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Theorem gasubg 19320
Description: The restriction of a group action to a subgroup is a group action. (Contributed by Mario Carneiro, 17-Jan-2015.)
Hypothesis
Ref Expression
gasubg.1 𝐻 = (𝐺s 𝑆)
Assertion
Ref Expression
gasubg (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ( ↾ (𝑆 × 𝑌)) ∈ (𝐻 GrpAct 𝑌))

Proof of Theorem gasubg
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gaset 19311 . . 3 ( ∈ (𝐺 GrpAct 𝑌) → 𝑌 ∈ V)
2 gasubg.1 . . . 4 𝐻 = (𝐺s 𝑆)
32subggrp 19147 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
41, 3anim12ci 614 . 2 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐻 ∈ Grp ∧ 𝑌 ∈ V))
5 eqid 2737 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
65gaf 19313 . . . . . 6 ( ∈ (𝐺 GrpAct 𝑌) → :((Base‘𝐺) × 𝑌)⟶𝑌)
76adantr 480 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → :((Base‘𝐺) × 𝑌)⟶𝑌)
8 simpr 484 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ∈ (SubGrp‘𝐺))
95subgss 19145 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
108, 9syl 17 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
11 xpss1 5704 . . . . . 6 (𝑆 ⊆ (Base‘𝐺) → (𝑆 × 𝑌) ⊆ ((Base‘𝐺) × 𝑌))
1210, 11syl 17 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑌) ⊆ ((Base‘𝐺) × 𝑌))
137, 12fssresd 6775 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ( ↾ (𝑆 × 𝑌)):(𝑆 × 𝑌)⟶𝑌)
142subgbas 19148 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
158, 14syl 17 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 = (Base‘𝐻))
1615xpeq1d 5714 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑌) = ((Base‘𝐻) × 𝑌))
1716feq2d 6722 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (( ↾ (𝑆 × 𝑌)):(𝑆 × 𝑌)⟶𝑌 ↔ ( ↾ (𝑆 × 𝑌)):((Base‘𝐻) × 𝑌)⟶𝑌))
1813, 17mpbid 232 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ( ↾ (𝑆 × 𝑌)):((Base‘𝐻) × 𝑌)⟶𝑌)
198adantr 480 . . . . . . . 8 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → 𝑆 ∈ (SubGrp‘𝐺))
20 eqid 2737 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
2120subg0cl 19152 . . . . . . . 8 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑆)
2219, 21syl 17 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → (0g𝐺) ∈ 𝑆)
23 simpr 484 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → 𝑥𝑌)
24 ovres 7599 . . . . . . 7 (((0g𝐺) ∈ 𝑆𝑥𝑌) → ((0g𝐺)( ↾ (𝑆 × 𝑌))𝑥) = ((0g𝐺) 𝑥))
2522, 23, 24syl2anc 584 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → ((0g𝐺)( ↾ (𝑆 × 𝑌))𝑥) = ((0g𝐺) 𝑥))
262, 20subg0 19150 . . . . . . . 8 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) = (0g𝐻))
2719, 26syl 17 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → (0g𝐺) = (0g𝐻))
2827oveq1d 7446 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → ((0g𝐺)( ↾ (𝑆 × 𝑌))𝑥) = ((0g𝐻)( ↾ (𝑆 × 𝑌))𝑥))
2920gagrpid 19312 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥𝑌) → ((0g𝐺) 𝑥) = 𝑥)
3029adantlr 715 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → ((0g𝐺) 𝑥) = 𝑥)
3125, 28, 303eqtr3d 2785 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → ((0g𝐻)( ↾ (𝑆 × 𝑌))𝑥) = 𝑥)
32 eqimss2 4043 . . . . . . . . . . 11 (𝑆 = (Base‘𝐻) → (Base‘𝐻) ⊆ 𝑆)
3315, 32syl 17 . . . . . . . . . 10 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (Base‘𝐻) ⊆ 𝑆)
3433adantr 480 . . . . . . . . 9 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → (Base‘𝐻) ⊆ 𝑆)
3534sselda 3983 . . . . . . . 8 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ 𝑦 ∈ (Base‘𝐻)) → 𝑦𝑆)
3634sselda 3983 . . . . . . . 8 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ 𝑧 ∈ (Base‘𝐻)) → 𝑧𝑆)
3735, 36anim12dan 619 . . . . . . 7 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦 ∈ (Base‘𝐻) ∧ 𝑧 ∈ (Base‘𝐻))) → (𝑦𝑆𝑧𝑆))
38 simprl 771 . . . . . . . . . 10 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → 𝑦𝑆)
397ad2antrr 726 . . . . . . . . . . 11 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → :((Base‘𝐺) × 𝑌)⟶𝑌)
409ad3antlr 731 . . . . . . . . . . . 12 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → 𝑆 ⊆ (Base‘𝐺))
41 simprr 773 . . . . . . . . . . . 12 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → 𝑧𝑆)
4240, 41sseldd 3984 . . . . . . . . . . 11 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → 𝑧 ∈ (Base‘𝐺))
4323adantr 480 . . . . . . . . . . 11 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → 𝑥𝑌)
4439, 42, 43fovcdmd 7605 . . . . . . . . . 10 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (𝑧 𝑥) ∈ 𝑌)
45 ovres 7599 . . . . . . . . . 10 ((𝑦𝑆 ∧ (𝑧 𝑥) ∈ 𝑌) → (𝑦( ↾ (𝑆 × 𝑌))(𝑧 𝑥)) = (𝑦 (𝑧 𝑥)))
4638, 44, 45syl2anc 584 . . . . . . . . 9 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (𝑦( ↾ (𝑆 × 𝑌))(𝑧 𝑥)) = (𝑦 (𝑧 𝑥)))
47 ovres 7599 . . . . . . . . . . 11 ((𝑧𝑆𝑥𝑌) → (𝑧( ↾ (𝑆 × 𝑌))𝑥) = (𝑧 𝑥))
4841, 43, 47syl2anc 584 . . . . . . . . . 10 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (𝑧( ↾ (𝑆 × 𝑌))𝑥) = (𝑧 𝑥))
4948oveq2d 7447 . . . . . . . . 9 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (𝑦( ↾ (𝑆 × 𝑌))(𝑧( ↾ (𝑆 × 𝑌))𝑥)) = (𝑦( ↾ (𝑆 × 𝑌))(𝑧 𝑥)))
50 simplll 775 . . . . . . . . . 10 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → ∈ (𝐺 GrpAct 𝑌))
5140, 38sseldd 3984 . . . . . . . . . 10 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → 𝑦 ∈ (Base‘𝐺))
52 eqid 2737 . . . . . . . . . . 11 (+g𝐺) = (+g𝐺)
535, 52gaass 19315 . . . . . . . . . 10 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺) ∧ 𝑥𝑌)) → ((𝑦(+g𝐺)𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))
5450, 51, 42, 43, 53syl13anc 1374 . . . . . . . . 9 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → ((𝑦(+g𝐺)𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))
5546, 49, 543eqtr4d 2787 . . . . . . . 8 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (𝑦( ↾ (𝑆 × 𝑌))(𝑧( ↾ (𝑆 × 𝑌))𝑥)) = ((𝑦(+g𝐺)𝑧) 𝑥))
5652subgcl 19154 . . . . . . . . . . 11 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑦𝑆𝑧𝑆) → (𝑦(+g𝐺)𝑧) ∈ 𝑆)
57563expb 1121 . . . . . . . . . 10 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑦𝑆𝑧𝑆)) → (𝑦(+g𝐺)𝑧) ∈ 𝑆)
5819, 57sylan 580 . . . . . . . . 9 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (𝑦(+g𝐺)𝑧) ∈ 𝑆)
59 ovres 7599 . . . . . . . . 9 (((𝑦(+g𝐺)𝑧) ∈ 𝑆𝑥𝑌) → ((𝑦(+g𝐺)𝑧)( ↾ (𝑆 × 𝑌))𝑥) = ((𝑦(+g𝐺)𝑧) 𝑥))
6058, 43, 59syl2anc 584 . . . . . . . 8 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → ((𝑦(+g𝐺)𝑧)( ↾ (𝑆 × 𝑌))𝑥) = ((𝑦(+g𝐺)𝑧) 𝑥))
612, 52ressplusg 17334 . . . . . . . . . . 11 (𝑆 ∈ (SubGrp‘𝐺) → (+g𝐺) = (+g𝐻))
6261ad3antlr 731 . . . . . . . . . 10 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (+g𝐺) = (+g𝐻))
6362oveqd 7448 . . . . . . . . 9 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (𝑦(+g𝐺)𝑧) = (𝑦(+g𝐻)𝑧))
6463oveq1d 7446 . . . . . . . 8 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → ((𝑦(+g𝐺)𝑧)( ↾ (𝑆 × 𝑌))𝑥) = ((𝑦(+g𝐻)𝑧)( ↾ (𝑆 × 𝑌))𝑥))
6555, 60, 643eqtr2rd 2784 . . . . . . 7 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → ((𝑦(+g𝐻)𝑧)( ↾ (𝑆 × 𝑌))𝑥) = (𝑦( ↾ (𝑆 × 𝑌))(𝑧( ↾ (𝑆 × 𝑌))𝑥)))
6637, 65syldan 591 . . . . . 6 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦 ∈ (Base‘𝐻) ∧ 𝑧 ∈ (Base‘𝐻))) → ((𝑦(+g𝐻)𝑧)( ↾ (𝑆 × 𝑌))𝑥) = (𝑦( ↾ (𝑆 × 𝑌))(𝑧( ↾ (𝑆 × 𝑌))𝑥)))
6766ralrimivva 3202 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → ∀𝑦 ∈ (Base‘𝐻)∀𝑧 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑧)( ↾ (𝑆 × 𝑌))𝑥) = (𝑦( ↾ (𝑆 × 𝑌))(𝑧( ↾ (𝑆 × 𝑌))𝑥)))
6831, 67jca 511 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → (((0g𝐻)( ↾ (𝑆 × 𝑌))𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐻)∀𝑧 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑧)( ↾ (𝑆 × 𝑌))𝑥) = (𝑦( ↾ (𝑆 × 𝑌))(𝑧( ↾ (𝑆 × 𝑌))𝑥))))
6968ralrimiva 3146 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ∀𝑥𝑌 (((0g𝐻)( ↾ (𝑆 × 𝑌))𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐻)∀𝑧 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑧)( ↾ (𝑆 × 𝑌))𝑥) = (𝑦( ↾ (𝑆 × 𝑌))(𝑧( ↾ (𝑆 × 𝑌))𝑥))))
7018, 69jca 511 . 2 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (( ↾ (𝑆 × 𝑌)):((Base‘𝐻) × 𝑌)⟶𝑌 ∧ ∀𝑥𝑌 (((0g𝐻)( ↾ (𝑆 × 𝑌))𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐻)∀𝑧 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑧)( ↾ (𝑆 × 𝑌))𝑥) = (𝑦( ↾ (𝑆 × 𝑌))(𝑧( ↾ (𝑆 × 𝑌))𝑥)))))
71 eqid 2737 . . 3 (Base‘𝐻) = (Base‘𝐻)
72 eqid 2737 . . 3 (+g𝐻) = (+g𝐻)
73 eqid 2737 . . 3 (0g𝐻) = (0g𝐻)
7471, 72, 73isga 19309 . 2 (( ↾ (𝑆 × 𝑌)) ∈ (𝐻 GrpAct 𝑌) ↔ ((𝐻 ∈ Grp ∧ 𝑌 ∈ V) ∧ (( ↾ (𝑆 × 𝑌)):((Base‘𝐻) × 𝑌)⟶𝑌 ∧ ∀𝑥𝑌 (((0g𝐻)( ↾ (𝑆 × 𝑌))𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐻)∀𝑧 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑧)( ↾ (𝑆 × 𝑌))𝑥) = (𝑦( ↾ (𝑆 × 𝑌))(𝑧( ↾ (𝑆 × 𝑌))𝑥))))))
754, 70, 74sylanbrc 583 1 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ( ↾ (𝑆 × 𝑌)) ∈ (𝐻 GrpAct 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  wss 3951   × cxp 5683  cres 5687  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  s cress 17274  +gcplusg 17297  0gc0g 17484  Grpcgrp 18951  SubGrpcsubg 19138   GrpAct cga 19307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-subg 19141  df-ga 19308
This theorem is referenced by:  sylow3lem5  19649
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