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Theorem gicabl 39679
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 18401 . 2 (𝐺𝑔 𝐻 ↔ (𝐺 GrpIso 𝐻) ≠ ∅)
2 n0 4308 . . 3 ((𝐺 GrpIso 𝐻) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐺 GrpIso 𝐻))
3 gimghm 18396 . . . . . . . 8 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝑥 ∈ (𝐺 GrpHom 𝐻))
4 ghmgrp1 18352 . . . . . . . 8 (𝑥 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
53, 4syl 17 . . . . . . 7 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝐺 ∈ Grp)
6 ghmgrp2 18353 . . . . . . . 8 (𝑥 ∈ (𝐺 GrpHom 𝐻) → 𝐻 ∈ Grp)
73, 6syl 17 . . . . . . 7 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝐻 ∈ Grp)
85, 72thd 267 . . . . . 6 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (𝐺 ∈ Grp ↔ 𝐻 ∈ Grp))
9 grpmnd 18102 . . . . . . . . . 10 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
105, 9syl 17 . . . . . . . . 9 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝐺 ∈ Mnd)
11 grpmnd 18102 . . . . . . . . . 10 (𝐻 ∈ Grp → 𝐻 ∈ Mnd)
127, 11syl 17 . . . . . . . . 9 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝐻 ∈ Mnd)
1310, 122thd 267 . . . . . . . 8 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (𝐺 ∈ Mnd ↔ 𝐻 ∈ Mnd))
14 eqid 2819 . . . . . . . . . . . . . . . 16 (Base‘𝐺) = (Base‘𝐺)
15 eqid 2819 . . . . . . . . . . . . . . . 16 (Base‘𝐻) = (Base‘𝐻)
1614, 15gimf1o 18395 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝑥:(Base‘𝐺)–1-1-onto→(Base‘𝐻))
17 f1of1 6607 . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → 𝑥:(Base‘𝐺)–1-1→(Base‘𝐻))
205adantr 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 2819 . . . . . . . . . . . . . . 15 (+g𝐺) = (+g𝐺)
2414, 23grpcl 18103 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝑦(+g𝐺)𝑧) ∈ (Base‘𝐺))
2520, 21, 22, 24syl3anc 1365 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑦(+g𝐺)𝑧) ∈ (Base‘𝐺))
2614, 23grpcl 18103 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝑧 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑧(+g𝐺)𝑦) ∈ (Base‘𝐺))
2720, 22, 21, 26syl3anc 1365 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑧(+g𝐺)𝑦) ∈ (Base‘𝐺))
28 f1fveq 7012 . . . . . . . . . . . . 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 834 . . . . . . . . . . . 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 483 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → 𝑥 ∈ (𝐺 GrpHom 𝐻))
31 eqid 2819 . . . . . . . . . . . . . . 15 (+g𝐻) = (+g𝐻)
3214, 23, 31ghmlin 18355 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝑥‘(𝑦(+g𝐺)𝑧)) = ((𝑥𝑦)(+g𝐻)(𝑥𝑧)))
3330, 21, 22, 32syl3anc 1365 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑥‘(𝑦(+g𝐺)𝑧)) = ((𝑥𝑦)(+g𝐻)(𝑥𝑧)))
3414, 23, 31ghmlin 18355 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥‘(𝑧(+g𝐺)𝑦)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦)))
3530, 22, 21, 34syl3anc 1365 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑥‘(𝑧(+g𝐺)𝑦)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦)))
3633, 35eqeq12d 2835 . . . . . . . . . . . 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 283 . . . . . . . . . . 11 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → ((𝑦(+g𝐺)𝑧) = (𝑧(+g𝐺)𝑦) ↔ ((𝑥𝑦)(+g𝐻)(𝑥𝑧)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦))))
38372ralbidva 3196 . . . . . . . . . 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 6615 . . . . . . . . . . . . . . 15 (𝑥:(Base‘𝐺)–1-1-onto→(Base‘𝐻) → 𝑥:(Base‘𝐺)–onto→(Base‘𝐻))
40 foima 6588 . . . . . . . . . . . . . . 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 3414 . . . . . . . . . . . 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 6609 . . . . . . . . . . . . . 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 3987 . . . . . . . . . . . . 13 (Base‘𝐺) ⊆ (Base‘𝐺)
47 oveq2 7156 . . . . . . . . . . . . . . 15 (𝑣 = (𝑥𝑧) → ((𝑥𝑦)(+g𝐻)𝑣) = ((𝑥𝑦)(+g𝐻)(𝑥𝑧)))
48 oveq1 7155 . . . . . . . . . . . . . . 15 (𝑣 = (𝑥𝑧) → (𝑣(+g𝐻)(𝑥𝑦)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦)))
4947, 48eqeq12d 2835 . . . . . . . . . . . . . 14 (𝑣 = (𝑥𝑧) → (((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦)) ↔ ((𝑥𝑦)(+g𝐻)(𝑥𝑧)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦))))
5049ralima 6992 . . . . . . . . . . . . 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 588 . . . . . . . . . . . 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 283 . . . . . . . . . . 11 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (∀𝑣 ∈ (Base‘𝐻)((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦)) ↔ ∀𝑧 ∈ (Base‘𝐺)((𝑥𝑦)(+g𝐻)(𝑥𝑧)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦))))
5352ralbidv 3195 . . . . . . . . . 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 284 . . . . . . . . 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 3414 . . . . . . . . . 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 7155 . . . . . . . . . . . . . 14 (𝑤 = (𝑥𝑦) → (𝑤(+g𝐻)𝑣) = ((𝑥𝑦)(+g𝐻)𝑣))
57 oveq2 7156 . . . . . . . . . . . . . 14 (𝑤 = (𝑥𝑦) → (𝑣(+g𝐻)𝑤) = (𝑣(+g𝐻)(𝑥𝑦)))
5856, 57eqeq12d 2835 . . . . . . . . . . . . 13 (𝑤 = (𝑥𝑦) → ((𝑤(+g𝐻)𝑣) = (𝑣(+g𝐻)𝑤) ↔ ((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦))))
5958ralbidv 3195 . . . . . . . . . . . 12 (𝑤 = (𝑥𝑦) → (∀𝑣 ∈ (Base‘𝐻)(𝑤(+g𝐻)𝑣) = (𝑣(+g𝐻)𝑤) ↔ ∀𝑣 ∈ (Base‘𝐻)((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦))))
6059ralima 6992 . . . . . . . . . . 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 588 . . . . . . . . . 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 283 . . . . . . . . 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 284 . . . . . . . 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 632 . . . . . . 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 18906 . . . . . . 7 (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)(𝑦(+g𝐺)𝑧) = (𝑧(+g𝐺)𝑦)))
6615, 31iscmn 18906 . . . . . . 7 (𝐻 ∈ CMnd ↔ (𝐻 ∈ Mnd ∧ ∀𝑤 ∈ (Base‘𝐻)∀𝑣 ∈ (Base‘𝐻)(𝑤(+g𝐻)𝑣) = (𝑣(+g𝐻)𝑤)))
6764, 65, 663bitr4g 316 . . . . . 6 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (𝐺 ∈ CMnd ↔ 𝐻 ∈ CMnd))
688, 67anbi12d 632 . . . . 5 (𝑥 ∈ (𝐺 GrpIso 𝐻) → ((𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd) ↔ (𝐻 ∈ Grp ∧ 𝐻 ∈ CMnd)))
69 isabl 18902 . . . . 5 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
70 isabl 18902 . . . . 5 (𝐻 ∈ Abel ↔ (𝐻 ∈ Grp ∧ 𝐻 ∈ CMnd))
7168, 69, 703bitr4g 316 . . . 4 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (𝐺 ∈ Abel ↔ 𝐻 ∈ Abel))
7271exlimiv 1924 . . 3 (∃𝑥 𝑥 ∈ (𝐺 GrpIso 𝐻) → (𝐺 ∈ Abel ↔ 𝐻 ∈ Abel))
732, 72sylbi 219 . 2 ((𝐺 GrpIso 𝐻) ≠ ∅ → (𝐺 ∈ Abel ↔ 𝐻 ∈ Abel))
741, 73sylbi 219 1 (𝐺𝑔 𝐻 → (𝐺 ∈ Abel ↔ 𝐻 ∈ Abel))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1530  wex 1773  wcel 2107  wne 3014  wral 3136  wss 3934  c0 4289   class class class wbr 5057  cima 5551   Fn wfn 6343  1-1wf1 6345  ontowfo 6346  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7148  Basecbs 16475  +gcplusg 16557  Mndcmnd 17903  Grpcgrp 18095   GrpHom cghm 18347   GrpIso cgim 18389  𝑔 cgic 18390  CMndccmn 18898  Abelcabl 18899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-1o 8094  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098  df-ghm 18348  df-gim 18391  df-gic 18392  df-cmn 18900  df-abl 18901
This theorem is referenced by:  isnumbasgrplem1  39681
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