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Theorem gasubg 17935
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 17926 . . 3 ( ∈ (𝐺 GrpAct 𝑌) → 𝑌 ∈ V)
2 gasubg.1 . . . 4 𝐻 = (𝐺s 𝑆)
32subggrp 17798 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
41, 3anim12ci 592 . 2 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐻 ∈ Grp ∧ 𝑌 ∈ V))
5 eqid 2760 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
65gaf 17928 . . . . . 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 17796 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
108, 9syl 17 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
11 xpss1 5284 . . . . . 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 6232 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ( ↾ (𝑆 × 𝑌)):(𝑆 × 𝑌)⟶𝑌)
142subgbas 17799 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
158, 14syl 17 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 = (Base‘𝐻))
1615xpeq1d 5295 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑌) = ((Base‘𝐻) × 𝑌))
1716feq2d 6192 . . . 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 2760 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
2120subg0cl 17803 . . . . . . . 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 6965 . . . . . . 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 17801 . . . . . . . 8 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) = (0g𝐻))
2719, 26syl 17 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → (0g𝐺) = (0g𝐻))
2827oveq1d 6828 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → ((0g𝐺)( ↾ (𝑆 × 𝑌))𝑥) = ((0g𝐻)( ↾ (𝑆 × 𝑌))𝑥))
2920gagrpid 17927 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥𝑌) → ((0g𝐺) 𝑥) = 𝑥)
3029adantlr 753 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → ((0g𝐺) 𝑥) = 𝑥)
3125, 28, 303eqtr3d 2802 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → ((0g𝐻)( ↾ (𝑆 × 𝑌))𝑥) = 𝑥)
32 eqimss2 3799 . . . . . . . . . . 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 3744 . . . . . . . 8 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ 𝑦 ∈ (Base‘𝐻)) → 𝑦𝑆)
3634sselda 3744 . . . . . . . 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 3745 . . . . . . . . . . 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 6971 . . . . . . . . . 10 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (𝑧 𝑥) ∈ 𝑌)
45 ovres 6965 . . . . . . . . . 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 6965 . . . . . . . . . . 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 6829 . . . . . . . . 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 3745 . . . . . . . . . 10 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → 𝑦 ∈ (Base‘𝐺))
52 eqid 2760 . . . . . . . . . . 11 (+g𝐺) = (+g𝐺)
535, 52gaass 17930 . . . . . . . . . 10 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺) ∧ 𝑥𝑌)) → ((𝑦(+g𝐺)𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))
5450, 51, 42, 43, 53syl13anc 1479 . . . . . . . . 9 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → ((𝑦(+g𝐺)𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))
5546, 49, 543eqtr4d 2804 . . . . . . . 8 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (𝑦( ↾ (𝑆 × 𝑌))(𝑧( ↾ (𝑆 × 𝑌))𝑥)) = ((𝑦(+g𝐺)𝑧) 𝑥))
5652subgcl 17805 . . . . . . . . . . 11 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑦𝑆𝑧𝑆) → (𝑦(+g𝐺)𝑧) ∈ 𝑆)
57563expb 1114 . . . . . . . . . 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 6965 . . . . . . . . 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 16195 . . . . . . . . . . 11 (𝑆 ∈ (SubGrp‘𝐺) → (+g𝐺) = (+g𝐻))
6261ad3antlr 769 . . . . . . . . . 10 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (+g𝐺) = (+g𝐻))
6362oveqd 6830 . . . . . . . . 9 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (𝑦(+g𝐺)𝑧) = (𝑦(+g𝐻)𝑧))
6463oveq1d 6828 . . . . . . . 8 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → ((𝑦(+g𝐺)𝑧)( ↾ (𝑆 × 𝑌))𝑥) = ((𝑦(+g𝐻)𝑧)( ↾ (𝑆 × 𝑌))𝑥))
6555, 60, 643eqtr2rd 2801 . . . . . . 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 3109 . . . . 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 3104 . . 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 2760 . . 3 (Base‘𝐻) = (Base‘𝐻)
72 eqid 2760 . . 3 (+g𝐻) = (+g𝐻)
73 eqid 2760 . . 3 (0g𝐻) = (0g𝐻)
7471, 72, 73isga 17924 . 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 1632  wcel 2139  wral 3050  Vcvv 3340  wss 3715   × cxp 5264  cres 5268  wf 6045  cfv 6049  (class class class)co 6813  Basecbs 16059  s cress 16060  +gcplusg 16143  0gc0g 16302  Grpcgrp 17623  SubGrpcsubg 17789   GrpAct cga 17922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-0g 16304  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-grp 17626  df-subg 17792  df-ga 17923
This theorem is referenced by:  sylow3lem5  18246
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