21	ax-mp 7	. . . . . . . . . . . 12 |- (B^0) = 1
23	20, 22	syl6eq 1526	. . . . . . . . . . 11 |- (k = 0 -> (B^k) = 1)
24	19, 23	opreq12d 3985	. . . . . . . . . 10 |- (k = 0 -> ((A^(0 - k)) x. (B^k)) = (1 x. 1))
25	5	mulid1 5351	. . . . . . . . . 10 |- (1 x. 1) = 1
26	24, 25	syl6eq 1526	. . . . . . . . 9 |- (k = 0 -> ((A^(0 - k)) x. (B^k)) = 1)
27	11, 26	opreq12d 3985	. . . . . . . 8 |- (k = 0 -> ((0 C. k) x. ((A^(0 - k)) x. (B^k))) = (1 x. 1))
28	27, 25	syl6eq 1526	. . . . . . 7 |- (k = 0 -> ((0 C. k) x. ((A^(0 - k)) x. (B^k))) = 1)
29	28	fsum1 7012	. . . . . 6 |- ((1 e. CC /\ 0 e. ZZ) -> sum_k e. (0...0)((0 C. k) x. ((A^(0 - k)) x. (B^k))) = 1)
30	5, 6, 29	mp2an 699	. . . . 5 |- sum_k e. (0...0)((0 C. k) x. ((A^(0 - k)) x. (B^k))) = 1
31	30	eqcomi 1482	. . . 4 |- 1 = sum_k e. (0...0)((0 C. k) x. ((A^(0 - k)) x. (B^k)))
32		opreq2 3976	. . . . 5 |- (N = 0 -> ((A + B)^N) = ((A + B)^0))
33	2, 3	addcl 5339	. . . . . 6 |- (A + B) e. CC
34		exp0t 6579	. . . . . 6 |- ((A + B) e. CC -> ((A + B)^0) = 1)
35	33, 34	ax-mp 7	. . . . 5 |- ((A + B)^0) = 1
36	32, 35	syl6eq 1526	. . . 4 |- (N = 0 -> ((A + B)^N) = 1)
37		opreq2 3976	. . . . 5 |- (N = 0 -> (0...N) = (0...0))
38		opreq1 3975	. . . . . . 7 |- (N = 0 -> (N C. k) = (0 C. k))
39		opreq1 3975	. . . . . . . . 9 |- (N = 0 -> (N - k) = (0 - k))
40	39	opreq2d 3983	. . . . . . . 8 |- (N = 0 -> (A^(N - k)) = (A^(0 - k)))
41	40	opreq1d 3982	. . . . . . 7 |- (N = 0 -> ((A^(N - k)) x. (B^k)) = ((A^(0 - k)) x. (B^k)))
42	38, 41	opreq12d 3985	. . . . . 6 |- (N = 0 -> ((N C. k) x. ((A^(N - k)) x. (B^k))) = ((0 C. k) x. ((A^(0 - k)) x. (B^k))))
43	42	adantr 391	. . . . 5 |- ((N = 0 /\ k e. (0...0)) -> ((N C. k) x. ((A^(N - k)) x. (B^k))) = ((0 C. k) x. ((A^(0 - k)) x. (B^k))))
44	37, 43	sumeq12rdv 7003	. . . 4 |- (N = 0 -> sum_k e. (0...N)((N C. k) x. ((A^(N - k)) x. (B^k))) = sum_k e. (0...0)((0 C. k) x. ((A^(0 - k)) x. (B^k))))
45	31, 36, 44	3eqtr4a 1535	. . 3 |- (N = 0 -> ((A + B)^N) = sum_k e. (0...N)((N C. k) x. ((A^(N - k)) x. (B^k))))
46	4, 45	jaoi 341	. 2 |- ((N e. NN \/ N = 0) -> ((A + B)^N) = sum_k e. (0...N)((N C. k) x. ((A^(N - k)) x. (B^k))))
47	1, 46	sylbi 199	1 |- (N e. NN0 -> ((A + B)^N) = sum_k e. (0...N)((N C. k) x. ((A^(N - k)) x. (B^k))))

Syntax hints:   -> wi 3   \/ wo 222   = wceq 958   e. wcel 960  (class class class)co 3970  CCcc 5251  0cc0 5253  1c1 5254   + caddc 5256   x. cmul 5258   - cmin 5311  NNcn 5315  NN0cn0 5316  ZZcz 5317  ...cfz 6475  ^cexp 6576   C. cbc 6963  sum_csu 6986

This theorem is referenced by:  binom1p 7080  efaddlem5 7349

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2699  ax-sep 2709  ax-nul 2716  ax-pow 2749  ax-pr 2786  ax-un 2873  ax-inf2 4641

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2006  df-dif 2053  df-un 2054  df-in 2055  df-ss 2057  df-pss 2059  df-nul 2285  df-if 2367  df-pw 2407  df-sn 2417  df-pr 2418  df-tp 2420  df-op 2421  df-uni 2509  df-int 2539  df-iun 2573  df-br 2626  df-opab 2673  df-tr 2687  df-eprel 2839  df-id 2842  df-po 2847  df-so 2857  df-fr 2924  df-we 2941  df-ord 2958  df-on 2959  df-lim 2960  df-suc 2961  df-om 3139  df-xp 3191  df-rel 3192  df-cnv 3193  df-co 3194  df-dm 3195  df-rn 3196  df-res 3197  df-ima 3198  df-fun 3199  df-fn 3200  df-f 3201  df-f1 3202  df-fo 3203  df-f1o 3204  df-fv 3205  df-rdg 3939  df-opr 3972  df-oprab 3973  df-1st 4086  df-2nd 4087  df-1o 4140  df-oadd 4142  df-omul 4143  df-er 4268  df-ec 4270  df-qs 4273  df-en 4375  df-dom 4376  df-sdom 4377  df-ni 5019  df-pli 5020  df-mi 5021  df-lti 5022  df-plpq 5054  df-mpq 5055  df-enq 5056  df-nq 5057  df-plq 5058  df-mq 5059  df-rq 5060  df-ltq 5061  df-1q 5062  df-np 5105  df-1p 5106  df-plp 5107  df-mp 5108  df-ltp 5109  df-plpr 5183  df-mpr 5184  df-enr 5185  df-nr 5186  df-plr 5187  df-mr 5188  df-ltr 5189  df-0r 5190  df-1r 5191  df-m1r 5192  df-c 5259  df-0 5260  df-1 5261  df-i 5262  df-r 5263  df-plus 5264  df-mul 5265  df-lt 5266  df-sub 5375  df-neg 5377  df-pnf 5506  df-mnf 5507  df-xr 5508  df-ltxr 5509  df-le 5510  df-div 5722  df-n 5934  df-n0 6109  df-z 6145  df-seq1 6316  df-shft 6349  df-uz 6426  df-fz 6476  df-seqz 6541  df-exp 6577  df-fac 6939  df-bc 6964  df-sum 6987

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