Theorem vcoprnelem 8193

Description: Lemma for vcoprne 8194.

Assertion

Ref	Expression
vcoprnelem	|- (<.G, G>. e. CVec -> G:(CC X. CC)-->CC)

Proof of Theorem vcoprnelem

Step	Hyp	Ref	Expression
1		vcrel 8162	. . . . 5 |- Rel CVec
2		df-rel 3191	. . . . 5 |- (Rel CVec <-> CVec (_ (V X. V))
3	1, 2	mpbi 189	. . . 4 |- CVec (_ (V X. V)
4	3	sseli 2068	. . 3 |- (<.G, G>. e. CVec -> <.G, G>. e. (V X. V))
5		opelxp1 3211	. . 3 |- (<.G, G>. e. (V X. V) -> G e. V)
6	4, 5	syl 10	. 2 |- (<.G, G>. e. CVec -> G e. V)
7		eqid 1478	. . . . . 6 |- ran G = ran G
8	7	isvclem 8192	. . . . 5 |- ((G e. V /\ G e. V) -> (<.G, G>. e. CVec <-> (G e. Abel /\ G:(CC X. ran G)-->ran G /\ A.x e. ran G((1Gx) = x /\ A.y e. CC (A.z e. ran G(yG(xGz)) = ((yGx)G(yGz)) /\ A.z e. CC (((y + z)Gx) = ((yGx)G(zGx)) /\ ((y x. z)Gx) = (yG(zGx))))))))
9	8	anidms 436	. . . 4 |- (G e. V -> (<.G, G>. e. CVec <-> (G e. Abel /\ G:(CC X. ran G)-->ran G /\ A.x e. ran G((1Gx) = x /\ A.y e. CC (A.z e. ran G(yG(xGz)) = ((yGx)G(yGz)) /\ A.z e. CC (((y + z)Gx) = ((yGx)G(zGx)) /\ ((y x. z)Gx) = (yG(zGx))))))))
10	9	biimpa 418	. . 3 |- ((G e. V /\ <.G, G>. e. CVec) -> (G e. Abel /\ G:(CC X. ran G)-->ran G /\ A.x e. ran G((1Gx) = x /\ A.y e. CC (A.z e. ran G(yG(xGz)) = ((yGx)G(yGz)) /\ A.z e. CC (((y + z)Gx) = ((yGx)G(zGx)) /\ ((y x. z)Gx) = (yG(zGx)))))))
11		pm3.27 323	. . . . 5 |- ((G e. Abel /\ G:(CC X. ran G)-->ran G) -> G:(CC X. ran G)-->ran G)
12		fndmu 3595	. . . . . . . . 9 |- ((G Fn (ran G X. ran G) /\ G Fn (CC X. ran G)) -> (ran G X. ran G) = (CC X. ran G))
13	7	grpfo 8040	. . . . . . . . . 10 |- (G e. Grp -> G:(ran G X. ran G)-onto->ran G)
14		fof 3678	. . . . . . . . . . 11 |- (G:(ran G X. ran G)-onto->ran G -> G:(ran G X. ran G)-->ran G)
15		ffn 3633	. . . . . . . . . . 11 |- (G:(ran G X. ran G)-->ran G -> G Fn (ran G X. ran G))
16	14, 15	syl 10	. . . . . . . . . 10 |- (G:(ran G X. ran G)-onto->ran G -> G Fn (ran G X. ran G))
17	13, 16	syl 10	. . . . . . . . 9 |- (G e. Grp -> G Fn (ran G X. ran G))
18		ffn 3633	. . . . . . . . 9 |- (G:(CC X. ran G)-->ran G -> G Fn (CC X. ran G))
19	12, 17, 18	syl2an 456	. . . . . . . 8 |- ((G e. Grp /\ G:(CC X. ran G)-->ran G) -> (ran G X. ran G) = (CC X. ran G))
20	7	grpn0 8043	. . . . . . . . . 10 |- (G e. Grp -> ran G =/= (/))
21		xp11a 3483	. . . . . . . . . 10 |- (ran G =/= (/) -> ((ran G X. ran G) = (CC X. ran G) <-> ran G = CC))
22	20, 21	syl 10	. . . . . . . . 9 |- (G e. Grp -> ((ran G X. ran G) = (CC X. ran G) <-> ran G = CC))
23	22	adantr 391	. . . . . . . 8 |- ((G e. Grp /\ G:(CC X. ran G)-->ran G) -> ((ran G X. ran G) = (CC X. ran G) <-> ran G = CC))
24	19, 23	mpbid 195	. . . . . . 7 |- ((G e. Grp /\ G:(CC X. ran G)-->ran G) -> ran G = CC)
25		ablgrp 8098	. . . . . . 7 |- (G e. Abel -> G e. Grp)
26	24, 25	sylan 450	. . . . . 6 |- ((G e. Abel /\ G:(CC X. ran G)-->ran G) -> ran G = CC)
27		xpeq2 3207	. . . . . . . 8 |- (ran G = CC -> (CC X. ran G) = (CC X. CC))
28		feq2 3627	. . . . . . . 8 |- ((CC X. ran G) = (CC X. CC) -> (G:(CC X. ran G)-->ran G <-> G:(CC X. CC)-->ran G))
29	27, 28	syl 10	. . . . . . 7 |- (ran G = CC -> (G:(CC X. ran G)-->ran G <-> G:(CC X. CC)-->ran G))
30		feq3 3628	. . . . . . 7 |- (ran G = CC -> (G:(CC X. CC)-->ran G <-> G:(CC X. CC)-->CC))
31	29, 30	bitrd 530	. . . . . 6 |- (ran G = CC -> (G:(CC X. ran G)-->ran G <-> G:(CC X. CC)-->CC))
32	26, 31	syl 10	. . . . 5 |- ((G e. Abel /\ G:(CC X. ran G)-->ran G) -> (G:(CC X. ran G)-->ran G <-> G:(CC X. CC)-->CC))
33	11, 32	mpbid 195	. . . 4 |- ((G e. Abel /\ G:(CC X. ran G)-->ran G) -> G:(CC X. CC)-->CC)
34	33	3adant3 801	. . 3 |- ((G e. Abel /\ G:(CC X. ran G)-->ran G /\ A.x e. ran G((1Gx) = x /\ A.y e. CC (A.z e. ran G(yG(xGz)) = ((yGx)G(yGz)) /\ A.z e. CC (((y + z)Gx) = ((yGx)G(zGx)) /\ ((y x. z)Gx) = (yG(zGx)))))) -> G:(CC X. CC)-->CC)
35	10, 34	syl 10	. 2 |- ((G e. V /\ <.G, G>. e. CVec) -> G:(CC X. CC)-->CC)
36	6, 35	mpancom 707	1 |- (<.G, G>. e. CVec -> G:(CC X. CC)-->CC)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960   =/= wne 1588  A.wral 1648  Vcvv 1814   (_ wss 2050  (/)c0 2283  <.cop 2415   X. cxp 3174  ran crn 3177  Rel wrel 3181   Fn wfn 3183  -->wf 3184  -onto->wfo 3186  (class class class)co 3969  CCcc 5244  1c1 5247   + caddc 5249   x. cmul 5251  Grpcgr 8030  Abelcabl 8095  CVeccvc 8160

This theorem is referenced by:  vcoprne 8194

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fo 3202  df-fv 3204  df-opr 3971  df-grp 8034  df-abl 8096  df-vc 8161

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