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Theorem ghmnsgima 19110
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 1136 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2 nsgsubg 19032 . . . 4 (𝑈 ∈ (NrmSGrp‘𝑆) → 𝑈 ∈ (SubGrp‘𝑆))
323ad2ant2 1134 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝑈 ∈ (SubGrp‘𝑆))
4 ghmima 19107 . . 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 19088 . . . . . . . . 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 769 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝑧 ∈ (Base‘𝑆))
10 eqid 2732 . . . . . . . . . . . 12 (Base‘𝑆) = (Base‘𝑆)
1110subgss 19001 . . . . . . . . . . 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 771 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝑥𝑈)
1513, 14sseldd 3982 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝑥 ∈ (Base‘𝑆))
16 eqid 2732 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
1710, 16grpcl 18823 . . . . . . . 8 ((𝑆 ∈ Grp ∧ 𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝑧(+g𝑆)𝑥) ∈ (Base‘𝑆))
188, 9, 15, 17syl3anc 1371 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → (𝑧(+g𝑆)𝑥) ∈ (Base‘𝑆))
19 eqid 2732 . . . . . . . 8 (-g𝑆) = (-g𝑆)
20 eqid 2732 . . . . . . . 8 (-g𝑇) = (-g𝑇)
2110, 19, 20ghmsub 19094 . . . . . . 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 1371 . . . . . 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 2732 . . . . . . . . 9 (+g𝑇) = (+g𝑇)
2410, 16, 23ghmlin 19091 . . . . . . . 8 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐹‘(𝑧(+g𝑆)𝑥)) = ((𝐹𝑧)(+g𝑇)(𝐹𝑥)))
256, 9, 15, 24syl3anc 1371 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → (𝐹‘(𝑧(+g𝑆)𝑥)) = ((𝐹𝑧)(+g𝑇)(𝐹𝑥)))
2625oveq1d 7420 . . . . . 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 2772 . . . . 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 19090 . . . . . . . . 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 6715 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝐹 Fn (Base‘𝑆))
33 simpl2 1192 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝑈 ∈ (NrmSGrp‘𝑆))
3410, 16, 19nsgconj 19033 . . . . . . 7 ((𝑈 ∈ (NrmSGrp‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈) → ((𝑧(+g𝑆)𝑥)(-g𝑆)𝑧) ∈ 𝑈)
3533, 9, 14, 34syl3anc 1371 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → ((𝑧(+g𝑆)𝑥)(-g𝑆)𝑧) ∈ 𝑈)
36 fnfvima 7231 . . . . . 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 1371 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → (𝐹‘((𝑧(+g𝑆)𝑥)(-g𝑆)𝑧)) ∈ (𝐹𝑈))
3827, 37eqeltrrd 2834 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → (((𝐹𝑧)(+g𝑇)(𝐹𝑥))(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈))
3938ralrimivva 3200 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → ∀𝑧 ∈ (Base‘𝑆)∀𝑥𝑈 (((𝐹𝑧)(+g𝑇)(𝐹𝑥))(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈))
4030ffnd 6715 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝐹 Fn (Base‘𝑆))
41 oveq1 7412 . . . . . . . . 9 (𝑥 = (𝐹𝑧) → (𝑥(+g𝑇)𝑦) = ((𝐹𝑧)(+g𝑇)𝑦))
42 id 22 . . . . . . . . 9 (𝑥 = (𝐹𝑧) → 𝑥 = (𝐹𝑧))
4341, 42oveq12d 7423 . . . . . . . 8 (𝑥 = (𝐹𝑧) → ((𝑥(+g𝑇)𝑦)(-g𝑇)𝑥) = (((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)))
4443eleq1d 2818 . . . . . . 7 (𝑥 = (𝐹𝑧) → (((𝑥(+g𝑇)𝑦)(-g𝑇)𝑥) ∈ (𝐹𝑈) ↔ (((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈)))
4544ralbidv 3177 . . . . . 6 (𝑥 = (𝐹𝑧) → (∀𝑦 ∈ (𝐹𝑈)((𝑥(+g𝑇)𝑦)(-g𝑇)𝑥) ∈ (𝐹𝑈) ↔ ∀𝑦 ∈ (𝐹𝑈)(((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈)))
4645ralrn 7086 . . . . 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 1138 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → ran 𝐹 = 𝑌)
4948raleqdv 3325 . . . 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 7413 . . . . . . . . 9 (𝑦 = (𝐹𝑥) → ((𝐹𝑧)(+g𝑇)𝑦) = ((𝐹𝑧)(+g𝑇)(𝐹𝑥)))
5150oveq1d 7420 . . . . . . . 8 (𝑦 = (𝐹𝑥) → (((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)) = (((𝐹𝑧)(+g𝑇)(𝐹𝑥))(-g𝑇)(𝐹𝑧)))
5251eleq1d 2818 . . . . . . 7 (𝑦 = (𝐹𝑥) → ((((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈) ↔ (((𝐹𝑧)(+g𝑇)(𝐹𝑥))(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈)))
5352ralima 7236 . . . . . 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 3177 . . . 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 308 . . 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 256 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → ∀𝑥𝑌𝑦 ∈ (𝐹𝑈)((𝑥(+g𝑇)𝑦)(-g𝑇)𝑥) ∈ (𝐹𝑈))
5828, 23, 20isnsg3 19034 . 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3061  wss 3947  ran crn 5676  cima 5678   Fn wfn 6535  wf 6536  cfv 6540  (class class class)co 7405  Basecbs 17140  +gcplusg 17193  Grpcgrp 18815  -gcsg 18817  SubGrpcsubg 18994  NrmSGrpcnsg 18995   GrpHom cghm 19083
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 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  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-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-minusg 18819  df-sbg 18820  df-subg 18997  df-nsg 18998  df-ghm 19084
This theorem is referenced by: (None)
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