Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  gicabl Structured version   Visualization version   GIF version

Theorem gicabl 37149
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 17632 . 2 (𝐺𝑔 𝐻 ↔ (𝐺 GrpIso 𝐻) ≠ ∅)
2 n0 3907 . . 3 ((𝐺 GrpIso 𝐻) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐺 GrpIso 𝐻))
3 gimghm 17627 . . . . . . . 8 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝑥 ∈ (𝐺 GrpHom 𝐻))
4 ghmgrp1 17583 . . . . . . . 8 (𝑥 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
53, 4syl 17 . . . . . . 7 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝐺 ∈ Grp)
6 ghmgrp2 17584 . . . . . . . 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 17350 . . . . . . . . . 10 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
105, 9syl 17 . . . . . . . . 9 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝐺 ∈ Mnd)
11 grpmnd 17350 . . . . . . . . . 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 2621 . . . . . . . . . . . . . . . 16 (Base‘𝐺) = (Base‘𝐺)
15 eqid 2621 . . . . . . . . . . . . . . . 16 (Base‘𝐻) = (Base‘𝐻)
1614, 15gimf1o 17626 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝑥:(Base‘𝐺)–1-1-onto→(Base‘𝐻))
17 f1of1 6093 . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → 𝑥:(Base‘𝐺)–1-1→(Base‘𝐻))
205adantr 481 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → 𝐺 ∈ Grp)
21 simprl 793 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → 𝑦 ∈ (Base‘𝐺))
22 simprr 795 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → 𝑧 ∈ (Base‘𝐺))
23 eqid 2621 . . . . . . . . . . . . . . 15 (+g𝐺) = (+g𝐺)
2414, 23grpcl 17351 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝑦(+g𝐺)𝑧) ∈ (Base‘𝐺))
2520, 21, 22, 24syl3anc 1323 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑦(+g𝐺)𝑧) ∈ (Base‘𝐺))
2614, 23grpcl 17351 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝑧 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑧(+g𝐺)𝑦) ∈ (Base‘𝐺))
2720, 22, 21, 26syl3anc 1323 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑧(+g𝐺)𝑦) ∈ (Base‘𝐺))
28 f1fveq 6473 . . . . . . . . . . . . 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 1321 . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → 𝑥 ∈ (𝐺 GrpHom 𝐻))
31 eqid 2621 . . . . . . . . . . . . . . 15 (+g𝐻) = (+g𝐻)
3214, 23, 31ghmlin 17586 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝑥‘(𝑦(+g𝐺)𝑧)) = ((𝑥𝑦)(+g𝐻)(𝑥𝑧)))
3330, 21, 22, 32syl3anc 1323 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑥‘(𝑦(+g𝐺)𝑧)) = ((𝑥𝑦)(+g𝐻)(𝑥𝑧)))
3414, 23, 31ghmlin 17586 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥‘(𝑧(+g𝐺)𝑦)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦)))
3530, 22, 21, 34syl3anc 1323 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑥‘(𝑧(+g𝐺)𝑦)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦)))
3633, 35eqeq12d 2636 . . . . . . . . . . . 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 2982 . . . . . . . . . 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 6101 . . . . . . . . . . . . . . 15 (𝑥:(Base‘𝐺)–1-1-onto→(Base‘𝐻) → 𝑥:(Base‘𝐺)–onto→(Base‘𝐻))
40 foima 6077 . . . . . . . . . . . . . . 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 3133 . . . . . . . . . . . 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 6095 . . . . . . . . . . . . . 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 3603 . . . . . . . . . . . . 13 (Base‘𝐺) ⊆ (Base‘𝐺)
47 oveq2 6612 . . . . . . . . . . . . . . 15 (𝑣 = (𝑥𝑧) → ((𝑥𝑦)(+g𝐻)𝑣) = ((𝑥𝑦)(+g𝐻)(𝑥𝑧)))
48 oveq1 6611 . . . . . . . . . . . . . . 15 (𝑣 = (𝑥𝑧) → (𝑣(+g𝐻)(𝑥𝑦)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦)))
4947, 48eqeq12d 2636 . . . . . . . . . . . . . 14 (𝑣 = (𝑥𝑧) → (((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦)) ↔ ((𝑥𝑦)(+g𝐻)(𝑥𝑧)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦))))
5049ralima 6452 . . . . . . . . . . . . 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 693 . . . . . . . . . . . 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 2980 . . . . . . . . . 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 3133 . . . . . . . . . 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 6611 . . . . . . . . . . . . . 14 (𝑤 = (𝑥𝑦) → (𝑤(+g𝐻)𝑣) = ((𝑥𝑦)(+g𝐻)𝑣))
57 oveq2 6612 . . . . . . . . . . . . . 14 (𝑤 = (𝑥𝑦) → (𝑣(+g𝐻)𝑤) = (𝑣(+g𝐻)(𝑥𝑦)))
5856, 57eqeq12d 2636 . . . . . . . . . . . . 13 (𝑤 = (𝑥𝑦) → ((𝑤(+g𝐻)𝑣) = (𝑣(+g𝐻)𝑤) ↔ ((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦))))
5958ralbidv 2980 . . . . . . . . . . . 12 (𝑤 = (𝑥𝑦) → (∀𝑣 ∈ (Base‘𝐻)(𝑤(+g𝐻)𝑣) = (𝑣(+g𝐻)𝑤) ↔ ∀𝑣 ∈ (Base‘𝐻)((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦))))
6059ralima 6452 . . . . . . . . . . 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 693 . . . . . . . . . 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 746 . . . . . . 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 18121 . . . . . . 7 (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)(𝑦(+g𝐺)𝑧) = (𝑧(+g𝐺)𝑦)))
6615, 31iscmn 18121 . . . . . . 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 746 . . . . 5 (𝑥 ∈ (𝐺 GrpIso 𝐻) → ((𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd) ↔ (𝐻 ∈ Grp ∧ 𝐻 ∈ CMnd)))
69 isabl 18118 . . . . 5 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
70 isabl 18118 . . . . 5 (𝐻 ∈ Abel ↔ (𝐻 ∈ Grp ∧ 𝐻 ∈ CMnd))
7168, 69, 703bitr4g 303 . . . 4 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (𝐺 ∈ Abel ↔ 𝐻 ∈ Abel))
7271exlimiv 1855 . . 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 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2907  wss 3555  c0 3891   class class class wbr 4613  cima 5077   Fn wfn 5842  1-1wf1 5844  ontowfo 5845  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  Mndcmnd 17215  Grpcgrp 17343   GrpHom cghm 17578   GrpIso cgim 17620  𝑔 cgic 17621  CMndccmn 18114  Abelcabl 18115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-1o 7505  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-grp 17346  df-ghm 17579  df-gim 17622  df-gic 17623  df-cmn 18116  df-abl 18117
This theorem is referenced by:  isnumbasgrplem1  37152
  Copyright terms: Public domain W3C validator