Theorem vcass 8173

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 3969	. . . . . 6 |- (x = C -> ((y x. z)Sx) = ((y x. z)SC))
2		opreq2 3969	. . . . . . 7 |- (x = C -> (zSx) = (zSC))
3	2	opreq2d 3976	. . . . . 6 |- (x = C -> (yS(zSx)) = (yS(zSC)))
4	1, 3	eqeq12d 1489	. . . . 5 |- (x = C -> (((y x. z)Sx) = (yS(zSx)) <-> ((y x. z)SC) = (yS(zSC))))
5		opreq1 3968	. . . . . . 7 |- (y = A -> (y x. z) = (A x. z))
6	5	opreq1d 3975	. . . . . 6 |- (y = A -> ((y x. z)SC) = ((A x. z)SC))
7		opreq1 3968	. . . . . 6 |- (y = A -> (yS(zSC)) = (AS(zSC)))
8	6, 7	eqeq12d 1489	. . . . 5 |- (y = A -> (((y x. z)SC) = (yS(zSC)) <-> ((A x. z)SC) = (AS(zSC))))
9		opreq2 3969	. . . . . . 7 |- (z = B -> (A x. z) = (A x. B))
10	9	opreq1d 3975	. . . . . 6 |- (z = B -> ((A x. z)SC) = ((A x. B)SC))
11		opreq1 3968	. . . . . . 7 |- (z = B -> (zSC) = (BSC))
12	11	opreq2d 3976	. . . . . 6 |- (z = B -> (AS(zSC)) = (AS(BSC)))
13	10, 12	eqeq12d 1489	. . . . 5 |- (z = B -> (((A x. z)SC) = (AS(zSC)) <-> ((A x. B)SC) = (AS(BSC))))
14	4, 8, 13	rcla43v 1882	. . . 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 8167	. . . . 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 1706	. . . . . . . . . 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 1706	. . . . . . . 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 1706	. . . . . 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 802	. . . . 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 840	. 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 775   = wceq 956   e. wcel 958  A.wral 1645   X. cxp 3168  ran crn 3171  -->wf 3178  ` cfv 3182  (class class class)co 3963  1stc1st 4077  2ndc2nd 4078  CCcc 5232  1c1 5235   + caddc 5237   x. cmul 5239  Abelcabl 8099  CVeccvc 8164

This theorem is referenced by:  vcsubdir 8175  vcz 8189  vcnegneg 8193  nvsass 8249

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-fv 3198  df-opr 3965  df-1st 4079  df-2nd 4080  df-vc 8165

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