Theorem vafval 8207

Description: Value of the function for the vector addition (group) operation on a normed complex vector space.

Hypothesis

Ref	Expression
vafval.2	|- G = (+v` U)

Assertion

Ref	Expression
vafval	|- G = (1st` (1st` U))

Proof of Theorem vafval

Step	Hyp	Ref	Expression
1		vafval.2	. 2 |- G = (+v` U)
2		fo1st 4091	. . . . . 6 |- 1st:V-onto->V
3		fofun 3673	. . . . . 6 |- (1st:V-onto->V -> Fun 1st)
4	2, 3	ax-mp 7	. . . . 5 |- Fun 1st
5		fof 3672	. . . . . 6 |- (1st:V-onto->V -> 1st:V-->V)
6	2, 5	ax-mp 7	. . . . 5 |- 1st:V-->V
7		fvco3 3776	. . . . 5 |- ((Fun 1st /\ 1st:V-->V /\ U e. V) -> ((1st o. 1st)` U) = (1st` (1st` U)))
8	4, 6, 7	mp3an12 906	. . . 4 |- (U e. V -> ((1st o. 1st)` U) = (1st` (1st` U)))
9		df-va 8199	. . . . 5 |- +v = (1st o. 1st)
10	9	fveq1i 3725	. . . 4 |- (+v` U) = ((1st o. 1st)` U)
11	8, 10	syl5eq 1519	. . 3 |- (U e. V -> (+v` U) = (1st` (1st` U)))
12		fvprc 3721	. . . 4 |- (-. U e. V -> (+v` U) = (/))
13		fvprc 3721	. . . . . 6 |- (-. U e. V -> (1st` U) = (/))
14	13	fveq2d 3728	. . . . 5 |- (-. U e. V -> (1st` (1st` U)) = (1st` (/)))
15		1st0 4083	. . . . 5 |- (1st` (/)) = (/)
16	14, 15	syl6req 1524	. . . 4 |- (-. U e. V -> (/) = (1st` (1st` U)))
17	12, 16	eqtrd 1507	. . 3 |- (-. U e. V -> (+v` U) = (1st` (1st` U)))
18	11, 17	pm2.61i 126	. 2 |- (+v` U) = (1st` (1st` U))
19	1, 18	eqtr 1495	1 |- G = (1st` (1st` U))

Syntax hints:  -. wn 2   = wceq 956   e. wcel 958  Vcvv 1811  (/)c0 2280   o. ccom 3174  Fun wfun 3176  -->wf 3178  -onto->wfo 3180  ` cfv 3182  1stc1st 4077  +vcpv 8189

This theorem is referenced by:  nvvop 8213  nvi 8218  nvvc 8219  nvabl 8220  nvsf 8223  nvscl 8232  nvsid 8233  nvsass 8234  nvdi 8236  nvdir 8237  nv2 8238  nv0 8243  nvsz 8244  nvinv 8245  cnnvg 8293  sm1cnilem 8332  ipfval 8337  ipid 8348  sspval 8367  phop 8462  phpar 8468  ip0i 8469  ipdirilem 8473  h2hva 8828  hhssva 9114  hhshsslem1 9122  hhsssh2 9125

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fo 3196  df-fv 3198  df-1st 4079  df-va 8199

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