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Theorem ghmnsgima 18127
Description: The image of a normal subgroup under a surjective homomorphism is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
Hypothesis
Ref Expression
ghmnsgima.1 𝑌 = (Base‘𝑇)
Assertion
Ref Expression
ghmnsgima ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (𝐹𝑈) ∈ (NrmSGrp‘𝑇))

Proof of Theorem ghmnsgima
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1129 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2 nsgsubg 18069 . . . 4 (𝑈 ∈ (NrmSGrp‘𝑆) → 𝑈 ∈ (SubGrp‘𝑆))
323ad2ant2 1127 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝑈 ∈ (SubGrp‘𝑆))
4 ghmima 18124 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (SubGrp‘𝑆)) → (𝐹𝑈) ∈ (SubGrp‘𝑇))
51, 3, 4syl2anc 584 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (𝐹𝑈) ∈ (SubGrp‘𝑇))
61adantr 481 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
7 ghmgrp1 18105 . . . . . . . . 9 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
86, 7syl 17 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝑆 ∈ Grp)
9 simprl 767 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝑧 ∈ (Base‘𝑆))
10 eqid 2797 . . . . . . . . . . . 12 (Base‘𝑆) = (Base‘𝑆)
1110subgss 18038 . . . . . . . . . . 11 (𝑈 ∈ (SubGrp‘𝑆) → 𝑈 ⊆ (Base‘𝑆))
123, 11syl 17 . . . . . . . . . 10 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝑈 ⊆ (Base‘𝑆))
1312adantr 481 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝑈 ⊆ (Base‘𝑆))
14 simprr 769 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝑥𝑈)
1513, 14sseldd 3896 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝑥 ∈ (Base‘𝑆))
16 eqid 2797 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
1710, 16grpcl 17873 . . . . . . . 8 ((𝑆 ∈ Grp ∧ 𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝑧(+g𝑆)𝑥) ∈ (Base‘𝑆))
188, 9, 15, 17syl3anc 1364 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → (𝑧(+g𝑆)𝑥) ∈ (Base‘𝑆))
19 eqid 2797 . . . . . . . 8 (-g𝑆) = (-g𝑆)
20 eqid 2797 . . . . . . . 8 (-g𝑇) = (-g𝑇)
2110, 19, 20ghmsub 18111 . . . . . . 7 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑧(+g𝑆)𝑥) ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆)) → (𝐹‘((𝑧(+g𝑆)𝑥)(-g𝑆)𝑧)) = ((𝐹‘(𝑧(+g𝑆)𝑥))(-g𝑇)(𝐹𝑧)))
226, 18, 9, 21syl3anc 1364 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → (𝐹‘((𝑧(+g𝑆)𝑥)(-g𝑆)𝑧)) = ((𝐹‘(𝑧(+g𝑆)𝑥))(-g𝑇)(𝐹𝑧)))
23 eqid 2797 . . . . . . . . 9 (+g𝑇) = (+g𝑇)
2410, 16, 23ghmlin 18108 . . . . . . . 8 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐹‘(𝑧(+g𝑆)𝑥)) = ((𝐹𝑧)(+g𝑇)(𝐹𝑥)))
256, 9, 15, 24syl3anc 1364 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → (𝐹‘(𝑧(+g𝑆)𝑥)) = ((𝐹𝑧)(+g𝑇)(𝐹𝑥)))
2625oveq1d 7038 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → ((𝐹‘(𝑧(+g𝑆)𝑥))(-g𝑇)(𝐹𝑧)) = (((𝐹𝑧)(+g𝑇)(𝐹𝑥))(-g𝑇)(𝐹𝑧)))
2722, 26eqtrd 2833 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → (𝐹‘((𝑧(+g𝑆)𝑥)(-g𝑆)𝑧)) = (((𝐹𝑧)(+g𝑇)(𝐹𝑥))(-g𝑇)(𝐹𝑧)))
28 ghmnsgima.1 . . . . . . . . . 10 𝑌 = (Base‘𝑇)
2910, 28ghmf 18107 . . . . . . . . 9 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶𝑌)
301, 29syl 17 . . . . . . . 8 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝐹:(Base‘𝑆)⟶𝑌)
3130adantr 481 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝐹:(Base‘𝑆)⟶𝑌)
3231ffnd 6390 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝐹 Fn (Base‘𝑆))
33 simpl2 1185 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝑈 ∈ (NrmSGrp‘𝑆))
3410, 16, 19nsgconj 18070 . . . . . . 7 ((𝑈 ∈ (NrmSGrp‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈) → ((𝑧(+g𝑆)𝑥)(-g𝑆)𝑧) ∈ 𝑈)
3533, 9, 14, 34syl3anc 1364 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → ((𝑧(+g𝑆)𝑥)(-g𝑆)𝑧) ∈ 𝑈)
36 fnfvima 6867 . . . . . 6 ((𝐹 Fn (Base‘𝑆) ∧ 𝑈 ⊆ (Base‘𝑆) ∧ ((𝑧(+g𝑆)𝑥)(-g𝑆)𝑧) ∈ 𝑈) → (𝐹‘((𝑧(+g𝑆)𝑥)(-g𝑆)𝑧)) ∈ (𝐹𝑈))
3732, 13, 35, 36syl3anc 1364 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → (𝐹‘((𝑧(+g𝑆)𝑥)(-g𝑆)𝑧)) ∈ (𝐹𝑈))
3827, 37eqeltrrd 2886 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → (((𝐹𝑧)(+g𝑇)(𝐹𝑥))(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈))
3938ralrimivva 3160 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → ∀𝑧 ∈ (Base‘𝑆)∀𝑥𝑈 (((𝐹𝑧)(+g𝑇)(𝐹𝑥))(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈))
4030ffnd 6390 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝐹 Fn (Base‘𝑆))
41 oveq1 7030 . . . . . . . . 9 (𝑥 = (𝐹𝑧) → (𝑥(+g𝑇)𝑦) = ((𝐹𝑧)(+g𝑇)𝑦))
42 id 22 . . . . . . . . 9 (𝑥 = (𝐹𝑧) → 𝑥 = (𝐹𝑧))
4341, 42oveq12d 7041 . . . . . . . 8 (𝑥 = (𝐹𝑧) → ((𝑥(+g𝑇)𝑦)(-g𝑇)𝑥) = (((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)))
4443eleq1d 2869 . . . . . . 7 (𝑥 = (𝐹𝑧) → (((𝑥(+g𝑇)𝑦)(-g𝑇)𝑥) ∈ (𝐹𝑈) ↔ (((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈)))
4544ralbidv 3166 . . . . . 6 (𝑥 = (𝐹𝑧) → (∀𝑦 ∈ (𝐹𝑈)((𝑥(+g𝑇)𝑦)(-g𝑇)𝑥) ∈ (𝐹𝑈) ↔ ∀𝑦 ∈ (𝐹𝑈)(((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈)))
4645ralrn 6726 . . . . 5 (𝐹 Fn (Base‘𝑆) → (∀𝑥 ∈ ran 𝐹𝑦 ∈ (𝐹𝑈)((𝑥(+g𝑇)𝑦)(-g𝑇)𝑥) ∈ (𝐹𝑈) ↔ ∀𝑧 ∈ (Base‘𝑆)∀𝑦 ∈ (𝐹𝑈)(((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈)))
4740, 46syl 17 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (∀𝑥 ∈ ran 𝐹𝑦 ∈ (𝐹𝑈)((𝑥(+g𝑇)𝑦)(-g𝑇)𝑥) ∈ (𝐹𝑈) ↔ ∀𝑧 ∈ (Base‘𝑆)∀𝑦 ∈ (𝐹𝑈)(((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈)))
48 simp3 1131 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → ran 𝐹 = 𝑌)
4948raleqdv 3377 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (∀𝑥 ∈ ran 𝐹𝑦 ∈ (𝐹𝑈)((𝑥(+g𝑇)𝑦)(-g𝑇)𝑥) ∈ (𝐹𝑈) ↔ ∀𝑥𝑌𝑦 ∈ (𝐹𝑈)((𝑥(+g𝑇)𝑦)(-g𝑇)𝑥) ∈ (𝐹𝑈)))
50 oveq2 7031 . . . . . . . . 9 (𝑦 = (𝐹𝑥) → ((𝐹𝑧)(+g𝑇)𝑦) = ((𝐹𝑧)(+g𝑇)(𝐹𝑥)))
5150oveq1d 7038 . . . . . . . 8 (𝑦 = (𝐹𝑥) → (((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)) = (((𝐹𝑧)(+g𝑇)(𝐹𝑥))(-g𝑇)(𝐹𝑧)))
5251eleq1d 2869 . . . . . . 7 (𝑦 = (𝐹𝑥) → ((((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈) ↔ (((𝐹𝑧)(+g𝑇)(𝐹𝑥))(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈)))
5352ralima 6872 . . . . . 6 ((𝐹 Fn (Base‘𝑆) ∧ 𝑈 ⊆ (Base‘𝑆)) → (∀𝑦 ∈ (𝐹𝑈)(((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈) ↔ ∀𝑥𝑈 (((𝐹𝑧)(+g𝑇)(𝐹𝑥))(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈)))
5440, 12, 53syl2anc 584 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (∀𝑦 ∈ (𝐹𝑈)(((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈) ↔ ∀𝑥𝑈 (((𝐹𝑧)(+g𝑇)(𝐹𝑥))(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈)))
5554ralbidv 3166 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (∀𝑧 ∈ (Base‘𝑆)∀𝑦 ∈ (𝐹𝑈)(((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈) ↔ ∀𝑧 ∈ (Base‘𝑆)∀𝑥𝑈 (((𝐹𝑧)(+g𝑇)(𝐹𝑥))(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈)))
5647, 49, 553bitr3d 310 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (∀𝑥𝑌𝑦 ∈ (𝐹𝑈)((𝑥(+g𝑇)𝑦)(-g𝑇)𝑥) ∈ (𝐹𝑈) ↔ ∀𝑧 ∈ (Base‘𝑆)∀𝑥𝑈 (((𝐹𝑧)(+g𝑇)(𝐹𝑥))(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈)))
5739, 56mpbird 258 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → ∀𝑥𝑌𝑦 ∈ (𝐹𝑈)((𝑥(+g𝑇)𝑦)(-g𝑇)𝑥) ∈ (𝐹𝑈))
5828, 23, 20isnsg3 18071 . 2 ((𝐹𝑈) ∈ (NrmSGrp‘𝑇) ↔ ((𝐹𝑈) ∈ (SubGrp‘𝑇) ∧ ∀𝑥𝑌𝑦 ∈ (𝐹𝑈)((𝑥(+g𝑇)𝑦)(-g𝑇)𝑥) ∈ (𝐹𝑈)))
595, 57, 58sylanbrc 583 1 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (𝐹𝑈) ∈ (NrmSGrp‘𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1080   = wceq 1525  wcel 2083  wral 3107  wss 3865  ran crn 5451  cima 5453   Fn wfn 6227  wf 6228  cfv 6232  (class class class)co 7023  Basecbs 16316  +gcplusg 16398  Grpcgrp 17865  -gcsg 17867  SubGrpcsubg 18031  NrmSGrpcnsg 18032   GrpHom cghm 18100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-er 8146  df-en 8365  df-dom 8366  df-sdom 8367  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-nn 11493  df-2 11554  df-ndx 16319  df-slot 16320  df-base 16322  df-sets 16323  df-ress 16324  df-plusg 16411  df-0g 16548  df-mgm 17685  df-sgrp 17727  df-mnd 17738  df-grp 17868  df-minusg 17869  df-sbg 17870  df-subg 18034  df-nsg 18035  df-ghm 18101
This theorem is referenced by: (None)
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