Theorem vcdir 8124

Description: Distributive 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
vcdir	|- ((W e. CVec /\ (A e. CC /\ B e. CC /\ C e. X)) -> ((A + B)SC) = ((ASC)G(BSC)))

Proof of Theorem vcdir

Step	Hyp	Ref	Expression
1		opreq2 3960	. . . . . 6 |- (x = C -> ((y + z)Sx) = ((y + z)SC))
2		opreq2 3960	. . . . . . 7 |- (x = C -> (ySx) = (ySC))
3		opreq2 3960	. . . . . . 7 |- (x = C -> (zSx) = (zSC))
4	2, 3	opreq12d 3969	. . . . . 6 |- (x = C -> ((ySx)G(zSx)) = ((ySC)G(zSC)))
5	1, 4	eqeq12d 1486	. . . . 5 |- (x = C -> (((y + z)Sx) = ((ySx)G(zSx)) <-> ((y + z)SC) = ((ySC)G(zSC))))
6		opreq1 3959	. . . . . . 7 |- (y = A -> (y + z) = (A + z))
7	6	opreq1d 3966	. . . . . 6 |- (y = A -> ((y + z)SC) = ((A + z)SC))
8		opreq1 3959	. . . . . . 7 |- (y = A -> (ySC) = (ASC))
9	8	opreq1d 3966	. . . . . 6 |- (y = A -> ((ySC)G(zSC)) = ((ASC)G(zSC)))
10	7, 9	eqeq12d 1486	. . . . 5 |- (y = A -> (((y + z)SC) = ((ySC)G(zSC)) <-> ((A + z)SC) = ((ASC)G(zSC))))
11		opreq2 3960	. . . . . . 7 |- (z = B -> (A + z) = (A + B))
12	11	opreq1d 3966	. . . . . 6 |- (z = B -> ((A + z)SC) = ((A + B)SC))
13		opreq1 3959	. . . . . . 7 |- (z = B -> (zSC) = (BSC))
14	13	opreq2d 3967	. . . . . 6 |- (z = B -> ((ASC)G(zSC)) = ((ASC)G(BSC)))
15	12, 14	eqeq12d 1486	. . . . 5 |- (z = B -> (((A + z)SC) = ((ASC)G(zSC)) <-> ((A + B)SC) = ((ASC)G(BSC))))
16	5, 10, 15	rcla43v 1878	. . . 4 |- ((C e. X /\ A e. CC /\ B e. CC) -> (A.x e. X A.y e. CC A.z e. CC ((y + z)Sx) = ((ySx)G(zSx)) -> ((A + B)SC) = ((ASC)G(BSC))))
17		vci.1	. . . . . 6 |- G = (1st` W)
18		vci.2	. . . . . 6 |- S = (2nd` W)
19		vci.3	. . . . . 6 |- X = ran G
20	17, 18, 19	vci 8119	. . . . 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)))))))
21		pm3.26 319	. . . . . . . . . . 11 |- ((((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx))) -> ((y + z)Sx) = ((ySx)G(zSx)))
22	21	r19.20si 1703	. . . . . . . . . 10 |- (A.z e. CC (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx))) -> A.z e. CC ((y + z)Sx) = ((ySx)G(zSx)))
23	22	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 + z)Sx) = ((ySx)G(zSx)))
24	23	r19.20si 1703	. . . . . . . 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 + z)Sx) = ((ySx)G(zSx)))
25	24	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 + z)Sx) = ((ySx)G(zSx)))
26	25	r19.20si 1703	. . . . . 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 + z)Sx) = ((ySx)G(zSx)))
27	26	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 + z)Sx) = ((ySx)G(zSx)))
28	20, 27	syl 10	. . . 4 |- (W e. CVec -> A.x e. X A.y e. CC A.z e. CC ((y + z)Sx) = ((ySx)G(zSx)))
29	16, 28	syl5 21	. . 3 |- ((C e. X /\ A e. CC /\ B e. CC) -> (W e. CVec -> ((A + B)SC) = ((ASC)G(BSC))))
30	29	3coml 839	. 2 |- ((A e. CC /\ B e. CC /\ C e. X) -> (W e. CVec -> ((A + B)SC) = ((ASC)G(BSC))))
31	30	impcom 351	1 |- ((W e. CVec /\ (A e. CC /\ B e. CC /\ C e. X)) -> ((A + B)SC) = ((ASC)G(BSC)))

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  vcsubdir 8127  vc0 8140  vcm 8142  nvdir 8204

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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