Theorem vafval 8174

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 4081	. . . . . 6 |- 1st:V-onto->V
3		fofun 3664	. . . . . 6 |- (1st:V-onto->V -> Fun 1st)
4	2, 3	ax-mp 7	. . . . 5 |- Fun 1st
5		fof 3663	. . . . . 6 |- (1st:V-onto->V -> 1st:V-->V)
6	2, 5	ax-mp 7	. . . . 5 |- 1st:V-->V
7		fvco3 3767	. . . . 5 |- ((Fun 1st /\ 1st:V-->V /\ U e. V) -> ((1st o. 1st)` U) = (1st` (1st` U)))
8	4, 6, 7	mp3an12 904	. . . 4 |- (U e. V -> ((1st o. 1st)` U) = (1st` (1st` U)))
9		df-va 8166	. . . . 5 |- +v = (1st o. 1st)
10	9	fveq1i 3716	. . . 4 |- (+v` U) = ((1st o. 1st)` U)
11	8, 10	syl5eq 1516	. . 3 |- (U e. V -> (+v` U) = (1st` (1st` U)))
12		fvprc 3712	. . . 4 |- (-. U e. V -> (+v` U) = (/))
13		fvprc 3712	. . . . . 6 |- (-. U e. V -> (1st` U) = (/))
14	13	fveq2d 3719	. . . . 5 |- (-. U e. V -> (1st` (1st` U)) = (1st` (/)))
15		1st0 4073	. . . . 5 |- (1st` (/)) = (/)
16	14, 15	syl6req 1521	. . . 4 |- (-. U e. V -> (/) = (1st` (1st` U)))
17	12, 16	eqtrd 1504	. . 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 1492	1 |- G = (1st` (1st` U))

Syntax hints:  -. wn 2   = wceq 954   e. wcel 956  Vcvv 1807  (/)c0 2276   o. ccom 3169  Fun wfun 3171  -->wf 3173  -onto->wfo 3175  ` cfv 3177  1stc1st 4067  +vcpv 8156

This theorem is referenced by:  nvvop 8180  nvi 8185  nvvc 8186  nvabl 8187  nvsf 8190  nvscl 8199  nvsid 8200  nvsass 8201  nvdi 8203  nvdir 8204  nv2 8205  nv0 8210  nvsz 8211  nvinv 8212  cnnvg 8259  sm1cnilem 8294  ipfval 8299  ipid 8310  sspval 8329  phop 8421  phpar 8427  ip0i 8428  ipdirilem 8432  h2hva 8782  hhssva 9068  hhshsslem1 9076  hhsssh2 9079

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fo 3191  df-fv 3193  df-1st 4069  df-va 8166

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