Theorem vcdir 9504

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 4890	. . . . . 6 |- (x = C -> ((y + z)Sx) = ((y + z)SC))
2		opreq2 4890	. . . . . . 7 |- (x = C -> (ySx) = (ySC))
3		opreq2 4890	. . . . . . 7 |- (x = C -> (zSx) = (zSC))
4	2, 3	opreq12d 4900	. . . . . 6 |- (x = C -> ((ySx)G(zSx)) = ((ySC)G(zSC)))
5	1, 4	eqeq12d 1899	. . . . 5 |- (x = C -> (((y + z)Sx) = ((ySx)G(zSx)) <-> ((y + z)SC) = ((ySC)G(zSC))))
6		opreq1 4889	. . . . . . 7 |- (y = A -> (y + z) = (A + z))
7	6	opreq1d 4897	. . . . . 6 |- (y = A -> ((y + z)SC) = ((A + z)SC))
8		opreq1 4889	. . . . . . 7 |- (y = A -> (ySC) = (ASC))
9	8	opreq1d 4897	. . . . . 6 |- (y = A -> ((ySC)G(zSC)) = ((ASC)G(zSC)))
10	7, 9	eqeq12d 1899	. . . . 5 |- (y = A -> (((y + z)SC) = ((ySC)G(zSC)) <-> ((A + z)SC) = ((ASC)G(zSC))))
11		opreq2 4890	. . . . . . 7 |- (z = B -> (A + z) = (A + B))
12	11	opreq1d 4897	. . . . . 6 |- (z = B -> ((A + z)SC) = ((A + B)SC))
13		opreq1 4889	. . . . . . 7 |- (z = B -> (zSC) = (BSC))
14	13	opreq2d 4898	. . . . . 6 |- (z = B -> ((ASC)G(zSC)) = ((ASC)G(BSC)))
15	12, 14	eqeq12d 1899	. . . . 5 |- (z = B -> (((A + z)SC) = ((ASC)G(zSC)) <-> ((A + B)SC) = ((ASC)G(BSC))))
16	5, 10, 15	rcla43v 2386	. . . 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 9499	. . . . 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		simpl 346	. . . . . . . . . . 11 |- ((((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx))) -> ((y + z)Sx) = ((ySx)G(zSx)))
22	21	ralimi 2168	. . . . . . . . . 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 424	. . . . . . . . 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	ralimi 2168	. . . . . . . 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 424	. . . . . . 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	ralimi 2168	. . . . . 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 899	. . . . 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 12	. . . 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 20	. . 3 |- ((C e. X /\ A e. CC /\ B e. CC) -> (W e. CVec -> ((A + B)SC) = ((ASC)G(BSC))))
30	29	3coml 1075	. 2 |- ((A e. CC /\ B e. CC /\ C e. X) -> (W e. CVec -> ((A + B)SC) = ((ASC)G(BSC))))
31	30	impcom 378	1 |- ((W e. CVec /\ (A e. CC /\ B e. CC /\ C e. X)) -> ((A + B)SC) = ((ASC)G(BSC)))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105   X. cxp 3984  ran crn 3987  -->wf 3994  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  CCcc 6384  1c1 6387   + caddc 6389   x. cmul 6391  Abelcabl 9407  CVeccvc 9496

This theorem is referenced by:  vc2 9506  vcsubdir 9507  vc0 9520  vcm 9522  nvdir 9584

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-1st 5020  df-2nd 5021  df-vc 9497

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