Theorem vcoprnelem 8197

Description: Lemma for vcoprne 8198.

Assertion

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

Proof of Theorem vcoprnelem

Step	Hyp	Ref	Expression
1		vcrel 8166	. . . . 5 |- Rel CVec
2		df-rel 3185	. . . . 5 |- (Rel CVec <-> CVec (_ (V X. V))
3	1, 2	mpbi 189	. . . 4 |- CVec (_ (V X. V)
4	3	sseli 2065	. . 3 |- (<.G, G>. e. CVec -> <.G, G>. e. (V X. V))
5		opelxp1 3205	. . 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 1475	. . . . . 6 |- ran G = ran G
8	7	isvclem 8196	. . . . 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 434	. . . 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 416	. . 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 3589	. . . . . . . . 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 8043	. . . . . . . . . 10 |- (G e. Grp -> G:(ran G X. ran G)-onto->ran G)
14		fof 3672	. . . . . . . . . . 11 |- (G:(ran G X. ran G)-onto->ran G -> G:(ran G X. ran G)-->ran G)
15		ffn 3627	. . . . . . . . . . 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 3627	. . . . . . . . 9 |- (G:(CC X. ran G)-->ran G -> G Fn (CC X. ran G))
19	12, 17, 18	syl2an 454	. . . . . . . 8 |- ((G e. Grp /\ G:(CC X. ran G)-->ran G) -> (ran G X. ran G) = (CC X. ran G))
20	7	grpn0 8046	. . . . . . . . . 10 |- (G e. Grp -> ran G =/= (/))
21		xp11a 3477	. . . . . . . . . 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 389	. . . . . . . 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 8102	. . . . . . 7 |- (G e. Abel -> G e. Grp)
26	24, 25	sylan 448	. . . . . 6 |- ((G e. Abel /\ G:(CC X. ran G)-->ran G) -> ran G = CC)
27		xpeq2 3201	. . . . . . . 8 |- (ran G = CC -> (CC X. ran G) = (CC X. CC))
28		feq2 3621	. . . . . . . 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 3622	. . . . . . 7 |- (ran G = CC -> (G:(CC X. CC)-->ran G <-> G:(CC X. CC)-->CC))
31	29, 30	bitrd 528	. . . . . 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 799	. . 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 705	1 |- (<.G, G>. e. CVec -> G:(CC X. CC)-->CC)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958   =/= wne 1585  A.wral 1645  Vcvv 1811   (_ wss 2047  (/)c0 2280  <.cop 2411   X. cxp 3168  ran crn 3171  Rel wrel 3175   Fn wfn 3177  -->wf 3178  -onto->wfo 3180  (class class class)co 3963  CCcc 5232  1c1 5235   + caddc 5237   x. cmul 5239  Grpcgr 8033  Abelcabl 8099  CVeccvc 8164

This theorem is referenced by:  vcoprne 8198

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fo 3196  df-fv 3198  df-opr 3965  df-grp 8037  df-abl 8100  df-vc 8165

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