Theorem vcoprnelem 8193

Description: Lemma for vcoprne 8194.

Assertion

Ref	Expression
vcoprnelem	⊢ (⟨G, G⟩ ∈ CVec → G:(ℂ × ℂ)–→ℂ)

Proof of Theorem vcoprnelem

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

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   ⋀ w3a 777   = wceq 958   ∈ wcel 960   ≠ wne 1588  ∀wral 1648  Vcvv 1814   ⊆ wss 2050  ∅c0 2283  ⟨cop 2415   × cxp 3174  ran crn 3177  Rel wrel 3181   Fn wfn 3183  –→wf 3184  –onto→wfo 3186  (class class class)co 3969  ℂcc 5244  1c1 5247   + caddc 5249   · 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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