Theorem vcass 8137

Description: Associative law 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
vcass	|- ((W e. CVec /\ (A e. CC /\ B e. CC /\ C e. X)) -> ((A x. B)SC) = (AS(BSC)))

Proof of Theorem vcass

Step	Hyp	Ref	Expression
1		opreq2 3964	. . . . . 6 |- (x = C -> ((y x. z)Sx) = ((y x. z)SC))
2		opreq2 3964	. . . . . . 7 |- (x = C -> (zSx) = (zSC))
3	2	opreq2d 3971	. . . . . 6 |- (x = C -> (yS(zSx)) = (yS(zSC)))
4	1, 3	eqeq12d 1487	. . . . 5 |- (x = C -> (((y x. z)Sx) = (yS(zSx)) <-> ((y x. z)SC) = (yS(zSC))))
5		opreq1 3963	. . . . . . 7 |- (y = A -> (y x. z) = (A x. z))
6	5	opreq1d 3970	. . . . . 6 |- (y = A -> ((y x. z)SC) = ((A x. z)SC))
7		opreq1 3963	. . . . . 6 |- (y = A -> (yS(zSC)) = (AS(zSC)))
8	6, 7	eqeq12d 1487	. . . . 5 |- (y = A -> (((y x. z)SC) = (yS(zSC)) <-> ((A x. z)SC) = (AS(zSC))))
9		opreq2 3964	. . . . . . 7 |- (z = B -> (A x. z) = (A x. B))
10	9	opreq1d 3970	. . . . . 6 |- (z = B -> ((A x. z)SC) = ((A x. B)SC))
11		opreq1 3963	. . . . . . 7 |- (z = B -> (zSC) = (BSC))
12	11	opreq2d 3971	. . . . . 6 |- (z = B -> (AS(zSC)) = (AS(BSC)))
13	10, 12	eqeq12d 1487	. . . . 5 |- (z = B -> (((A x. z)SC) = (AS(zSC)) <-> ((A x. B)SC) = (AS(BSC))))
14	4, 8, 13	rcla43v 1879	. . . 4 |- ((C e. X /\ A e. CC /\ B e. CC) -> (A.x e. X A.y e. CC A.z e. CC ((y x. z)Sx) = (yS(zSx)) -> ((A x. B)SC) = (AS(BSC))))
15		vci.1	. . . . . 6 |- G = (1st` W)
16		vci.2	. . . . . 6 |- S = (2nd` W)
17		vci.3	. . . . . 6 |- X = ran G
18	15, 16, 17	vci 8131	. . . . 5 |- (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)))))))
19		pm3.27 323	. . . . . . . . . . 11 |- ((((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx))) -> ((y x. z)Sx) = (yS(zSx)))
20	19	r19.20si 1704	. . . . . . . . . 10 |- (A.z e. CC (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx))) -> A.z e. CC ((y x. z)Sx) = (yS(zSx)))
21	20	adantl 388	. . . . . . . . 9 |- ((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.z e. CC ((y x. z)Sx) = (yS(zSx)))
22	21	r19.20si 1704	. . . . . . . 8 |- (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.y e. CC A.z e. CC ((y x. z)Sx) = (yS(zSx)))
23	22	adantl 388	. . . . . . 7 |- (((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.y e. CC A.z e. CC ((y x. z)Sx) = (yS(zSx)))
24	23	r19.20si 1704	. . . . . 6 |- (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 A.y e. CC A.z e. CC ((y x. z)Sx) = (yS(zSx)))
25	24	3ad2ant3 801	. . . . 5 |- ((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 A.y e. CC A.z e. CC ((y x. z)Sx) = (yS(zSx)))
26	18, 25	syl 10	. . . 4 |- (W e. CVec -> A.x e. X A.y e. CC A.z e. CC ((y x. z)Sx) = (yS(zSx)))
27	14, 26	syl5 21	. . 3 |- ((C e. X /\ A e. CC /\ B e. CC) -> (W e. CVec -> ((A x. B)SC) = (AS(BSC))))
28	27	3coml 839	. 2 |- ((A e. CC /\ B e. CC /\ C e. X) -> (W e. CVec -> ((A x. B)SC) = (AS(BSC))))
29	28	impcom 351	1 |- ((W e. CVec /\ (A e. CC /\ B e. CC /\ C e. X)) -> ((A x. B)SC) = (AS(BSC)))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 774   = wceq 955   e. wcel 957  A.wral 1643   X. cxp 3164  ran crn 3167  -->wf 3174  ` cfv 3178  (class class class)co 3958  1stc1st 4070  2ndc2nd 4071  CCcc 5215  1c1 5218   + caddc 5220   x. cmul 5222  Abelcabl 8062  CVeccvc 8128

This theorem is referenced by:  vcsubdir 8139  vcz 8153  vcnegneg 8157  nvsass 8213

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-fv 3194  df-opr 3960  df-1st 4072  df-2nd 4073  df-vc 8129

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