Theorem gicabl 39706

Description: Being Abelian is a group invariant. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.)

Assertion

Ref	Expression
gicabl	⊢ (𝐺 ≃𝑔 𝐻 → (𝐺 ∈ Abel ↔ 𝐻 ∈ Abel))

Proof of Theorem gicabl

Dummy variables 𝑤 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brgic 18411	. 2 ⊢ (𝐺 ≃𝑔 𝐻 ↔ (𝐺 GrpIso 𝐻) ≠ ∅)
2		n0 4312	. . 3 ⊢ ((𝐺 GrpIso 𝐻) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐺 GrpIso 𝐻))
3		gimghm 18406	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝑥 ∈ (𝐺 GrpHom 𝐻))
4		ghmgrp1 18362	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
5	3, 4	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝐺 ∈ Grp)
6		ghmgrp2 18363	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐺 GrpHom 𝐻) → 𝐻 ∈ Grp)
7	3, 6	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝐻 ∈ Grp)
8	5, 7	2thd 267	. . . . . 6 ⊢ (𝑥 ∈ (𝐺 GrpIso 𝐻) → (𝐺 ∈ Grp ↔ 𝐻 ∈ Grp))
9		grpmnd 18112	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
10	5, 9	syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝐺 ∈ Mnd)
11		grpmnd 18112	. . . . . . . . . 10 ⊢ (𝐻 ∈ Grp → 𝐻 ∈ Mnd)
12	7, 11	syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝐻 ∈ Mnd)
13	10, 12	2thd 267	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐺 GrpIso 𝐻) → (𝐺 ∈ Mnd ↔ 𝐻 ∈ Mnd))
14		eqid 2823	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐺) = (Base‘𝐺)
15		eqid 2823	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐻) = (Base‘𝐻)
16	14, 15	gimf1o 18405	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝑥:(Base‘𝐺)–1-1-onto→(Base‘𝐻))
17		f1of1 6616	. . . . . . . . . . . . . . 15 ⊢ (𝑥:(Base‘𝐺)–1-1-onto→(Base‘𝐻) → 𝑥:(Base‘𝐺)–1-1→(Base‘𝐻))
18	16, 17	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝑥:(Base‘𝐺)–1-1→(Base‘𝐻))
19	18	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → 𝑥:(Base‘𝐺)–1-1→(Base‘𝐻))
20	5	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → 𝐺 ∈ Grp)
21		simprl 769	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → 𝑦 ∈ (Base‘𝐺))
22		simprr 771	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → 𝑧 ∈ (Base‘𝐺))
23		eqid 2823	. . . . . . . . . . . . . . 15 ⊢ (+g‘𝐺) = (+g‘𝐺)
24	14, 23	grpcl 18113	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝑦(+g‘𝐺)𝑧) ∈ (Base‘𝐺))
25	20, 21, 22, 24	syl3anc 1367	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑦(+g‘𝐺)𝑧) ∈ (Base‘𝐺))
26	14, 23	grpcl 18113	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑧(+g‘𝐺)𝑦) ∈ (Base‘𝐺))
27	20, 22, 21, 26	syl3anc 1367	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑧(+g‘𝐺)𝑦) ∈ (Base‘𝐺))
28		f1fveq 7022	. . . . . . . . . . . . 13 ⊢ ((𝑥:(Base‘𝐺)–1-1→(Base‘𝐻) ∧ ((𝑦(+g‘𝐺)𝑧) ∈ (Base‘𝐺) ∧ (𝑧(+g‘𝐺)𝑦) ∈ (Base‘𝐺))) → ((𝑥‘(𝑦(+g‘𝐺)𝑧)) = (𝑥‘(𝑧(+g‘𝐺)𝑦)) ↔ (𝑦(+g‘𝐺)𝑧) = (𝑧(+g‘𝐺)𝑦)))
29	19, 25, 27, 28	syl12anc 834	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → ((𝑥‘(𝑦(+g‘𝐺)𝑧)) = (𝑥‘(𝑧(+g‘𝐺)𝑦)) ↔ (𝑦(+g‘𝐺)𝑧) = (𝑧(+g‘𝐺)𝑦)))
30	3	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → 𝑥 ∈ (𝐺 GrpHom 𝐻))
31		eqid 2823	. . . . . . . . . . . . . . 15 ⊢ (+g‘𝐻) = (+g‘𝐻)
32	14, 23, 31	ghmlin 18365	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝑥‘(𝑦(+g‘𝐺)𝑧)) = ((𝑥‘𝑦)(+g‘𝐻)(𝑥‘𝑧)))
33	30, 21, 22, 32	syl3anc 1367	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑥‘(𝑦(+g‘𝐺)𝑧)) = ((𝑥‘𝑦)(+g‘𝐻)(𝑥‘𝑧)))
34	14, 23, 31	ghmlin 18365	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥‘(𝑧(+g‘𝐺)𝑦)) = ((𝑥‘𝑧)(+g‘𝐻)(𝑥‘𝑦)))
35	30, 22, 21, 34	syl3anc 1367	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑥‘(𝑧(+g‘𝐺)𝑦)) = ((𝑥‘𝑧)(+g‘𝐻)(𝑥‘𝑦)))
36	33, 35	eqeq12d 2839	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → ((𝑥‘(𝑦(+g‘𝐺)𝑧)) = (𝑥‘(𝑧(+g‘𝐺)𝑦)) ↔ ((𝑥‘𝑦)(+g‘𝐻)(𝑥‘𝑧)) = ((𝑥‘𝑧)(+g‘𝐻)(𝑥‘𝑦))))
37	29, 36	bitr3d 283	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → ((𝑦(+g‘𝐺)𝑧) = (𝑧(+g‘𝐺)𝑦) ↔ ((𝑥‘𝑦)(+g‘𝐻)(𝑥‘𝑧)) = ((𝑥‘𝑧)(+g‘𝐻)(𝑥‘𝑦))))
38	37	2ralbidva 3200	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐺 GrpIso 𝐻) → (∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)(𝑦(+g‘𝐺)𝑧) = (𝑧(+g‘𝐺)𝑦) ↔ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑥‘𝑦)(+g‘𝐻)(𝑥‘𝑧)) = ((𝑥‘𝑧)(+g‘𝐻)(𝑥‘𝑦))))
39		f1ofo 6624	. . . . . . . . . . . . . . 15 ⊢ (𝑥:(Base‘𝐺)–1-1-onto→(Base‘𝐻) → 𝑥:(Base‘𝐺)–onto→(Base‘𝐻))
40		foima 6597	. . . . . . . . . . . . . . 15 ⊢ (𝑥:(Base‘𝐺)–onto→(Base‘𝐻) → (𝑥 “ (Base‘𝐺)) = (Base‘𝐻))
41	39, 40	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑥:(Base‘𝐺)–1-1-onto→(Base‘𝐻) → (𝑥 “ (Base‘𝐺)) = (Base‘𝐻))
42	16, 41	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐺 GrpIso 𝐻) → (𝑥 “ (Base‘𝐺)) = (Base‘𝐻))
43	42	raleqdv 3417	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐺 GrpIso 𝐻) → (∀𝑣 ∈ (𝑥 “ (Base‘𝐺))((𝑥‘𝑦)(+g‘𝐻)𝑣) = (𝑣(+g‘𝐻)(𝑥‘𝑦)) ↔ ∀𝑣 ∈ (Base‘𝐻)((𝑥‘𝑦)(+g‘𝐻)𝑣) = (𝑣(+g‘𝐻)(𝑥‘𝑦))))
44		f1ofn 6618	. . . . . . . . . . . . . 14 ⊢ (𝑥:(Base‘𝐺)–1-1-onto→(Base‘𝐻) → 𝑥 Fn (Base‘𝐺))
45	16, 44	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝑥 Fn (Base‘𝐺))
46		ssid 3991	. . . . . . . . . . . . 13 ⊢ (Base‘𝐺) ⊆ (Base‘𝐺)
47		oveq2 7166	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑥‘𝑧) → ((𝑥‘𝑦)(+g‘𝐻)𝑣) = ((𝑥‘𝑦)(+g‘𝐻)(𝑥‘𝑧)))
48		oveq1 7165	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑥‘𝑧) → (𝑣(+g‘𝐻)(𝑥‘𝑦)) = ((𝑥‘𝑧)(+g‘𝐻)(𝑥‘𝑦)))
49	47, 48	eqeq12d 2839	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (𝑥‘𝑧) → (((𝑥‘𝑦)(+g‘𝐻)𝑣) = (𝑣(+g‘𝐻)(𝑥‘𝑦)) ↔ ((𝑥‘𝑦)(+g‘𝐻)(𝑥‘𝑧)) = ((𝑥‘𝑧)(+g‘𝐻)(𝑥‘𝑦))))
50	49	ralima 7002	. . . . . . . . . . . . 13 ⊢ ((𝑥 Fn (Base‘𝐺) ∧ (Base‘𝐺) ⊆ (Base‘𝐺)) → (∀𝑣 ∈ (𝑥 “ (Base‘𝐺))((𝑥‘𝑦)(+g‘𝐻)𝑣) = (𝑣(+g‘𝐻)(𝑥‘𝑦)) ↔ ∀𝑧 ∈ (Base‘𝐺)((𝑥‘𝑦)(+g‘𝐻)(𝑥‘𝑧)) = ((𝑥‘𝑧)(+g‘𝐻)(𝑥‘𝑦))))
51	45, 46, 50	sylancl 588	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐺 GrpIso 𝐻) → (∀𝑣 ∈ (𝑥 “ (Base‘𝐺))((𝑥‘𝑦)(+g‘𝐻)𝑣) = (𝑣(+g‘𝐻)(𝑥‘𝑦)) ↔ ∀𝑧 ∈ (Base‘𝐺)((𝑥‘𝑦)(+g‘𝐻)(𝑥‘𝑧)) = ((𝑥‘𝑧)(+g‘𝐻)(𝑥‘𝑦))))
52	43, 51	bitr3d 283	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐺 GrpIso 𝐻) → (∀𝑣 ∈ (Base‘𝐻)((𝑥‘𝑦)(+g‘𝐻)𝑣) = (𝑣(+g‘𝐻)(𝑥‘𝑦)) ↔ ∀𝑧 ∈ (Base‘𝐺)((𝑥‘𝑦)(+g‘𝐻)(𝑥‘𝑧)) = ((𝑥‘𝑧)(+g‘𝐻)(𝑥‘𝑦))))
53	52	ralbidv 3199	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐺 GrpIso 𝐻) → (∀𝑦 ∈ (Base‘𝐺)∀𝑣 ∈ (Base‘𝐻)((𝑥‘𝑦)(+g‘𝐻)𝑣) = (𝑣(+g‘𝐻)(𝑥‘𝑦)) ↔ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑥‘𝑦)(+g‘𝐻)(𝑥‘𝑧)) = ((𝑥‘𝑧)(+g‘𝐻)(𝑥‘𝑦))))
54	38, 53	bitr4d 284	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐺 GrpIso 𝐻) → (∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)(𝑦(+g‘𝐺)𝑧) = (𝑧(+g‘𝐺)𝑦) ↔ ∀𝑦 ∈ (Base‘𝐺)∀𝑣 ∈ (Base‘𝐻)((𝑥‘𝑦)(+g‘𝐻)𝑣) = (𝑣(+g‘𝐻)(𝑥‘𝑦))))
55	42	raleqdv 3417	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐺 GrpIso 𝐻) → (∀𝑤 ∈ (𝑥 “ (Base‘𝐺))∀𝑣 ∈ (Base‘𝐻)(𝑤(+g‘𝐻)𝑣) = (𝑣(+g‘𝐻)𝑤) ↔ ∀𝑤 ∈ (Base‘𝐻)∀𝑣 ∈ (Base‘𝐻)(𝑤(+g‘𝐻)𝑣) = (𝑣(+g‘𝐻)𝑤)))
56		oveq1 7165	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑥‘𝑦) → (𝑤(+g‘𝐻)𝑣) = ((𝑥‘𝑦)(+g‘𝐻)𝑣))
57		oveq2 7166	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑥‘𝑦) → (𝑣(+g‘𝐻)𝑤) = (𝑣(+g‘𝐻)(𝑥‘𝑦)))
58	56, 57	eqeq12d 2839	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑥‘𝑦) → ((𝑤(+g‘𝐻)𝑣) = (𝑣(+g‘𝐻)𝑤) ↔ ((𝑥‘𝑦)(+g‘𝐻)𝑣) = (𝑣(+g‘𝐻)(𝑥‘𝑦))))
59	58	ralbidv 3199	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑥‘𝑦) → (∀𝑣 ∈ (Base‘𝐻)(𝑤(+g‘𝐻)𝑣) = (𝑣(+g‘𝐻)𝑤) ↔ ∀𝑣 ∈ (Base‘𝐻)((𝑥‘𝑦)(+g‘𝐻)𝑣) = (𝑣(+g‘𝐻)(𝑥‘𝑦))))
60	59	ralima 7002	. . . . . . . . . . 11 ⊢ ((𝑥 Fn (Base‘𝐺) ∧ (Base‘𝐺) ⊆ (Base‘𝐺)) → (∀𝑤 ∈ (𝑥 “ (Base‘𝐺))∀𝑣 ∈ (Base‘𝐻)(𝑤(+g‘𝐻)𝑣) = (𝑣(+g‘𝐻)𝑤) ↔ ∀𝑦 ∈ (Base‘𝐺)∀𝑣 ∈ (Base‘𝐻)((𝑥‘𝑦)(+g‘𝐻)𝑣) = (𝑣(+g‘𝐻)(𝑥‘𝑦))))
61	45, 46, 60	sylancl 588	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐺 GrpIso 𝐻) → (∀𝑤 ∈ (𝑥 “ (Base‘𝐺))∀𝑣 ∈ (Base‘𝐻)(𝑤(+g‘𝐻)𝑣) = (𝑣(+g‘𝐻)𝑤) ↔ ∀𝑦 ∈ (Base‘𝐺)∀𝑣 ∈ (Base‘𝐻)((𝑥‘𝑦)(+g‘𝐻)𝑣) = (𝑣(+g‘𝐻)(𝑥‘𝑦))))
62	55, 61	bitr3d 283	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐺 GrpIso 𝐻) → (∀𝑤 ∈ (Base‘𝐻)∀𝑣 ∈ (Base‘𝐻)(𝑤(+g‘𝐻)𝑣) = (𝑣(+g‘𝐻)𝑤) ↔ ∀𝑦 ∈ (Base‘𝐺)∀𝑣 ∈ (Base‘𝐻)((𝑥‘𝑦)(+g‘𝐻)𝑣) = (𝑣(+g‘𝐻)(𝑥‘𝑦))))
63	54, 62	bitr4d 284	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐺 GrpIso 𝐻) → (∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)(𝑦(+g‘𝐺)𝑧) = (𝑧(+g‘𝐺)𝑦) ↔ ∀𝑤 ∈ (Base‘𝐻)∀𝑣 ∈ (Base‘𝐻)(𝑤(+g‘𝐻)𝑣) = (𝑣(+g‘𝐻)𝑤)))
64	13, 63	anbi12d 632	. . . . . . 7 ⊢ (𝑥 ∈ (𝐺 GrpIso 𝐻) → ((𝐺 ∈ Mnd ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)(𝑦(+g‘𝐺)𝑧) = (𝑧(+g‘𝐺)𝑦)) ↔ (𝐻 ∈ Mnd ∧ ∀𝑤 ∈ (Base‘𝐻)∀𝑣 ∈ (Base‘𝐻)(𝑤(+g‘𝐻)𝑣) = (𝑣(+g‘𝐻)𝑤))))
65	14, 23	iscmn 18916	. . . . . . 7 ⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)(𝑦(+g‘𝐺)𝑧) = (𝑧(+g‘𝐺)𝑦)))
66	15, 31	iscmn 18916	. . . . . . 7 ⊢ (𝐻 ∈ CMnd ↔ (𝐻 ∈ Mnd ∧ ∀𝑤 ∈ (Base‘𝐻)∀𝑣 ∈ (Base‘𝐻)(𝑤(+g‘𝐻)𝑣) = (𝑣(+g‘𝐻)𝑤)))
67	64, 65, 66	3bitr4g 316	. . . . . 6 ⊢ (𝑥 ∈ (𝐺 GrpIso 𝐻) → (𝐺 ∈ CMnd ↔ 𝐻 ∈ CMnd))
68	8, 67	anbi12d 632	. . . . 5 ⊢ (𝑥 ∈ (𝐺 GrpIso 𝐻) → ((𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd) ↔ (𝐻 ∈ Grp ∧ 𝐻 ∈ CMnd)))
69		isabl 18912	. . . . 5 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
70		isabl 18912	. . . . 5 ⊢ (𝐻 ∈ Abel ↔ (𝐻 ∈ Grp ∧ 𝐻 ∈ CMnd))
71	68, 69, 70	3bitr4g 316	. . . 4 ⊢ (𝑥 ∈ (𝐺 GrpIso 𝐻) → (𝐺 ∈ Abel ↔ 𝐻 ∈ Abel))
72	71	exlimiv 1931	. . 3 ⊢ (∃𝑥 𝑥 ∈ (𝐺 GrpIso 𝐻) → (𝐺 ∈ Abel ↔ 𝐻 ∈ Abel))
73	2, 72	sylbi 219	. 2 ⊢ ((𝐺 GrpIso 𝐻) ≠ ∅ → (𝐺 ∈ Abel ↔ 𝐻 ∈ Abel))
74	1, 73	sylbi 219	1 ⊢ (𝐺 ≃𝑔 𝐻 → (𝐺 ∈ Abel ↔ 𝐻 ∈ Abel))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114   ≠ wne 3018  ∀wral 3140   ⊆ wss 3938  ∅c0 4293   class class class wbr 5068   “ cima 5560   Fn wfn 6352  –1-1→wf1 6354  –onto→wfo 6355  –1-1-onto→wf1o 6356  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  +_gcplusg 16567  Mndcmnd 17913  Grpcgrp 18105   GrpHom cghm 18357   GrpIso cgim 18399   ≃_𝑔 cgic 18400  CMndccmn 18908  Abelcabl 18909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-1st 7691  df-2nd 7692  df-1o 8104  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-grp 18108  df-ghm 18358  df-gim 18401  df-gic 18402  df-cmn 18910  df-abl 18911

This theorem is referenced by:  isnumbasgrplem1  39708