Theorem isvclem 9528

Description: Lemma for isvc 9532.

Hypothesis

Ref	Expression
isvclem.1	|- X = ran G

Assertion

Ref	Expression
isvclem	|- ((G e. _V /\ S e. _V) -> (<.G, S>. 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,z

Proof of Theorem isvclem

Step	Hyp	Ref	Expression
1		eleq1 1957	. . . 4 |- (g = G -> (g e. Abel <-> G e. Abel))
2		rneq 4186	. . . . . 6 |- (g = G -> ran g = ran G)
3		isvclem.1	. . . . . 6 |- X = ran G
4	2, 3	syl6eqr 1946	. . . . 5 |- (g = G -> ran g = X)
5		xpeq2 4017	. . . . . . 7 |- (ran g = X -> (CC X. ran g) = (CC X. X))
6	5	feq2d 4557	. . . . . 6 |- (ran g = X -> (s:(CC X. ran g)-->ran g <-> s:(CC X. X)-->ran g))
7		feq3 4553	. . . . . 6 |- (ran g = X -> (s:(CC X. X)-->ran g <-> s:(CC X. X)-->X))
8	6, 7	bitrd 587	. . . . 5 |- (ran g = X -> (s:(CC X. ran g)-->ran g <-> s:(CC X. X)-->X))
9	4, 8	syl 12	. . . 4 |- (g = G -> (s:(CC X. ran g)-->ran g <-> s:(CC X. X)-->X))
10		opreq 4888	. . . . . . . . . . 11 |- (g = G -> (xgz) = (xGz))
11	10	opreq2d 4898	. . . . . . . . . 10 |- (g = G -> (ys(xgz)) = (ys(xGz)))
12		opreq 4888	. . . . . . . . . 10 |- (g = G -> ((ysx)g(ysz)) = ((ysx)G(ysz)))
13	11, 12	eqeq12d 1899	. . . . . . . . 9 |- (g = G -> ((ys(xgz)) = ((ysx)g(ysz)) <-> (ys(xGz)) = ((ysx)G(ysz))))
14	4, 13	raleqbidv 2274	. . . . . . . 8 |- (g = G -> (A.z e. ran g(ys(xgz)) = ((ysx)g(ysz)) <-> A.z e. X (ys(xGz)) = ((ysx)G(ysz))))
15		opreq 4888	. . . . . . . . . . 11 |- (g = G -> ((ysx)g(zsx)) = ((ysx)G(zsx)))
16	15	eqeq2d 1895	. . . . . . . . . 10 |- (g = G -> (((y + z)sx) = ((ysx)g(zsx)) <-> ((y + z)sx) = ((ysx)G(zsx))))
17	16	anbi1d 679	. . . . . . . . 9 |- (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)))))
18	17	ralbidv 2123	. . . . . . . 8 |- (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)))))
19	14, 18	anbi12d 690	. . . . . . 7 |- (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))))))
20	19	ralbidv 2123	. . . . . 6 |- (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))))))
21	20	anbi2d 678	. . . . 5 |- (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)))))))
22	4, 21	raleqbidv 2274	. . . 4 |- (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)))))))
23	1, 9, 22	3anbi123d 1168	. . 3 |- (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))))))))
24		feq1 4551	. . . 4 |- (s = S -> (s:(CC X. X)-->X <-> S:(CC X. X)-->X))
25		opreq 4888	. . . . . . 7 |- (s = S -> (1sx) = (1Sx))
26	25	eqeq1d 1892	. . . . . 6 |- (s = S -> ((1sx) = x <-> (1Sx) = x))
27		opreq 4888	. . . . . . . . . 10 |- (s = S -> (ys(xGz)) = (yS(xGz)))
28		opreq 4888	. . . . . . . . . . 11 |- (s = S -> (ysx) = (ySx))
29		opreq 4888	. . . . . . . . . . 11 |- (s = S -> (ysz) = (ySz))
30	28, 29	opreq12d 4900	. . . . . . . . . 10 |- (s = S -> ((ysx)G(ysz)) = ((ySx)G(ySz)))
31	27, 30	eqeq12d 1899	. . . . . . . . 9 |- (s = S -> ((ys(xGz)) = ((ysx)G(ysz)) <-> (yS(xGz)) = ((ySx)G(ySz))))
32	31	ralbidv 2123	. . . . . . . 8 |- (s = S -> (A.z e. X (ys(xGz)) = ((ysx)G(ysz)) <-> A.z e. X (yS(xGz)) = ((ySx)G(ySz))))
33		opreq 4888	. . . . . . . . . . 11 |- (s = S -> ((y + z)sx) = ((y + z)Sx))
34		opreq 4888	. . . . . . . . . . . 12 |- (s = S -> (zsx) = (zSx))
35	28, 34	opreq12d 4900	. . . . . . . . . . 11 |- (s = S -> ((ysx)G(zsx)) = ((ySx)G(zSx)))
36	33, 35	eqeq12d 1899	. . . . . . . . . 10 |- (s = S -> (((y + z)sx) = ((ysx)G(zsx)) <-> ((y + z)Sx) = ((ySx)G(zSx))))
37		opreq 4888	. . . . . . . . . . 11 |- (s = S -> ((y x. z)sx) = ((y x. z)Sx))
38		opreq 4888	. . . . . . . . . . . 12 |- (s = S -> (ys(zsx)) = (yS(zsx)))
39	34	opreq2d 4898	. . . . . . . . . . . 12 |- (s = S -> (yS(zsx)) = (yS(zSx)))
40	38, 39	eqtrd 1925	. . . . . . . . . . 11 |- (s = S -> (ys(zsx)) = (yS(zSx)))
41	37, 40	eqeq12d 1899	. . . . . . . . . 10 |- (s = S -> (((y x. z)sx) = (ys(zsx)) <-> ((y x. z)Sx) = (yS(zSx))))
42	36, 41	anbi12d 690	. . . . . . . . 9 |- (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)))))
43	42	ralbidv 2123	. . . . . . . 8 |- (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)))))
44	32, 43	anbi12d 690	. . . . . . 7 |- (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))))))
45	44	ralbidv 2123	. . . . . 6 |- (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))))))
46	26, 45	anbi12d 690	. . . . 5 |- (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)))))))
47	46	ralbidv 2123	. . . 4 |- (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)))))))
48	24, 47	3anbi23d 1171	. . 3 |- (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))))))))
49	23, 48	opelopabg 3567	. 2 |- ((G e. _V /\ S e. _V) -> (<.G, S>. 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))))))))
50		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))))))}
51	50	eleq2i 1961	. 2 |- (<.G, S>. e. CVec <-> <.G, S>. 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))))))})
52	49, 51	syl5bb 591	1 |- ((G e. _V /\ S e. _V) -> (<.G, S>. 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  _Vcvv 2292  <.cop 3046  {copab 3395   X. cxp 3984  ran crn 3987  -->wf 3994  (class class class)co 4884  CCcc 6384  1c1 6387   + caddc 6389   x. cmul 6391  Abelcabl 9407  CVeccvc 9496

This theorem is referenced by:  vcoprnelem 9529  isvc 9532

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-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

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-vc 9497

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