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Theorem ghmnsgima 18985
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 18913 . . . 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 18982 . . 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 18963 . . . . . . . . 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 2736 . . . . . . . . . . . 12 (Base‘𝑆) = (Base‘𝑆)
1110subgss 18882 . . . . . . . . . . 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 3943 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝑥 ∈ (Base‘𝑆))
16 eqid 2736 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
1710, 16grpcl 18710 . . . . . . . 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 2736 . . . . . . . 8 (-g𝑆) = (-g𝑆)
20 eqid 2736 . . . . . . . 8 (-g𝑇) = (-g𝑇)
2110, 19, 20ghmsub 18969 . . . . . . 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 2736 . . . . . . . . 9 (+g𝑇) = (+g𝑇)
2410, 16, 23ghmlin 18966 . . . . . . . 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 7366 . . . . . 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 2776 . . . . 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 18965 . . . . . . . . 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 6666 . . . . . 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 18914 . . . . . . 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 7179 . . . . . 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 2839 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → (((𝐹𝑧)(+g𝑇)(𝐹𝑥))(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈))
3938ralrimivva 3195 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → ∀𝑧 ∈ (Base‘𝑆)∀𝑥𝑈 (((𝐹𝑧)(+g𝑇)(𝐹𝑥))(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈))
4030ffnd 6666 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝐹 Fn (Base‘𝑆))
41 oveq1 7358 . . . . . . . . 9 (𝑥 = (𝐹𝑧) → (𝑥(+g𝑇)𝑦) = ((𝐹𝑧)(+g𝑇)𝑦))
42 id 22 . . . . . . . . 9 (𝑥 = (𝐹𝑧) → 𝑥 = (𝐹𝑧))
4341, 42oveq12d 7369 . . . . . . . 8 (𝑥 = (𝐹𝑧) → ((𝑥(+g𝑇)𝑦)(-g𝑇)𝑥) = (((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)))
4443eleq1d 2822 . . . . . . 7 (𝑥 = (𝐹𝑧) → (((𝑥(+g𝑇)𝑦)(-g𝑇)𝑥) ∈ (𝐹𝑈) ↔ (((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈)))
4544ralbidv 3172 . . . . . 6 (𝑥 = (𝐹𝑧) → (∀𝑦 ∈ (𝐹𝑈)((𝑥(+g𝑇)𝑦)(-g𝑇)𝑥) ∈ (𝐹𝑈) ↔ ∀𝑦 ∈ (𝐹𝑈)(((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈)))
4645ralrn 7034 . . . . 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 3311 . . . 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 7359 . . . . . . . . 9 (𝑦 = (𝐹𝑥) → ((𝐹𝑧)(+g𝑇)𝑦) = ((𝐹𝑧)(+g𝑇)(𝐹𝑥)))
5150oveq1d 7366 . . . . . . . 8 (𝑦 = (𝐹𝑥) → (((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)) = (((𝐹𝑧)(+g𝑇)(𝐹𝑥))(-g𝑇)(𝐹𝑧)))
5251eleq1d 2822 . . . . . . 7 (𝑦 = (𝐹𝑥) → ((((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈) ↔ (((𝐹𝑧)(+g𝑇)(𝐹𝑥))(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈)))
5352ralima 7184 . . . . . 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 3172 . . . 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 18915 . 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 3062  wss 3908  ran crn 5632  cima 5634   Fn wfn 6488  wf 6489  cfv 6493  (class class class)co 7351  Basecbs 17037  +gcplusg 17087  Grpcgrp 18702  -gcsg 18704  SubGrpcsubg 18875  NrmSGrpcnsg 18876   GrpHom cghm 18958
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 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7307  df-ov 7354  df-oprab 7355  df-mpo 7356  df-om 7795  df-1st 7913  df-2nd 7914  df-frecs 8204  df-wrecs 8235  df-recs 8309  df-rdg 8348  df-er 8606  df-en 8842  df-dom 8843  df-sdom 8844  df-pnf 11149  df-mnf 11150  df-xr 11151  df-ltxr 11152  df-le 11153  df-sub 11345  df-neg 11346  df-nn 12112  df-2 12174  df-sets 16990  df-slot 17008  df-ndx 17020  df-base 17038  df-ress 17067  df-plusg 17100  df-0g 17277  df-mgm 18451  df-sgrp 18500  df-mnd 18511  df-grp 18705  df-minusg 18706  df-sbg 18707  df-subg 18878  df-nsg 18879  df-ghm 18959
This theorem is referenced by: (None)
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