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Theorem gicabl 42355
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 19187 . 2 (𝐺𝑔 𝐻 ↔ (𝐺 GrpIso 𝐻) ≠ ∅)
2 n0 4339 . . 3 ((𝐺 GrpIso 𝐻) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐺 GrpIso 𝐻))
3 gimghm 19181 . . . . . . . 8 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝑥 ∈ (𝐺 GrpHom 𝐻))
4 ghmgrp1 19135 . . . . . . . 8 (𝑥 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
53, 4syl 17 . . . . . . 7 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝐺 ∈ Grp)
6 ghmgrp2 19136 . . . . . . . 8 (𝑥 ∈ (𝐺 GrpHom 𝐻) → 𝐻 ∈ Grp)
73, 6syl 17 . . . . . . 7 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝐻 ∈ Grp)
85, 72thd 265 . . . . . 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 265 . . . . . . . 8 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (𝐺 ∈ Mnd ↔ 𝐻 ∈ Mnd))
12 eqid 2724 . . . . . . . . . . . . . . . 16 (Base‘𝐺) = (Base‘𝐺)
13 eqid 2724 . . . . . . . . . . . . . . . 16 (Base‘𝐻) = (Base‘𝐻)
1412, 13gimf1o 19180 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝐺 GrpIso 𝐻) → 𝑥:(Base‘𝐺)–1-1-onto→(Base‘𝐻))
15 f1of1 6823 . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → 𝑥:(Base‘𝐺)–1-1→(Base‘𝐻))
185adantr 480 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → 𝐺 ∈ Grp)
19 simprl 768 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → 𝑦 ∈ (Base‘𝐺))
20 simprr 770 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → 𝑧 ∈ (Base‘𝐺))
21 eqid 2724 . . . . . . . . . . . . . . 15 (+g𝐺) = (+g𝐺)
2212, 21grpcl 18863 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝑦(+g𝐺)𝑧) ∈ (Base‘𝐺))
2318, 19, 20, 22syl3anc 1368 . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑧(+g𝐺)𝑦) ∈ (Base‘𝐺))
26 f1fveq 7254 . . . . . . . . . . . . 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 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)))
283adantr 480 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → 𝑥 ∈ (𝐺 GrpHom 𝐻))
29 eqid 2724 . . . . . . . . . . . . . . 15 (+g𝐻) = (+g𝐻)
3012, 21, 29ghmlin 19138 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝑥‘(𝑦(+g𝐺)𝑧)) = ((𝑥𝑦)(+g𝐻)(𝑥𝑧)))
3128, 19, 20, 30syl3anc 1368 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑥‘(𝑦(+g𝐺)𝑧)) = ((𝑥𝑦)(+g𝐻)(𝑥𝑧)))
3212, 21, 29ghmlin 19138 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥‘(𝑧(+g𝐺)𝑦)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦)))
3328, 20, 19, 32syl3anc 1368 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑥‘(𝑧(+g𝐺)𝑦)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦)))
3431, 33eqeq12d 2740 . . . . . . . . . . . 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 281 . . . . . . . . . . 11 ((𝑥 ∈ (𝐺 GrpIso 𝐻) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → ((𝑦(+g𝐺)𝑧) = (𝑧(+g𝐺)𝑦) ↔ ((𝑥𝑦)(+g𝐻)(𝑥𝑧)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦))))
36352ralbidva 3208 . . . . . . . . . 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 6831 . . . . . . . . . . . . . . 15 (𝑥:(Base‘𝐺)–1-1-onto→(Base‘𝐻) → 𝑥:(Base‘𝐺)–onto→(Base‘𝐻))
38 foima 6801 . . . . . . . . . . . . . . 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 3317 . . . . . . . . . . . 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 6825 . . . . . . . . . . . . . 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 3997 . . . . . . . . . . . . 13 (Base‘𝐺) ⊆ (Base‘𝐺)
45 oveq2 7410 . . . . . . . . . . . . . . 15 (𝑣 = (𝑥𝑧) → ((𝑥𝑦)(+g𝐻)𝑣) = ((𝑥𝑦)(+g𝐻)(𝑥𝑧)))
46 oveq1 7409 . . . . . . . . . . . . . . 15 (𝑣 = (𝑥𝑧) → (𝑣(+g𝐻)(𝑥𝑦)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦)))
4745, 46eqeq12d 2740 . . . . . . . . . . . . . 14 (𝑣 = (𝑥𝑧) → (((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦)) ↔ ((𝑥𝑦)(+g𝐻)(𝑥𝑧)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦))))
4847ralima 7232 . . . . . . . . . . . . 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 585 . . . . . . . . . . . 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 281 . . . . . . . . . . 11 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (∀𝑣 ∈ (Base‘𝐻)((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦)) ↔ ∀𝑧 ∈ (Base‘𝐺)((𝑥𝑦)(+g𝐻)(𝑥𝑧)) = ((𝑥𝑧)(+g𝐻)(𝑥𝑦))))
5150ralbidv 3169 . . . . . . . . . 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 282 . . . . . . . . 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 3317 . . . . . . . . . 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 7409 . . . . . . . . . . . . . 14 (𝑤 = (𝑥𝑦) → (𝑤(+g𝐻)𝑣) = ((𝑥𝑦)(+g𝐻)𝑣))
55 oveq2 7410 . . . . . . . . . . . . . 14 (𝑤 = (𝑥𝑦) → (𝑣(+g𝐻)𝑤) = (𝑣(+g𝐻)(𝑥𝑦)))
5654, 55eqeq12d 2740 . . . . . . . . . . . . 13 (𝑤 = (𝑥𝑦) → ((𝑤(+g𝐻)𝑣) = (𝑣(+g𝐻)𝑤) ↔ ((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦))))
5756ralbidv 3169 . . . . . . . . . . . 12 (𝑤 = (𝑥𝑦) → (∀𝑣 ∈ (Base‘𝐻)(𝑤(+g𝐻)𝑣) = (𝑣(+g𝐻)𝑤) ↔ ∀𝑣 ∈ (Base‘𝐻)((𝑥𝑦)(+g𝐻)𝑣) = (𝑣(+g𝐻)(𝑥𝑦))))
5857ralima 7232 . . . . . . . . . . 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 585 . . . . . . . . . 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 281 . . . . . . . . 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 282 . . . . . . . 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 630 . . . . . . 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 19701 . . . . . . 7 (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)(𝑦(+g𝐺)𝑧) = (𝑧(+g𝐺)𝑦)))
6413, 29iscmn 19701 . . . . . . 7 (𝐻 ∈ CMnd ↔ (𝐻 ∈ Mnd ∧ ∀𝑤 ∈ (Base‘𝐻)∀𝑣 ∈ (Base‘𝐻)(𝑤(+g𝐻)𝑣) = (𝑣(+g𝐻)𝑤)))
6562, 63, 643bitr4g 314 . . . . . 6 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (𝐺 ∈ CMnd ↔ 𝐻 ∈ CMnd))
668, 65anbi12d 630 . . . . 5 (𝑥 ∈ (𝐺 GrpIso 𝐻) → ((𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd) ↔ (𝐻 ∈ Grp ∧ 𝐻 ∈ CMnd)))
67 isabl 19696 . . . . 5 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
68 isabl 19696 . . . . 5 (𝐻 ∈ Abel ↔ (𝐻 ∈ Grp ∧ 𝐻 ∈ CMnd))
6966, 67, 683bitr4g 314 . . . 4 (𝑥 ∈ (𝐺 GrpIso 𝐻) → (𝐺 ∈ Abel ↔ 𝐻 ∈ Abel))
7069exlimiv 1925 . . 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 395   = wceq 1533  wex 1773  wcel 2098  wne 2932  wral 3053  wss 3941  c0 4315   class class class wbr 5139  cima 5670   Fn wfn 6529  1-1wf1 6531  ontowfo 6532  1-1-ontowf1o 6533  cfv 6534  (class class class)co 7402  Basecbs 17145  +gcplusg 17198  Mndcmnd 18659  Grpcgrp 18855   GrpHom cghm 19130   GrpIso cgim 19174  𝑔 cgic 19175  CMndccmn 19692  Abelcabl 19693
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-1st 7969  df-2nd 7970  df-1o 8462  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858  df-ghm 19131  df-gim 19176  df-gic 19177  df-cmn 19694  df-abl 19695
This theorem is referenced by:  isnumbasgrplem1  42357
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