Theorem vcid 8122

Description: Identity element for the scalar product of a complex vector space.

Hypotheses

Ref	Expression
vci.1	|- G = (1st` W)
vci.2	|- S = (2nd` W)
vci.3	|- X = ran G

Assertion

Ref	Expression
vcid	|- ((W e. CVec /\ A e. X) -> (1SA) = A)

Proof of Theorem vcid

Step	Hyp	Ref	Expression
1		opreq2 3960	. . . 4 |- (x = A -> (1Sx) = (1SA))
2		id 59	. . . 4 |- (x = A -> x = A)
3	1, 2	eqeq12d 1486	. . 3 |- (x = A -> ((1Sx) = x <-> (1SA) = A))
4	3	rcla4cva 1872	. 2 |- ((A.x e. X (1Sx) = x /\ A e. X) -> (1SA) = A)
5		vci.1	. . . 4 |- G = (1st` W)
6		vci.2	. . . 4 |- S = (2nd` W)
7		vci.3	. . . 4 |- X = ran G
8	5, 6, 7	vci 8119	. . 3 |- (W e. CVec -> (G e. Abel /\ S:(CC X. X)-->X /\ A.x e. X ((1Sx) = x /\ A.y e. CC (A.z e. X (yS(xGz)) = ((ySx)G(ySz)) /\ A.z e. CC (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx)))))))
9		pm3.26 319	. . . . 5 |- (((1Sx) = x /\ A.y e. CC (A.z e. X (yS(xGz)) = ((ySx)G(ySz)) /\ A.z e. CC (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx))))) -> (1Sx) = x)
10	9	r19.20si 1703	. . . 4 |- (A.x e. X ((1Sx) = x /\ A.y e. CC (A.z e. X (yS(xGz)) = ((ySx)G(ySz)) /\ A.z e. CC (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx))))) -> A.x e. X (1Sx) = x)
11	10	3ad2ant3 801	. . 3 |- ((G e. Abel /\ S:(CC X. X)-->X /\ A.x e. X ((1Sx) = x /\ A.y e. CC (A.z e. X (yS(xGz)) = ((ySx)G(ySz)) /\ A.z e. CC (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx)))))) -> A.x e. X (1Sx) = x)
12	8, 11	syl 10	. 2 |- (W e. CVec -> A.x e. X (1Sx) = x)
13	4, 12	sylan 448	1 |- ((W e. CVec /\ A e. X) -> (1SA) = A)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 774   = wceq 954   e. wcel 956  A.wral 1642   X. cxp 3163  ran crn 3166  -->wf 3173  ` cfv 3177  (class class class)co 3954  1stc1st 4067  2ndc2nd 4068  CCcc 5212  1c1 5215   + caddc 5217   x. cmul 5219  Abelcabl 8050  CVeccvc 8116

This theorem is referenced by:  vc2 8126  vc0 8140  vcm 8142  vcnegneg 8145  vcoprne 8150  nvsid 8200

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-fv 3193  df-opr 3956  df-1st 4069  df-2nd 4070  df-vc 8117

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