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Theorem gicabl 37986
 Description: Being Abelian is a group invariant. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.)
Assertion
Ref Expression
gicabl (𝐺𝑔 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.
StepHypRef Expression
1 brgic 17758 . 2 (𝐺𝑔 𝐻 ↔ (𝐺 GrpIso 𝐻) ≠ ∅)
2 n0 3964 . . 3 ((𝐺 GrpIso 𝐻) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐺 GrpIso 𝐻))
3 gimghm 17753 . . . . . . . 8 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝑥 ∈ (𝐺 GrpHom 𝐻))
4 ghmgrp1 17709 . . . . . . . 8 (𝑥 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
53, 4syl 17 . . . . . . 7 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝐺 ∈ Grp)
6 ghmgrp2 17710 . . . . . . . 8 (𝑥 ∈ (𝐺 GrpHom 𝐻) → 𝐻 ∈ Grp)
73, 6syl 17 . . . . . . 7 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝐻 ∈ Grp)
85, 72thd 255 . . . . . 6 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (𝐺 ∈ Grp ↔ 𝐻 ∈ Grp))
9 grpmnd 17476 . . . . . . . . . 10 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
105, 9syl 17 . . . . . . . . 9 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝐺 ∈ Mnd)
11 grpmnd 17476 . . . . . . . . . 10 (𝐻 ∈ Grp → 𝐻 ∈ Mnd)
127, 11syl 17 . . . . . . . . 9 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝐻 ∈ Mnd)
1310, 122thd 255 . . . . . . . 8 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (𝐺 ∈ Mnd ↔ 𝐻 ∈ Mnd))
14 eqid 2651 . . . . . . . . . . . . . . . 16 (Base‘𝐺) = (Base‘𝐺)
15 eqid 2651 . . . . . . . . . . . . . . . 16 (Base‘𝐻) = (Base‘𝐻)
1614, 15gimf1o 17752 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝑥:(Base‘𝐺)–1-1-onto→(Base‘𝐻))
17 f1of1 6174 . . . . . . . . . . . . . . 15 (𝑥:(Base‘𝐺)–1-1-onto→(Base‘𝐻) → 𝑥:(Base‘𝐺)–1-1→(Base‘𝐻))
1816, 17syl 17 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝑥:(Base‘𝐺)–1-1→(Base‘𝐻))
1918adantr 480 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → 𝑥:(Base‘𝐺)–1-1→(Base‘𝐻))
205adantr 480 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → 𝐺 ∈ Grp)
21 simprl 809 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → 𝑦 ∈ (Base‘𝐺))
22 simprr 811 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → 𝑧 ∈ (Base‘𝐺))
23 eqid 2651 . . . . . . . . . . . . . . 15 (+g𝐺) = (+g𝐺)
2414, 23grpcl 17477 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝑦(+g𝐺)𝑧) ∈ (Base‘𝐺))
2520, 21, 22, 24syl3anc 1366 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑦(+g𝐺)𝑧) ∈ (Base‘𝐺))
2614, 23grpcl 17477 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝑧 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑧(+g𝐺)𝑦) ∈ (Base‘𝐺))
2720, 22, 21, 26syl3anc 1366 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑧(+g𝐺)𝑦) ∈ (Base‘𝐺))
28 f1fveq 6559 . . . . . . . . . . . . 13 ((𝑥:(Base‘𝐺)–1-1→(Base‘𝐻) ∧ ((𝑦(+g𝐺)𝑧) ∈ (Base‘𝐺) ∧ (𝑧(+g𝐺)𝑦) ∈ (Base‘𝐺))) → ((𝑥‘(𝑦(+g𝐺)𝑧)) = (𝑥‘(𝑧(+g𝐺)𝑦)) ↔ (𝑦(+g𝐺)𝑧) = (𝑧(+g𝐺)𝑦)))
2919, 25, 27, 28syl12anc 1364 . . . . . . . . . . . 12 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → ((𝑥‘(𝑦(+g𝐺)𝑧)) = (𝑥‘(𝑧(+g𝐺)𝑦)) ↔ (𝑦(+g𝐺)𝑧) = (𝑧(+g𝐺)𝑦)))
303adantr 480 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → 𝑥 ∈ (𝐺 GrpHom 𝐻))
31 eqid 2651 . . . . . . . . . . . . . . 15 (+g𝐻) = (+g𝐻)
3214, 23, 31ghmlin 17712 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝑥‘(𝑦(+g𝐺)𝑧)) = ((𝑥𝑦)(+g𝐻)(𝑥𝑧)))
3330, 21, 22, 32syl3anc 1366 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑥‘(𝑦(+g𝐺)𝑧)) = ((𝑥𝑦)(+g𝐻)(𝑥𝑧)))
3414, 23, 31ghmlin 17712 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥‘(𝑧(+g𝐺)𝑦)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦)))
3530, 22, 21, 34syl3anc 1366 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑥‘(𝑧(+g𝐺)𝑦)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦)))
3633, 35eqeq12d 2666 . . . . . . . . . . . 12 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → ((𝑥‘(𝑦(+g𝐺)𝑧)) = (𝑥‘(𝑧(+g𝐺)𝑦)) ↔ ((𝑥𝑦)(+g𝐻)(𝑥𝑧)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦))))
3729, 36bitr3d 270 . . . . . . . . . . 11 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → ((𝑦(+g𝐺)𝑧) = (𝑧(+g𝐺)𝑦) ↔ ((𝑥𝑦)(+g𝐻)(𝑥𝑧)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦))))
38372ralbidva 3017 . . . . . . . . . 10 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)(𝑦(+g𝐺)𝑧) = (𝑧(+g𝐺)𝑦) ↔ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑥𝑦)(+g𝐻)(𝑥𝑧)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦))))
39 f1ofo 6182 . . . . . . . . . . . . . . 15 (𝑥:(Base‘𝐺)–1-1-onto→(Base‘𝐻) → 𝑥:(Base‘𝐺)–onto→(Base‘𝐻))
40 foima 6158 . . . . . . . . . . . . . . 15 (𝑥:(Base‘𝐺)–onto→(Base‘𝐻) → (𝑥 “ (Base‘𝐺)) = (Base‘𝐻))
4139, 40syl 17 . . . . . . . . . . . . . 14 (𝑥:(Base‘𝐺)–1-1-onto→(Base‘𝐻) → (𝑥 “ (Base‘𝐺)) = (Base‘𝐻))
4216, 41syl 17 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (𝑥 “ (Base‘𝐺)) = (Base‘𝐻))
4342raleqdv 3174 . . . . . . . . . . . 12 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (∀𝑣 ∈ (𝑥 “ (Base‘𝐺))((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦)) ↔ ∀𝑣 ∈ (Base‘𝐻)((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦))))
44 f1ofn 6176 . . . . . . . . . . . . . 14 (𝑥:(Base‘𝐺)–1-1-onto→(Base‘𝐻) → 𝑥 Fn (Base‘𝐺))
4516, 44syl 17 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝑥 Fn (Base‘𝐺))
46 ssid 3657 . . . . . . . . . . . . 13 (Base‘𝐺) ⊆ (Base‘𝐺)
47 oveq2 6698 . . . . . . . . . . . . . . 15 (𝑣 = (𝑥𝑧) → ((𝑥𝑦)(+g𝐻)𝑣) = ((𝑥𝑦)(+g𝐻)(𝑥𝑧)))
48 oveq1 6697 . . . . . . . . . . . . . . 15 (𝑣 = (𝑥𝑧) → (𝑣(+g𝐻)(𝑥𝑦)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦)))
4947, 48eqeq12d 2666 . . . . . . . . . . . . . 14 (𝑣 = (𝑥𝑧) → (((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦)) ↔ ((𝑥𝑦)(+g𝐻)(𝑥𝑧)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦))))
5049ralima 6538 . . . . . . . . . . . . 13 ((𝑥 Fn (Base‘𝐺) ∧ (Base‘𝐺) ⊆ (Base‘𝐺)) → (∀𝑣 ∈ (𝑥 “ (Base‘𝐺))((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦)) ↔ ∀𝑧 ∈ (Base‘𝐺)((𝑥𝑦)(+g𝐻)(𝑥𝑧)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦))))
5145, 46, 50sylancl 695 . . . . . . . . . . . 12 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (∀𝑣 ∈ (𝑥 “ (Base‘𝐺))((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦)) ↔ ∀𝑧 ∈ (Base‘𝐺)((𝑥𝑦)(+g𝐻)(𝑥𝑧)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦))))
5243, 51bitr3d 270 . . . . . . . . . . 11 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (∀𝑣 ∈ (Base‘𝐻)((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦)) ↔ ∀𝑧 ∈ (Base‘𝐺)((𝑥𝑦)(+g𝐻)(𝑥𝑧)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦))))
5352ralbidv 3015 . . . . . . . . . 10 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (∀𝑦 ∈ (Base‘𝐺)∀𝑣 ∈ (Base‘𝐻)((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦)) ↔ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑥𝑦)(+g𝐻)(𝑥𝑧)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦))))
5438, 53bitr4d 271 . . . . . . . . 9 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)(𝑦(+g𝐺)𝑧) = (𝑧(+g𝐺)𝑦) ↔ ∀𝑦 ∈ (Base‘𝐺)∀𝑣 ∈ (Base‘𝐻)((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦))))
5542raleqdv 3174 . . . . . . . . . 10 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (∀𝑤 ∈ (𝑥 “ (Base‘𝐺))∀𝑣 ∈ (Base‘𝐻)(𝑤(+g𝐻)𝑣) = (𝑣(+g𝐻)𝑤) ↔ ∀𝑤 ∈ (Base‘𝐻)∀𝑣 ∈ (Base‘𝐻)(𝑤(+g𝐻)𝑣) = (𝑣(+g𝐻)𝑤)))
56 oveq1 6697 . . . . . . . . . . . . . 14 (𝑤 = (𝑥𝑦) → (𝑤(+g𝐻)𝑣) = ((𝑥𝑦)(+g𝐻)𝑣))
57 oveq2 6698 . . . . . . . . . . . . . 14 (𝑤 = (𝑥𝑦) → (𝑣(+g𝐻)𝑤) = (𝑣(+g𝐻)(𝑥𝑦)))
5856, 57eqeq12d 2666 . . . . . . . . . . . . 13 (𝑤 = (𝑥𝑦) → ((𝑤(+g𝐻)𝑣) = (𝑣(+g𝐻)𝑤) ↔ ((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦))))
5958ralbidv 3015 . . . . . . . . . . . 12 (𝑤 = (𝑥𝑦) → (∀𝑣 ∈ (Base‘𝐻)(𝑤(+g𝐻)𝑣) = (𝑣(+g𝐻)𝑤) ↔ ∀𝑣 ∈ (Base‘𝐻)((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦))))
6059ralima 6538 . . . . . . . . . . 11 ((𝑥 Fn (Base‘𝐺) ∧ (Base‘𝐺) ⊆ (Base‘𝐺)) → (∀𝑤 ∈ (𝑥 “ (Base‘𝐺))∀𝑣 ∈ (Base‘𝐻)(𝑤(+g𝐻)𝑣) = (𝑣(+g𝐻)𝑤) ↔ ∀𝑦 ∈ (Base‘𝐺)∀𝑣 ∈ (Base‘𝐻)((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦))))
6145, 46, 60sylancl 695 . . . . . . . . . 10 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (∀𝑤 ∈ (𝑥 “ (Base‘𝐺))∀𝑣 ∈ (Base‘𝐻)(𝑤(+g𝐻)𝑣) = (𝑣(+g𝐻)𝑤) ↔ ∀𝑦 ∈ (Base‘𝐺)∀𝑣 ∈ (Base‘𝐻)((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦))))
6255, 61bitr3d 270 . . . . . . . . 9 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (∀𝑤 ∈ (Base‘𝐻)∀𝑣 ∈ (Base‘𝐻)(𝑤(+g𝐻)𝑣) = (𝑣(+g𝐻)𝑤) ↔ ∀𝑦 ∈ (Base‘𝐺)∀𝑣 ∈ (Base‘𝐻)((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦))))
6354, 62bitr4d 271 . . . . . . . 8 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)(𝑦(+g𝐺)𝑧) = (𝑧(+g𝐺)𝑦) ↔ ∀𝑤 ∈ (Base‘𝐻)∀𝑣 ∈ (Base‘𝐻)(𝑤(+g𝐻)𝑣) = (𝑣(+g𝐻)𝑤)))
6413, 63anbi12d 747 . . . . . . 7 (𝑥 ∈ (𝐺 GrpIso 𝐻) → ((𝐺 ∈ Mnd ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)(𝑦(+g𝐺)𝑧) = (𝑧(+g𝐺)𝑦)) ↔ (𝐻 ∈ Mnd ∧ ∀𝑤 ∈ (Base‘𝐻)∀𝑣 ∈ (Base‘𝐻)(𝑤(+g𝐻)𝑣) = (𝑣(+g𝐻)𝑤))))
6514, 23iscmn 18246 . . . . . . 7 (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)(𝑦(+g𝐺)𝑧) = (𝑧(+g𝐺)𝑦)))
6615, 31iscmn 18246 . . . . . . 7 (𝐻 ∈ CMnd ↔ (𝐻 ∈ Mnd ∧ ∀𝑤 ∈ (Base‘𝐻)∀𝑣 ∈ (Base‘𝐻)(𝑤(+g𝐻)𝑣) = (𝑣(+g𝐻)𝑤)))
6764, 65, 663bitr4g 303 . . . . . 6 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (𝐺 ∈ CMnd ↔ 𝐻 ∈ CMnd))
688, 67anbi12d 747 . . . . 5 (𝑥 ∈ (𝐺 GrpIso 𝐻) → ((𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd) ↔ (𝐻 ∈ Grp ∧ 𝐻 ∈ CMnd)))
69 isabl 18243 . . . . 5 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
70 isabl 18243 . . . . 5 (𝐻 ∈ Abel ↔ (𝐻 ∈ Grp ∧ 𝐻 ∈ CMnd))
7168, 69, 703bitr4g 303 . . . 4 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (𝐺 ∈ Abel ↔ 𝐻 ∈ Abel))
7271exlimiv 1898 . . 3 (∃𝑥 𝑥 ∈ (𝐺 GrpIso 𝐻) → (𝐺 ∈ Abel ↔ 𝐻 ∈ Abel))
732, 72sylbi 207 . 2 ((𝐺 GrpIso 𝐻) ≠ ∅ → (𝐺 ∈ Abel ↔ 𝐻 ∈ Abel))
741, 73sylbi 207 1 (𝐺𝑔 𝐻 → (𝐺 ∈ Abel ↔ 𝐻 ∈ Abel))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523  ∃wex 1744   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941   ⊆ wss 3607  ∅c0 3948   class class class wbr 4685   “ cima 5146   Fn wfn 5921  –1-1→wf1 5923  –onto→wfo 5924  –1-1-onto→wf1o 5925  ‘cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  Mndcmnd 17341  Grpcgrp 17469   GrpHom cghm 17704   GrpIso cgim 17746   ≃𝑔 cgic 17747  CMndccmn 18239  Abelcabl 18240 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-1o 7605  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-ghm 17705  df-gim 17748  df-gic 17749  df-cmn 18241  df-abl 18242 This theorem is referenced by:  isnumbasgrplem1  37988
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