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Theorem gicabl 42143
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 19184 . 2 (𝐺𝑔 𝐻 ↔ (𝐺 GrpIso 𝐻) ≠ ∅)
2 n0 4346 . . 3 ((𝐺 GrpIso 𝐻) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐺 GrpIso 𝐻))
3 gimghm 19178 . . . . . . . 8 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝑥 ∈ (𝐺 GrpHom 𝐻))
4 ghmgrp1 19132 . . . . . . . 8 (𝑥 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
53, 4syl 17 . . . . . . 7 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝐺 ∈ Grp)
6 ghmgrp2 19133 . . . . . . . 8 (𝑥 ∈ (𝐺 GrpHom 𝐻) → 𝐻 ∈ Grp)
73, 6syl 17 . . . . . . 7 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝐻 ∈ Grp)
85, 72thd 264 . . . . . 6 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (𝐺 ∈ Grp ↔ 𝐻 ∈ Grp))
95grpmndd 18868 . . . . . . . . 9 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝐺 ∈ Mnd)
107grpmndd 18868 . . . . . . . . 9 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝐻 ∈ Mnd)
119, 102thd 264 . . . . . . . 8 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (𝐺 ∈ Mnd ↔ 𝐻 ∈ Mnd))
12 eqid 2732 . . . . . . . . . . . . . . . 16 (Base‘𝐺) = (Base‘𝐺)
13 eqid 2732 . . . . . . . . . . . . . . . 16 (Base‘𝐻) = (Base‘𝐻)
1412, 13gimf1o 19177 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝑥:(Base‘𝐺)–1-1-onto→(Base‘𝐻))
15 f1of1 6832 . . . . . . . . . . . . . . 15 (𝑥:(Base‘𝐺)–1-1-onto→(Base‘𝐻) → 𝑥:(Base‘𝐺)–1-1→(Base‘𝐻))
1614, 15syl 17 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝑥:(Base‘𝐺)–1-1→(Base‘𝐻))
1716adantr 481 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → 𝑥:(Base‘𝐺)–1-1→(Base‘𝐻))
185adantr 481 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → 𝐺 ∈ Grp)
19 simprl 769 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → 𝑦 ∈ (Base‘𝐺))
20 simprr 771 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → 𝑧 ∈ (Base‘𝐺))
21 eqid 2732 . . . . . . . . . . . . . . 15 (+g𝐺) = (+g𝐺)
2212, 21grpcl 18863 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝑦(+g𝐺)𝑧) ∈ (Base‘𝐺))
2318, 19, 20, 22syl3anc 1371 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑦(+g𝐺)𝑧) ∈ (Base‘𝐺))
2412, 21grpcl 18863 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝑧 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑧(+g𝐺)𝑦) ∈ (Base‘𝐺))
2518, 20, 19, 24syl3anc 1371 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑧(+g𝐺)𝑦) ∈ (Base‘𝐺))
26 f1fveq 7263 . . . . . . . . . . . . 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)))
2717, 23, 25, 26syl12anc 835 . . . . . . . . . . . 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)))
283adantr 481 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → 𝑥 ∈ (𝐺 GrpHom 𝐻))
29 eqid 2732 . . . . . . . . . . . . . . 15 (+g𝐻) = (+g𝐻)
3012, 21, 29ghmlin 19135 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝑥‘(𝑦(+g𝐺)𝑧)) = ((𝑥𝑦)(+g𝐻)(𝑥𝑧)))
3128, 19, 20, 30syl3anc 1371 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑥‘(𝑦(+g𝐺)𝑧)) = ((𝑥𝑦)(+g𝐻)(𝑥𝑧)))
3212, 21, 29ghmlin 19135 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥‘(𝑧(+g𝐺)𝑦)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦)))
3328, 20, 19, 32syl3anc 1371 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑥‘(𝑧(+g𝐺)𝑦)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦)))
3431, 33eqeq12d 2748 . . . . . . . . . . . 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))))
3527, 34bitr3d 280 . . . . . . . . . . 11 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → ((𝑦(+g𝐺)𝑧) = (𝑧(+g𝐺)𝑦) ↔ ((𝑥𝑦)(+g𝐻)(𝑥𝑧)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦))))
36352ralbidva 3216 . . . . . . . . . 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))))
37 f1ofo 6840 . . . . . . . . . . . . . . 15 (𝑥:(Base‘𝐺)–1-1-onto→(Base‘𝐻) → 𝑥:(Base‘𝐺)–onto→(Base‘𝐻))
38 foima 6810 . . . . . . . . . . . . . . 15 (𝑥:(Base‘𝐺)–onto→(Base‘𝐻) → (𝑥 “ (Base‘𝐺)) = (Base‘𝐻))
3937, 38syl 17 . . . . . . . . . . . . . 14 (𝑥:(Base‘𝐺)–1-1-onto→(Base‘𝐻) → (𝑥 “ (Base‘𝐺)) = (Base‘𝐻))
4014, 39syl 17 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (𝑥 “ (Base‘𝐺)) = (Base‘𝐻))
4140raleqdv 3325 . . . . . . . . . . . 12 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (∀𝑣 ∈ (𝑥 “ (Base‘𝐺))((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦)) ↔ ∀𝑣 ∈ (Base‘𝐻)((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦))))
42 f1ofn 6834 . . . . . . . . . . . . . 14 (𝑥:(Base‘𝐺)–1-1-onto→(Base‘𝐻) → 𝑥 Fn (Base‘𝐺))
4314, 42syl 17 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝑥 Fn (Base‘𝐺))
44 ssid 4004 . . . . . . . . . . . . 13 (Base‘𝐺) ⊆ (Base‘𝐺)
45 oveq2 7419 . . . . . . . . . . . . . . 15 (𝑣 = (𝑥𝑧) → ((𝑥𝑦)(+g𝐻)𝑣) = ((𝑥𝑦)(+g𝐻)(𝑥𝑧)))
46 oveq1 7418 . . . . . . . . . . . . . . 15 (𝑣 = (𝑥𝑧) → (𝑣(+g𝐻)(𝑥𝑦)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦)))
4745, 46eqeq12d 2748 . . . . . . . . . . . . . 14 (𝑣 = (𝑥𝑧) → (((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦)) ↔ ((𝑥𝑦)(+g𝐻)(𝑥𝑧)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦))))
4847ralima 7242 . . . . . . . . . . . . 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))))
4943, 44, 48sylancl 586 . . . . . . . . . . . 12 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (∀𝑣 ∈ (𝑥 “ (Base‘𝐺))((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦)) ↔ ∀𝑧 ∈ (Base‘𝐺)((𝑥𝑦)(+g𝐻)(𝑥𝑧)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦))))
5041, 49bitr3d 280 . . . . . . . . . . 11 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (∀𝑣 ∈ (Base‘𝐻)((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦)) ↔ ∀𝑧 ∈ (Base‘𝐺)((𝑥𝑦)(+g𝐻)(𝑥𝑧)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦))))
5150ralbidv 3177 . . . . . . . . . 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))))
5236, 51bitr4d 281 . . . . . . . . 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))))
5340raleqdv 3325 . . . . . . . . . 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)))
54 oveq1 7418 . . . . . . . . . . . . . 14 (𝑤 = (𝑥𝑦) → (𝑤(+g𝐻)𝑣) = ((𝑥𝑦)(+g𝐻)𝑣))
55 oveq2 7419 . . . . . . . . . . . . . 14 (𝑤 = (𝑥𝑦) → (𝑣(+g𝐻)𝑤) = (𝑣(+g𝐻)(𝑥𝑦)))
5654, 55eqeq12d 2748 . . . . . . . . . . . . 13 (𝑤 = (𝑥𝑦) → ((𝑤(+g𝐻)𝑣) = (𝑣(+g𝐻)𝑤) ↔ ((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦))))
5756ralbidv 3177 . . . . . . . . . . . 12 (𝑤 = (𝑥𝑦) → (∀𝑣 ∈ (Base‘𝐻)(𝑤(+g𝐻)𝑣) = (𝑣(+g𝐻)𝑤) ↔ ∀𝑣 ∈ (Base‘𝐻)((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦))))
5857ralima 7242 . . . . . . . . . . 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))))
5943, 44, 58sylancl 586 . . . . . . . . . 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))))
6053, 59bitr3d 280 . . . . . . . . 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))))
6152, 60bitr4d 281 . . . . . . . 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)))
6211, 61anbi12d 631 . . . . . . 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))))
6312, 21iscmn 19698 . . . . . . 7 (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)(𝑦(+g𝐺)𝑧) = (𝑧(+g𝐺)𝑦)))
6413, 29iscmn 19698 . . . . . . 7 (𝐻 ∈ CMnd ↔ (𝐻 ∈ Mnd ∧ ∀𝑤 ∈ (Base‘𝐻)∀𝑣 ∈ (Base‘𝐻)(𝑤(+g𝐻)𝑣) = (𝑣(+g𝐻)𝑤)))
6562, 63, 643bitr4g 313 . . . . . 6 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (𝐺 ∈ CMnd ↔ 𝐻 ∈ CMnd))
668, 65anbi12d 631 . . . . 5 (𝑥 ∈ (𝐺 GrpIso 𝐻) → ((𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd) ↔ (𝐻 ∈ Grp ∧ 𝐻 ∈ CMnd)))
67 isabl 19693 . . . . 5 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
68 isabl 19693 . . . . 5 (𝐻 ∈ Abel ↔ (𝐻 ∈ Grp ∧ 𝐻 ∈ CMnd))
6966, 67, 683bitr4g 313 . . . 4 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (𝐺 ∈ Abel ↔ 𝐻 ∈ Abel))
7069exlimiv 1933 . . 3 (∃𝑥 𝑥 ∈ (𝐺 GrpIso 𝐻) → (𝐺 ∈ Abel ↔ 𝐻 ∈ Abel))
712, 70sylbi 216 . 2 ((𝐺 GrpIso 𝐻) ≠ ∅ → (𝐺 ∈ Abel ↔ 𝐻 ∈ Abel))
721, 71sylbi 216 1 (𝐺𝑔 𝐻 → (𝐺 ∈ Abel ↔ 𝐻 ∈ Abel))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  wne 2940  wral 3061  wss 3948  c0 4322   class class class wbr 5148  cima 5679   Fn wfn 6538  1-1wf1 6540  ontowfo 6541  1-1-ontowf1o 6542  cfv 6543  (class class class)co 7411  Basecbs 17148  +gcplusg 17201  Mndcmnd 18659  Grpcgrp 18855   GrpHom cghm 19127   GrpIso cgim 19171  𝑔 cgic 19172  CMndccmn 19689  Abelcabl 19690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-1o 8468  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858  df-ghm 19128  df-gim 19173  df-gic 19174  df-cmn 19691  df-abl 19692
This theorem is referenced by:  isnumbasgrplem1  42145
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