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Theorem gasubg 17927
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 17918 . . 3 ( ∈ (𝐺 GrpAct 𝑌) → 𝑌 ∈ V)
2 gasubg.1 . . . 4 𝐻 = (𝐺s 𝑆)
32subggrp 17790 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
41, 3anim12ci 592 . 2 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐻 ∈ Grp ∧ 𝑌 ∈ V))
5 eqid 2752 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
65gaf 17920 . . . . . 6 ( ∈ (𝐺 GrpAct 𝑌) → :((Base‘𝐺) × 𝑌)⟶𝑌)
76adantr 472 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → :((Base‘𝐺) × 𝑌)⟶𝑌)
8 simpr 479 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ∈ (SubGrp‘𝐺))
95subgss 17788 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
108, 9syl 17 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
11 xpss1 5276 . . . . . 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 6224 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ( ↾ (𝑆 × 𝑌)):(𝑆 × 𝑌)⟶𝑌)
142subgbas 17791 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
158, 14syl 17 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 = (Base‘𝐻))
1615xpeq1d 5287 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑌) = ((Base‘𝐻) × 𝑌))
1716feq2d 6184 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (( ↾ (𝑆 × 𝑌)):(𝑆 × 𝑌)⟶𝑌 ↔ ( ↾ (𝑆 × 𝑌)):((Base‘𝐻) × 𝑌)⟶𝑌))
1813, 17mpbid 222 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ( ↾ (𝑆 × 𝑌)):((Base‘𝐻) × 𝑌)⟶𝑌)
198adantr 472 . . . . . . . 8 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → 𝑆 ∈ (SubGrp‘𝐺))
20 eqid 2752 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
2120subg0cl 17795 . . . . . . . 8 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑆)
2219, 21syl 17 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → (0g𝐺) ∈ 𝑆)
23 simpr 479 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → 𝑥𝑌)
24 ovres 6957 . . . . . . 7 (((0g𝐺) ∈ 𝑆𝑥𝑌) → ((0g𝐺)( ↾ (𝑆 × 𝑌))𝑥) = ((0g𝐺) 𝑥))
2522, 23, 24syl2anc 696 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → ((0g𝐺)( ↾ (𝑆 × 𝑌))𝑥) = ((0g𝐺) 𝑥))
262, 20subg0 17793 . . . . . . . 8 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) = (0g𝐻))
2719, 26syl 17 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → (0g𝐺) = (0g𝐻))
2827oveq1d 6820 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → ((0g𝐺)( ↾ (𝑆 × 𝑌))𝑥) = ((0g𝐻)( ↾ (𝑆 × 𝑌))𝑥))
2920gagrpid 17919 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥𝑌) → ((0g𝐺) 𝑥) = 𝑥)
3029adantlr 753 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → ((0g𝐺) 𝑥) = 𝑥)
3125, 28, 303eqtr3d 2794 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → ((0g𝐻)( ↾ (𝑆 × 𝑌))𝑥) = 𝑥)
32 eqimss2 3791 . . . . . . . . . . 11 (𝑆 = (Base‘𝐻) → (Base‘𝐻) ⊆ 𝑆)
3315, 32syl 17 . . . . . . . . . 10 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (Base‘𝐻) ⊆ 𝑆)
3433adantr 472 . . . . . . . . 9 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → (Base‘𝐻) ⊆ 𝑆)
3534sselda 3736 . . . . . . . 8 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ 𝑦 ∈ (Base‘𝐻)) → 𝑦𝑆)
3634sselda 3736 . . . . . . . 8 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ 𝑧 ∈ (Base‘𝐻)) → 𝑧𝑆)
3735, 36anim12dan 918 . . . . . . 7 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦 ∈ (Base‘𝐻) ∧ 𝑧 ∈ (Base‘𝐻))) → (𝑦𝑆𝑧𝑆))
38 simprl 811 . . . . . . . . . 10 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → 𝑦𝑆)
397ad2antrr 764 . . . . . . . . . . 11 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → :((Base‘𝐺) × 𝑌)⟶𝑌)
409ad3antlr 769 . . . . . . . . . . . 12 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → 𝑆 ⊆ (Base‘𝐺))
41 simprr 813 . . . . . . . . . . . 12 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → 𝑧𝑆)
4240, 41sseldd 3737 . . . . . . . . . . 11 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → 𝑧 ∈ (Base‘𝐺))
4323adantr 472 . . . . . . . . . . 11 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → 𝑥𝑌)
4439, 42, 43fovrnd 6963 . . . . . . . . . 10 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (𝑧 𝑥) ∈ 𝑌)
45 ovres 6957 . . . . . . . . . 10 ((𝑦𝑆 ∧ (𝑧 𝑥) ∈ 𝑌) → (𝑦( ↾ (𝑆 × 𝑌))(𝑧 𝑥)) = (𝑦 (𝑧 𝑥)))
4638, 44, 45syl2anc 696 . . . . . . . . 9 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (𝑦( ↾ (𝑆 × 𝑌))(𝑧 𝑥)) = (𝑦 (𝑧 𝑥)))
47 ovres 6957 . . . . . . . . . . 11 ((𝑧𝑆𝑥𝑌) → (𝑧( ↾ (𝑆 × 𝑌))𝑥) = (𝑧 𝑥))
4841, 43, 47syl2anc 696 . . . . . . . . . 10 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (𝑧( ↾ (𝑆 × 𝑌))𝑥) = (𝑧 𝑥))
4948oveq2d 6821 . . . . . . . . 9 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (𝑦( ↾ (𝑆 × 𝑌))(𝑧( ↾ (𝑆 × 𝑌))𝑥)) = (𝑦( ↾ (𝑆 × 𝑌))(𝑧 𝑥)))
50 simplll 815 . . . . . . . . . 10 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → ∈ (𝐺 GrpAct 𝑌))
5140, 38sseldd 3737 . . . . . . . . . 10 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → 𝑦 ∈ (Base‘𝐺))
52 eqid 2752 . . . . . . . . . . 11 (+g𝐺) = (+g𝐺)
535, 52gaass 17922 . . . . . . . . . 10 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺) ∧ 𝑥𝑌)) → ((𝑦(+g𝐺)𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))
5450, 51, 42, 43, 53syl13anc 1475 . . . . . . . . 9 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → ((𝑦(+g𝐺)𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))
5546, 49, 543eqtr4d 2796 . . . . . . . 8 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (𝑦( ↾ (𝑆 × 𝑌))(𝑧( ↾ (𝑆 × 𝑌))𝑥)) = ((𝑦(+g𝐺)𝑧) 𝑥))
5652subgcl 17797 . . . . . . . . . . 11 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑦𝑆𝑧𝑆) → (𝑦(+g𝐺)𝑧) ∈ 𝑆)
57563expb 1113 . . . . . . . . . 10 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑦𝑆𝑧𝑆)) → (𝑦(+g𝐺)𝑧) ∈ 𝑆)
5819, 57sylan 489 . . . . . . . . 9 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (𝑦(+g𝐺)𝑧) ∈ 𝑆)
59 ovres 6957 . . . . . . . . 9 (((𝑦(+g𝐺)𝑧) ∈ 𝑆𝑥𝑌) → ((𝑦(+g𝐺)𝑧)( ↾ (𝑆 × 𝑌))𝑥) = ((𝑦(+g𝐺)𝑧) 𝑥))
6058, 43, 59syl2anc 696 . . . . . . . 8 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → ((𝑦(+g𝐺)𝑧)( ↾ (𝑆 × 𝑌))𝑥) = ((𝑦(+g𝐺)𝑧) 𝑥))
612, 52ressplusg 16187 . . . . . . . . . . 11 (𝑆 ∈ (SubGrp‘𝐺) → (+g𝐺) = (+g𝐻))
6261ad3antlr 769 . . . . . . . . . 10 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (+g𝐺) = (+g𝐻))
6362oveqd 6822 . . . . . . . . 9 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (𝑦(+g𝐺)𝑧) = (𝑦(+g𝐻)𝑧))
6463oveq1d 6820 . . . . . . . 8 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → ((𝑦(+g𝐺)𝑧)( ↾ (𝑆 × 𝑌))𝑥) = ((𝑦(+g𝐻)𝑧)( ↾ (𝑆 × 𝑌))𝑥))
6555, 60, 643eqtr2rd 2793 . . . . . . 7 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → ((𝑦(+g𝐻)𝑧)( ↾ (𝑆 × 𝑌))𝑥) = (𝑦( ↾ (𝑆 × 𝑌))(𝑧( ↾ (𝑆 × 𝑌))𝑥)))
6637, 65syldan 488 . . . . . 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 3101 . . . . 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 555 . . . 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 3096 . . 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 555 . 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 2752 . . 3 (Base‘𝐻) = (Base‘𝐻)
72 eqid 2752 . . 3 (+g𝐻) = (+g𝐻)
73 eqid 2752 . . 3 (0g𝐻) = (0g𝐻)
7471, 72, 73isga 17916 . 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 701 1 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ( ↾ (𝑆 × 𝑌)) ∈ (𝐻 GrpAct 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1624  wcel 2131  wral 3042  Vcvv 3332  wss 3707   × cxp 5256  cres 5260  wf 6037  cfv 6041  (class class class)co 6805  Basecbs 16051  s cress 16052  +gcplusg 16135  0gc0g 16294  Grpcgrp 17615  SubGrpcsubg 17781   GrpAct cga 17914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-er 7903  df-map 8017  df-en 8114  df-dom 8115  df-sdom 8116  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-2 11263  df-ndx 16054  df-slot 16055  df-base 16057  df-sets 16058  df-ress 16059  df-plusg 16148  df-0g 16296  df-mgm 17435  df-sgrp 17477  df-mnd 17488  df-grp 17618  df-subg 17784  df-ga 17915
This theorem is referenced by:  sylow3lem5  18238
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