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Theorem ghmnsgima 19271
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 1135 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2 nsgsubg 19189 . . . 4 (𝑈 ∈ (NrmSGrp‘𝑆) → 𝑈 ∈ (SubGrp‘𝑆))
323ad2ant2 1133 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝑈 ∈ (SubGrp‘𝑆))
4 ghmima 19268 . . 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 480 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
7 ghmgrp1 19249 . . . . . . . . 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 771 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝑧 ∈ (Base‘𝑆))
10 eqid 2735 . . . . . . . . . . . 12 (Base‘𝑆) = (Base‘𝑆)
1110subgss 19158 . . . . . . . . . . 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 480 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝑈 ⊆ (Base‘𝑆))
14 simprr 773 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝑥𝑈)
1513, 14sseldd 3996 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝑥 ∈ (Base‘𝑆))
16 eqid 2735 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
1710, 16grpcl 18972 . . . . . . . 8 ((𝑆 ∈ Grp ∧ 𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝑧(+g𝑆)𝑥) ∈ (Base‘𝑆))
188, 9, 15, 17syl3anc 1370 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → (𝑧(+g𝑆)𝑥) ∈ (Base‘𝑆))
19 eqid 2735 . . . . . . . 8 (-g𝑆) = (-g𝑆)
20 eqid 2735 . . . . . . . 8 (-g𝑇) = (-g𝑇)
2110, 19, 20ghmsub 19255 . . . . . . 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 1370 . . . . . 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 2735 . . . . . . . . 9 (+g𝑇) = (+g𝑇)
2410, 16, 23ghmlin 19252 . . . . . . . 8 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐹‘(𝑧(+g𝑆)𝑥)) = ((𝐹𝑧)(+g𝑇)(𝐹𝑥)))
256, 9, 15, 24syl3anc 1370 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → (𝐹‘(𝑧(+g𝑆)𝑥)) = ((𝐹𝑧)(+g𝑇)(𝐹𝑥)))
2625oveq1d 7446 . . . . . 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 2775 . . . . 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 19251 . . . . . . . . 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 480 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝐹:(Base‘𝑆)⟶𝑌)
3231ffnd 6738 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝐹 Fn (Base‘𝑆))
33 simpl2 1191 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝑈 ∈ (NrmSGrp‘𝑆))
3410, 16, 19nsgconj 19190 . . . . . . 7 ((𝑈 ∈ (NrmSGrp‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈) → ((𝑧(+g𝑆)𝑥)(-g𝑆)𝑧) ∈ 𝑈)
3533, 9, 14, 34syl3anc 1370 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → ((𝑧(+g𝑆)𝑥)(-g𝑆)𝑧) ∈ 𝑈)
36 fnfvima 7253 . . . . . 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 1370 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → (𝐹‘((𝑧(+g𝑆)𝑥)(-g𝑆)𝑧)) ∈ (𝐹𝑈))
3827, 37eqeltrrd 2840 . . . 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 6738 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝐹 Fn (Base‘𝑆))
41 oveq1 7438 . . . . . . . . 9 (𝑥 = (𝐹𝑧) → (𝑥(+g𝑇)𝑦) = ((𝐹𝑧)(+g𝑇)𝑦))
42 id 22 . . . . . . . . 9 (𝑥 = (𝐹𝑧) → 𝑥 = (𝐹𝑧))
4341, 42oveq12d 7449 . . . . . . . 8 (𝑥 = (𝐹𝑧) → ((𝑥(+g𝑇)𝑦)(-g𝑇)𝑥) = (((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)))
4443eleq1d 2824 . . . . . . 7 (𝑥 = (𝐹𝑧) → (((𝑥(+g𝑇)𝑦)(-g𝑇)𝑥) ∈ (𝐹𝑈) ↔ (((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈)))
4544ralbidv 3176 . . . . . 6 (𝑥 = (𝐹𝑧) → (∀𝑦 ∈ (𝐹𝑈)((𝑥(+g𝑇)𝑦)(-g𝑇)𝑥) ∈ (𝐹𝑈) ↔ ∀𝑦 ∈ (𝐹𝑈)(((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈)))
4645ralrn 7108 . . . . 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 1137 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → ran 𝐹 = 𝑌)
4948raleqdv 3324 . . . 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 7439 . . . . . . . . 9 (𝑦 = (𝐹𝑥) → ((𝐹𝑧)(+g𝑇)𝑦) = ((𝐹𝑧)(+g𝑇)(𝐹𝑥)))
5150oveq1d 7446 . . . . . . . 8 (𝑦 = (𝐹𝑥) → (((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)) = (((𝐹𝑧)(+g𝑇)(𝐹𝑥))(-g𝑇)(𝐹𝑧)))
5251eleq1d 2824 . . . . . . 7 (𝑦 = (𝐹𝑥) → ((((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈) ↔ (((𝐹𝑧)(+g𝑇)(𝐹𝑥))(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈)))
5352ralima 7257 . . . . . 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 3176 . . . 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 309 . . 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 257 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → ∀𝑥𝑌𝑦 ∈ (𝐹𝑈)((𝑥(+g𝑇)𝑦)(-g𝑇)𝑥) ∈ (𝐹𝑈))
5828, 23, 20isnsg3 19191 . 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wss 3963  ran crn 5690  cima 5692   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  Grpcgrp 18964  -gcsg 18966  SubGrpcsubg 19151  NrmSGrpcnsg 19152   GrpHom cghm 19243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-nsg 19155  df-ghm 19244
This theorem is referenced by: (None)
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