Theorem vci 9499

Description: The properties of a complex vector space, which is an Abelian group (i.e. the vectors, with the operation of vector addition) accompanied by a scalar multiplication operation on the field of complex numbers. The variable W was chosen because _V is already used for the universal class.

Hypotheses

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

Assertion

Ref	Expression
vci	|- (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)))))))

Distinct variable groups:   x,y,z,G   x,S,y,z   x,X,y,z

Proof of Theorem vci

Step	Hyp	Ref	Expression
1		df-vc 9497	. . 3 |- CVec = {<.g, s>. | (g e. Abel /\ s:(CC X. ran g)-->ran g /\ A.x e. ran g((1sx) = x /\ A.y e. CC (A.z e. ran g(ys(xgz)) = ((ysx)g(ysz)) /\ A.z e. CC (((y + z)sx) = ((ysx)g(zsx)) /\ ((y x. z)sx) = (ys(zsx))))))}
2	1	eleq2i 1961	. 2 |- (W e. CVec <-> W e. {<.g, s>. | (g e. Abel /\ s:(CC X. ran g)-->ran g /\ A.x e. ran g((1sx) = x /\ A.y e. CC (A.z e. ran g(ys(xgz)) = ((ysx)g(ysz)) /\ A.z e. CC (((y + z)sx) = ((ysx)g(zsx)) /\ ((y x. z)sx) = (ys(zsx))))))})
3		vci.1	. . . . 5 |- G = (1st` W)
4	3	eqeq2i 1894	. . . 4 |- (g = G <-> g = (1st` W))
5		eleq1 1957	. . . . 5 |- (g = G -> (g e. Abel <-> G e. Abel))
6		rneq 4186	. . . . . . 7 |- (g = G -> ran g = ran G)
7		vci.3	. . . . . . 7 |- X = ran G
8	6, 7	syl6eqr 1946	. . . . . 6 |- (g = G -> ran g = X)
9		xpeq2 4017	. . . . . . . 8 |- (ran g = X -> (CC X. ran g) = (CC X. X))
10	9	feq2d 4557	. . . . . . 7 |- (ran g = X -> (s:(CC X. ran g)-->ran g <-> s:(CC X. X)-->ran g))
11		feq3 4553	. . . . . . 7 |- (ran g = X -> (s:(CC X. X)-->ran g <-> s:(CC X. X)-->X))
12	10, 11	bitrd 587	. . . . . 6 |- (ran g = X -> (s:(CC X. ran g)-->ran g <-> s:(CC X. X)-->X))
13	8, 12	syl 12	. . . . 5 |- (g = G -> (s:(CC X. ran g)-->ran g <-> s:(CC X. X)-->X))
14		opreq 4888	. . . . . . . . . . . 12 |- (g = G -> (xgz) = (xGz))
15	14	opreq2d 4898	. . . . . . . . . . 11 |- (g = G -> (ys(xgz)) = (ys(xGz)))
16		opreq 4888	. . . . . . . . . . 11 |- (g = G -> ((ysx)g(ysz)) = ((ysx)G(ysz)))
17	15, 16	eqeq12d 1899	. . . . . . . . . 10 |- (g = G -> ((ys(xgz)) = ((ysx)g(ysz)) <-> (ys(xGz)) = ((ysx)G(ysz))))
18	8, 17	raleqbidv 2274	. . . . . . . . 9 |- (g = G -> (A.z e. ran g(ys(xgz)) = ((ysx)g(ysz)) <-> A.z e. X (ys(xGz)) = ((ysx)G(ysz))))
19		opreq 4888	. . . . . . . . . . . 12 |- (g = G -> ((ysx)g(zsx)) = ((ysx)G(zsx)))
20	19	eqeq2d 1895	. . . . . . . . . . 11 |- (g = G -> (((y + z)sx) = ((ysx)g(zsx)) <-> ((y + z)sx) = ((ysx)G(zsx))))
21	20	anbi1d 679	. . . . . . . . . 10 |- (g = G -> ((((y + z)sx) = ((ysx)g(zsx)) /\ ((y x. z)sx) = (ys(zsx))) <-> (((y + z)sx) = ((ysx)G(zsx)) /\ ((y x. z)sx) = (ys(zsx)))))
22	21	ralbidv 2123	. . . . . . . . 9 |- (g = G -> (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)) /\ ((y x. z)sx) = (ys(zsx)))))
23	18, 22	anbi12d 690	. . . . . . . 8 |- (g = G -> ((A.z e. ran g(ys(xgz)) = ((ysx)g(ysz)) /\ A.z e. CC (((y + z)sx) = ((ysx)g(zsx)) /\ ((y x. z)sx) = (ys(zsx)))) <-> (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))))))
24	23	ralbidv 2123	. . . . . . 7 |- (g = G -> (A.y e. CC (A.z e. ran g(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. X (ys(xGz)) = ((ysx)G(ysz)) /\ A.z e. CC (((y + z)sx) = ((ysx)G(zsx)) /\ ((y x. z)sx) = (ys(zsx))))))
25	24	anbi2d 678	. . . . . 6 |- (g = G -> (((1sx) = x /\ A.y e. CC (A.z e. ran g(ys(xgz)) = ((ysx)g(ysz)) /\ A.z e. CC (((y + z)sx) = ((ysx)g(zsx)) /\ ((y x. z)sx) = (ys(zsx))))) <-> ((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)))))))
26	8, 25	raleqbidv 2274	. . . . 5 |- (g = G -> (A.x e. ran g((1sx) = x /\ A.y e. CC (A.z e. ran g(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 /\ 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)))))))
27	5, 13, 26	3anbi123d 1168	. . . 4 |- (g = G -> ((g e. Abel /\ s:(CC X. ran g)-->ran g /\ A.x e. ran g((1sx) = x /\ A.y e. CC (A.z e. ran g(ys(xgz)) = ((ysx)g(ysz)) /\ A.z e. CC (((y + z)sx) = ((ysx)g(zsx)) /\ ((y x. z)sx) = (ys(zsx)))))) <-> (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))))))))
28	4, 27	sylbir 218	. . 3 |- (g = (1st` W) -> ((g e. Abel /\ s:(CC X. ran g)-->ran g /\ A.x e. ran g((1sx) = x /\ A.y e. CC (A.z e. ran g(ys(xgz)) = ((ysx)g(ysz)) /\ A.z e. CC (((y + z)sx) = ((ysx)g(zsx)) /\ ((y x. z)sx) = (ys(zsx)))))) <-> (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))))))))
29		vci.2	. . . . 5 |- S = (2nd` W)
30	29	eqeq2i 1894	. . . 4 |- (s = S <-> s = (2nd` W))
31		feq1 4551	. . . . 5 |- (s = S -> (s:(CC X. X)-->X <-> S:(CC X. X)-->X))
32		opreq 4888	. . . . . . . 8 |- (s = S -> (1sx) = (1Sx))
33	32	eqeq1d 1892	. . . . . . 7 |- (s = S -> ((1sx) = x <-> (1Sx) = x))
34		opreq 4888	. . . . . . . . . . 11 |- (s = S -> (ys(xGz)) = (yS(xGz)))
35		opreq 4888	. . . . . . . . . . . 12 |- (s = S -> (ysx) = (ySx))
36		opreq 4888	. . . . . . . . . . . 12 |- (s = S -> (ysz) = (ySz))
37	35, 36	opreq12d 4900	. . . . . . . . . . 11 |- (s = S -> ((ysx)G(ysz)) = ((ySx)G(ySz)))
38	34, 37	eqeq12d 1899	. . . . . . . . . 10 |- (s = S -> ((ys(xGz)) = ((ysx)G(ysz)) <-> (yS(xGz)) = ((ySx)G(ySz))))
39	38	ralbidv 2123	. . . . . . . . 9 |- (s = S -> (A.z e. X (ys(xGz)) = ((ysx)G(ysz)) <-> A.z e. X (yS(xGz)) = ((ySx)G(ySz))))
40		opreq 4888	. . . . . . . . . . . 12 |- (s = S -> ((y + z)sx) = ((y + z)Sx))
41		opreq 4888	. . . . . . . . . . . . 13 |- (s = S -> (zsx) = (zSx))
42	35, 41	opreq12d 4900	. . . . . . . . . . . 12 |- (s = S -> ((ysx)G(zsx)) = ((ySx)G(zSx)))
43	40, 42	eqeq12d 1899	. . . . . . . . . . 11 |- (s = S -> (((y + z)sx) = ((ysx)G(zsx)) <-> ((y + z)Sx) = ((ySx)G(zSx))))
44		opreq 4888	. . . . . . . . . . . 12 |- (s = S -> ((y x. z)sx) = ((y x. z)Sx))
45	41	opreq2d 4898	. . . . . . . . . . . . 13 |- (s = S -> (ys(zsx)) = (ys(zSx)))
46		opreq 4888	. . . . . . . . . . . . 13 |- (s = S -> (ys(zSx)) = (yS(zSx)))
47	45, 46	eqtrd 1925	. . . . . . . . . . . 12 |- (s = S -> (ys(zsx)) = (yS(zSx)))
48	44, 47	eqeq12d 1899	. . . . . . . . . . 11 |- (s = S -> (((y x. z)sx) = (ys(zsx)) <-> ((y x. z)Sx) = (yS(zSx))))
49	43, 48	anbi12d 690	. . . . . . . . . 10 |- (s = S -> ((((y + z)sx) = ((ysx)G(zsx)) /\ ((y x. z)sx) = (ys(zsx))) <-> (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx)))))
50	49	ralbidv 2123	. . . . . . . . 9 |- (s = S -> (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)) /\ ((y x. z)Sx) = (yS(zSx)))))
51	39, 50	anbi12d 690	. . . . . . . 8 |- (s = S -> ((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. X (yS(xGz)) = ((ySx)G(ySz)) /\ A.z e. CC (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx))))))
52	51	ralbidv 2123	. . . . . . 7 |- (s = S -> (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. X (yS(xGz)) = ((ySx)G(ySz)) /\ A.z e. CC (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx))))))
53	33, 52	anbi12d 690	. . . . . 6 |- (s = S -> (((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 /\ 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)))))))
54	53	ralbidv 2123	. . . . 5 |- (s = S -> (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 /\ 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)))))))
55	31, 54	3anbi23d 1171	. . . 4 |- (s = S -> ((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)))))) <-> (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))))))))
56	30, 55	sylbir 218	. . 3 |- (s = (2nd` W) -> ((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)))))) <-> (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))))))))
57	28, 56	elopabi 5059	. 2 |- (W e. {<.g, s>. | (g e. Abel /\ s:(CC X. ran g)-->ran g /\ A.x e. ran g((1sx) = x /\ A.y e. CC (A.z e. ran g(ys(xgz)) = ((ysx)g(ysz)) /\ A.z e. CC (((y + z)sx) = ((ysx)g(zsx)) /\ ((y x. z)sx) = (ys(zsx))))))} -> (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)))))))
58	2, 57	sylbi 216	1 |- (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)))))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  {copab 3395   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:  vcsm 9500  vcid 9502  vcdi 9503  vcdir 9504  vcass 9505  vcabl 9508

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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