Theorem nn0ind-raph 6223

Description: Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis. (Contributed by Raph Levien, 10-Apr-2004. Raph says: "This seems a bit painful. I wonder if an explicit substitution version would be easier.")

Hypotheses

Ref	Expression
nn0ind-raph.1	|- (x = 0 -> (ph <-> ps))
nn0ind-raph.2	|- (x = y -> (ph <-> ch))
nn0ind-raph.3	|- (x = (y + 1) -> (ph <-> th))
nn0ind-raph.4	|- (x = A -> (ph <-> ta))
nn0ind-raph.5	|- ps
nn0ind-raph.6	|- (y e. NN0 -> (ch -> th))

Assertion

Ref	Expression
nn0ind-raph	|- (A e. NN0 -> ta)

Distinct variable groups:   x,y   x,A   ps,x   ch,x   th,x   ta,x   ph,y

Proof of Theorem nn0ind-raph

Step	Hyp	Ref	Expression
1		elnn0 6110	. 2 |- (A e. NN0 <-> (A e. NN \/ A = 0))
2		dfsbcq 1946	. . . 4 |- (z = 1 -> ([z / x]ph <-> [1 / x]ph))
3		nn0ind-raph.2	. . . . 5 |- (x = y -> (ph <-> ch))
4	3	sbhyp 1943	. . . 4 |- (z = y -> ([z / x]ph <-> ch))
5		nn0ind-raph.3	. . . . 5 |- (x = (y + 1) -> (ph <-> th))
6	5	sbhyp 1943	. . . 4 |- (z = (y + 1) -> ([z / x]ph <-> th))
7		nn0ind-raph.4	. . . . 5 |- (x = A -> (ph <-> ta))
8	7	sbhyp 1943	. . . 4 |- (z = A -> ([z / x]ph <-> ta))
9		1re 5454	. . . . . . 7 |- 1 e. RR
10	9	elisseti 1821	. . . . . 6 |- 1 e. V
11	10	hbsbc1v 1953	. . . . 5 |- ([1 / x]ph -> A.x[1 / x]ph)
12		0nn0 6122	. . . . . . . 8 |- 0 e. NN0
13	12	elisseti 1821	. . . . . . 7 |- 0 e. V
14		nn0ind-raph.6	. . . . . . . . . . 11 |- (y e. NN0 -> (ch -> th))
15		eleq1a 1546	. . . . . . . . . . . 12 |- (0 e. NN0 -> (y = 0 -> y e. NN0))
16	12, 15	ax-mp 7	. . . . . . . . . . 11 |- (y = 0 -> y e. NN0)
17		nn0ind-raph.5	. . . . . . . . . . . . . . 15 |- ps
18		nn0ind-raph.1	. . . . . . . . . . . . . . 15 |- (x = 0 -> (ph <-> ps))
19	17, 18	mpbiri 194	. . . . . . . . . . . . . 14 |- (x = 0 -> ph)
20		eqeq2 1487	. . . . . . . . . . . . . . . 16 |- (y = 0 -> (x = y <-> x = 0))
21	20, 3	syl6bir 215	. . . . . . . . . . . . . . 15 |- (y = 0 -> (x = 0 -> (ph <-> ch)))
22	21	pm5.74d 587	. . . . . . . . . . . . . 14 |- (y = 0 -> ((x = 0 -> ph) <-> (x = 0 -> ch)))
23	19, 22	mpbii 193	. . . . . . . . . . . . 13 |- (y = 0 -> (x = 0 -> ch))
24	23	com12 11	. . . . . . . . . . . 12 |- (x = 0 -> (y = 0 -> ch))
25	13, 24	vtocle 1861	. . . . . . . . . . 11 |- (y = 0 -> ch)
26	14, 16, 25	sylc 68	. . . . . . . . . 10 |- (y = 0 -> th)
27	26	adantr 391	. . . . . . . . 9 |- ((y = 0 /\ x = 1) -> th)
28		opreq1 3975	. . . . . . . . . . . . 13 |- (y = 0 -> (y + 1) = (0 + 1))
29		ax1cn 5288	. . . . . . . . . . . . . 14 |- 1 e. CC
30	29	addid2 5350	. . . . . . . . . . . . 13 |- (0 + 1) = 1
31	28, 30	syl6eq 1526	. . . . . . . . . . . 12 |- (y = 0 -> (y + 1) = 1)
32	31	eqeq2d 1489	. . . . . . . . . . 11 |- (y = 0 -> (x = (y + 1) <-> x = 1))
33	32, 5	syl6bir 215	. . . . . . . . . 10 |- (y = 0 -> (x = 1 -> (ph <-> th)))
34	33	imp 350	. . . . . . . . 9 |- ((y = 0 /\ x = 1) -> (ph <-> th))
35	27, 34	mpbird 196	. . . . . . . 8 |- ((y = 0 /\ x = 1) -> ph)
36	35	ex 373	. . . . . . 7 |- (y = 0 -> (x = 1 -> ph))
37	13, 36	vtocle 1861	. . . . . 6 |- (x = 1 -> ph)
38		sbceq1a 1947	. . . . . 6 |- (x = 1 -> (ph <-> [1 / x]ph))
39	37, 38	mpbid 195	. . . . 5 |- (x = 1 -> [1 / x]ph)
40	11, 10, 39	vtoclef 1860	. . . 4 |- [1 / x]ph
41		nnnn0t 6115	. . . . 5 |- (y e. NN -> y e. NN0)
42	41, 14	syl 10	. . . 4 |- (y e. NN -> (ch -> th))
43	2, 4, 6, 8, 40, 42	nnind 5946	. . 3 |- (A e. NN -> ta)
44		ax-17 973	. . . . . 6 |- (0 = A -> A.x0 = A)
45		ax-17 973	. . . . . 6 |- (ta -> A.xta)
46	44, 45	hbim 1009	. . . . 5 |- ((0 = A -> ta) -> A.x(0 = A -> ta))
47		eqeq1 1484	. . . . . 6 |- (x = 0 -> (x = A <-> 0 = A))
48	18	bicomd 523	. . . . . . . . 9 |- (x = 0 -> (ps <-> ph))
49	48, 7	sylan9bb 542	. . . . . . . 8 |- ((x = 0 /\ x = A) -> (ps <-> ta))
50	17, 49	mpbii 193	. . . . . . 7 |- ((x = 0 /\ x = A) -> ta)
51	50	ex 373	. . . . . 6 |- (x = 0 -> (x = A -> ta))
52	47, 51	sylbird 205	. . . . 5 |- (x = 0 -> (0 = A -> ta))
53	46, 13, 52	vtoclef 1860	. . . 4 |- (0 = A -> ta)
54	53	eqcoms 1481	. . 3 |- (A = 0 -> ta)
55	43, 54	jaoi 341	. 2 |- ((A e. NN \/ A = 0) -> ta)
56	1, 55	sylbi 199	1 |- (A e. NN0 -> ta)

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 958   e. wcel 960  [wsbc 1172  (class class class)co 3970  RRcr 5252  0cc0 5253  1c1 5254   + caddc 5256  NNcn 5315  NN0cn0 5316

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-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-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-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-sub 5375  df-neg 5377  df-n 5934  df-n0 6109

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