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Theorem ghmnsgima 18395
 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 1133 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2 nsgsubg 18323 . . . 4 (𝑈 ∈ (NrmSGrp‘𝑆) → 𝑈 ∈ (SubGrp‘𝑆))
323ad2ant2 1131 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝑈 ∈ (SubGrp‘𝑆))
4 ghmima 18392 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (SubGrp‘𝑆)) → (𝐹𝑈) ∈ (SubGrp‘𝑇))
51, 3, 4syl2anc 587 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (𝐹𝑈) ∈ (SubGrp‘𝑇))
61adantr 484 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
7 ghmgrp1 18373 . . . . . . . . 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 770 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝑧 ∈ (Base‘𝑆))
10 eqid 2798 . . . . . . . . . . . 12 (Base‘𝑆) = (Base‘𝑆)
1110subgss 18293 . . . . . . . . . . 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 484 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝑈 ⊆ (Base‘𝑆))
14 simprr 772 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝑥𝑈)
1513, 14sseldd 3918 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝑥 ∈ (Base‘𝑆))
16 eqid 2798 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
1710, 16grpcl 18123 . . . . . . . 8 ((𝑆 ∈ Grp ∧ 𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝑧(+g𝑆)𝑥) ∈ (Base‘𝑆))
188, 9, 15, 17syl3anc 1368 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → (𝑧(+g𝑆)𝑥) ∈ (Base‘𝑆))
19 eqid 2798 . . . . . . . 8 (-g𝑆) = (-g𝑆)
20 eqid 2798 . . . . . . . 8 (-g𝑇) = (-g𝑇)
2110, 19, 20ghmsub 18379 . . . . . . 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 1368 . . . . . 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 2798 . . . . . . . . 9 (+g𝑇) = (+g𝑇)
2410, 16, 23ghmlin 18376 . . . . . . . 8 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐹‘(𝑧(+g𝑆)𝑥)) = ((𝐹𝑧)(+g𝑇)(𝐹𝑥)))
256, 9, 15, 24syl3anc 1368 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → (𝐹‘(𝑧(+g𝑆)𝑥)) = ((𝐹𝑧)(+g𝑇)(𝐹𝑥)))
2625oveq1d 7160 . . . . . 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 18375 . . . . . . . . 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 484 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝐹:(Base‘𝑆)⟶𝑌)
3231ffnd 6496 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝐹 Fn (Base‘𝑆))
33 simpl2 1189 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → 𝑈 ∈ (NrmSGrp‘𝑆))
3410, 16, 19nsgconj 18324 . . . . . . 7 ((𝑈 ∈ (NrmSGrp‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈) → ((𝑧(+g𝑆)𝑥)(-g𝑆)𝑧) ∈ 𝑈)
3533, 9, 14, 34syl3anc 1368 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → ((𝑧(+g𝑆)𝑥)(-g𝑆)𝑧) ∈ 𝑈)
36 fnfvima 6983 . . . . . 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 1368 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → (𝐹‘((𝑧(+g𝑆)𝑥)(-g𝑆)𝑧)) ∈ (𝐹𝑈))
3827, 37eqeltrrd 2891 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥𝑈)) → (((𝐹𝑧)(+g𝑇)(𝐹𝑥))(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈))
3938ralrimivva 3156 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → ∀𝑧 ∈ (Base‘𝑆)∀𝑥𝑈 (((𝐹𝑧)(+g𝑇)(𝐹𝑥))(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈))
4030ffnd 6496 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝐹 Fn (Base‘𝑆))
41 oveq1 7152 . . . . . . . . 9 (𝑥 = (𝐹𝑧) → (𝑥(+g𝑇)𝑦) = ((𝐹𝑧)(+g𝑇)𝑦))
42 id 22 . . . . . . . . 9 (𝑥 = (𝐹𝑧) → 𝑥 = (𝐹𝑧))
4341, 42oveq12d 7163 . . . . . . . 8 (𝑥 = (𝐹𝑧) → ((𝑥(+g𝑇)𝑦)(-g𝑇)𝑥) = (((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)))
4443eleq1d 2874 . . . . . . 7 (𝑥 = (𝐹𝑧) → (((𝑥(+g𝑇)𝑦)(-g𝑇)𝑥) ∈ (𝐹𝑈) ↔ (((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈)))
4544ralbidv 3162 . . . . . 6 (𝑥 = (𝐹𝑧) → (∀𝑦 ∈ (𝐹𝑈)((𝑥(+g𝑇)𝑦)(-g𝑇)𝑥) ∈ (𝐹𝑈) ↔ ∀𝑦 ∈ (𝐹𝑈)(((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈)))
4645ralrn 6841 . . . . 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 1135 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → ran 𝐹 = 𝑌)
4948raleqdv 3365 . . . 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 7153 . . . . . . . . 9 (𝑦 = (𝐹𝑥) → ((𝐹𝑧)(+g𝑇)𝑦) = ((𝐹𝑧)(+g𝑇)(𝐹𝑥)))
5150oveq1d 7160 . . . . . . . 8 (𝑦 = (𝐹𝑥) → (((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)) = (((𝐹𝑧)(+g𝑇)(𝐹𝑥))(-g𝑇)(𝐹𝑧)))
5251eleq1d 2874 . . . . . . 7 (𝑦 = (𝐹𝑥) → ((((𝐹𝑧)(+g𝑇)𝑦)(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈) ↔ (((𝐹𝑧)(+g𝑇)(𝐹𝑥))(-g𝑇)(𝐹𝑧)) ∈ (𝐹𝑈)))
5352ralima 6988 . . . . . 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 587 . . . . 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 3162 . . . 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 312 . . 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 260 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → ∀𝑥𝑌𝑦 ∈ (𝐹𝑈)((𝑥(+g𝑇)𝑦)(-g𝑇)𝑥) ∈ (𝐹𝑈))
5828, 23, 20isnsg3 18325 . 2 ((𝐹𝑈) ∈ (NrmSGrp‘𝑇) ↔ ((𝐹𝑈) ∈ (SubGrp‘𝑇) ∧ ∀𝑥𝑌𝑦 ∈ (𝐹𝑈)((𝑥(+g𝑇)𝑦)(-g𝑇)𝑥) ∈ (𝐹𝑈)))
595, 57, 58sylanbrc 586 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ⊆ wss 3883  ran crn 5524   “ cima 5526   Fn wfn 6327  ⟶wf 6328  ‘cfv 6332  (class class class)co 7145  Basecbs 16495  +gcplusg 16577  Grpcgrp 18115  -gcsg 18117  SubGrpcsubg 18286  NrmSGrpcnsg 18287   GrpHom cghm 18368 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-cnex 10600  ax-resscn 10601  ax-1cn 10602  ax-icn 10603  ax-addcl 10604  ax-addrcl 10605  ax-mulcl 10606  ax-mulrcl 10607  ax-mulcom 10608  ax-addass 10609  ax-mulass 10610  ax-distr 10611  ax-i2m1 10612  ax-1ne0 10613  ax-1rid 10614  ax-rnegex 10615  ax-rrecex 10616  ax-cnre 10617  ax-pre-lttri 10618  ax-pre-lttrn 10619  ax-pre-ltadd 10620  ax-pre-mulgt0 10621 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7574  df-1st 7684  df-2nd 7685  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-er 8290  df-en 8511  df-dom 8512  df-sdom 8513  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688  df-sub 10879  df-neg 10880  df-nn 11644  df-2 11706  df-ndx 16498  df-slot 16499  df-base 16501  df-sets 16502  df-ress 16503  df-plusg 16590  df-0g 16727  df-mgm 17864  df-sgrp 17913  df-mnd 17924  df-grp 18118  df-minusg 18119  df-sbg 18120  df-subg 18289  df-nsg 18290  df-ghm 18369 This theorem is referenced by: (None)
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