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Theorem gasubg 19214
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 19205 . . 3 ( ∈ (𝐺 GrpAct 𝑌) → 𝑌 ∈ V)
2 gasubg.1 . . . 4 𝐻 = (𝐺s 𝑆)
32subggrp 19052 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
41, 3anim12ci 613 . 2 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐻 ∈ Grp ∧ 𝑌 ∈ V))
5 eqid 2731 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
65gaf 19207 . . . . . 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 19050 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
108, 9syl 17 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
11 xpss1 5695 . . . . . 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 6758 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ( ↾ (𝑆 × 𝑌)):(𝑆 × 𝑌)⟶𝑌)
142subgbas 19053 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
158, 14syl 17 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 = (Base‘𝐻))
1615xpeq1d 5705 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑌) = ((Base‘𝐻) × 𝑌))
1716feq2d 6703 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (( ↾ (𝑆 × 𝑌)):(𝑆 × 𝑌)⟶𝑌 ↔ ( ↾ (𝑆 × 𝑌)):((Base‘𝐻) × 𝑌)⟶𝑌))
1813, 17mpbid 231 . . 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 2731 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
2120subg0cl 19057 . . . . . . . 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 7577 . . . . . . 7 (((0g𝐺) ∈ 𝑆𝑥𝑌) → ((0g𝐺)( ↾ (𝑆 × 𝑌))𝑥) = ((0g𝐺) 𝑥))
2522, 23, 24syl2anc 583 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → ((0g𝐺)( ↾ (𝑆 × 𝑌))𝑥) = ((0g𝐺) 𝑥))
262, 20subg0 19055 . . . . . . . 8 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) = (0g𝐻))
2719, 26syl 17 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → (0g𝐺) = (0g𝐻))
2827oveq1d 7427 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → ((0g𝐺)( ↾ (𝑆 × 𝑌))𝑥) = ((0g𝐻)( ↾ (𝑆 × 𝑌))𝑥))
2920gagrpid 19206 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥𝑌) → ((0g𝐺) 𝑥) = 𝑥)
3029adantlr 712 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → ((0g𝐺) 𝑥) = 𝑥)
3125, 28, 303eqtr3d 2779 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → ((0g𝐻)( ↾ (𝑆 × 𝑌))𝑥) = 𝑥)
32 eqimss2 4041 . . . . . . . . . . 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 3982 . . . . . . . 8 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ 𝑦 ∈ (Base‘𝐻)) → 𝑦𝑆)
3634sselda 3982 . . . . . . . 8 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ 𝑧 ∈ (Base‘𝐻)) → 𝑧𝑆)
3735, 36anim12dan 618 . . . . . . 7 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦 ∈ (Base‘𝐻) ∧ 𝑧 ∈ (Base‘𝐻))) → (𝑦𝑆𝑧𝑆))
38 simprl 768 . . . . . . . . . 10 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → 𝑦𝑆)
397ad2antrr 723 . . . . . . . . . . 11 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → :((Base‘𝐺) × 𝑌)⟶𝑌)
409ad3antlr 728 . . . . . . . . . . . 12 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → 𝑆 ⊆ (Base‘𝐺))
41 simprr 770 . . . . . . . . . . . 12 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → 𝑧𝑆)
4240, 41sseldd 3983 . . . . . . . . . . 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 7583 . . . . . . . . . 10 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (𝑧 𝑥) ∈ 𝑌)
45 ovres 7577 . . . . . . . . . 10 ((𝑦𝑆 ∧ (𝑧 𝑥) ∈ 𝑌) → (𝑦( ↾ (𝑆 × 𝑌))(𝑧 𝑥)) = (𝑦 (𝑧 𝑥)))
4638, 44, 45syl2anc 583 . . . . . . . . 9 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (𝑦( ↾ (𝑆 × 𝑌))(𝑧 𝑥)) = (𝑦 (𝑧 𝑥)))
47 ovres 7577 . . . . . . . . . . 11 ((𝑧𝑆𝑥𝑌) → (𝑧( ↾ (𝑆 × 𝑌))𝑥) = (𝑧 𝑥))
4841, 43, 47syl2anc 583 . . . . . . . . . 10 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (𝑧( ↾ (𝑆 × 𝑌))𝑥) = (𝑧 𝑥))
4948oveq2d 7428 . . . . . . . . 9 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (𝑦( ↾ (𝑆 × 𝑌))(𝑧( ↾ (𝑆 × 𝑌))𝑥)) = (𝑦( ↾ (𝑆 × 𝑌))(𝑧 𝑥)))
50 simplll 772 . . . . . . . . . 10 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → ∈ (𝐺 GrpAct 𝑌))
5140, 38sseldd 3983 . . . . . . . . . 10 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → 𝑦 ∈ (Base‘𝐺))
52 eqid 2731 . . . . . . . . . . 11 (+g𝐺) = (+g𝐺)
535, 52gaass 19209 . . . . . . . . . 10 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺) ∧ 𝑥𝑌)) → ((𝑦(+g𝐺)𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))
5450, 51, 42, 43, 53syl13anc 1371 . . . . . . . . 9 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → ((𝑦(+g𝐺)𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))
5546, 49, 543eqtr4d 2781 . . . . . . . 8 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (𝑦( ↾ (𝑆 × 𝑌))(𝑧( ↾ (𝑆 × 𝑌))𝑥)) = ((𝑦(+g𝐺)𝑧) 𝑥))
5652subgcl 19059 . . . . . . . . . . 11 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑦𝑆𝑧𝑆) → (𝑦(+g𝐺)𝑧) ∈ 𝑆)
57563expb 1119 . . . . . . . . . 10 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑦𝑆𝑧𝑆)) → (𝑦(+g𝐺)𝑧) ∈ 𝑆)
5819, 57sylan 579 . . . . . . . . 9 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (𝑦(+g𝐺)𝑧) ∈ 𝑆)
59 ovres 7577 . . . . . . . . 9 (((𝑦(+g𝐺)𝑧) ∈ 𝑆𝑥𝑌) → ((𝑦(+g𝐺)𝑧)( ↾ (𝑆 × 𝑌))𝑥) = ((𝑦(+g𝐺)𝑧) 𝑥))
6058, 43, 59syl2anc 583 . . . . . . . 8 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → ((𝑦(+g𝐺)𝑧)( ↾ (𝑆 × 𝑌))𝑥) = ((𝑦(+g𝐺)𝑧) 𝑥))
612, 52ressplusg 17242 . . . . . . . . . . 11 (𝑆 ∈ (SubGrp‘𝐺) → (+g𝐺) = (+g𝐻))
6261ad3antlr 728 . . . . . . . . . 10 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (+g𝐺) = (+g𝐻))
6362oveqd 7429 . . . . . . . . 9 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (𝑦(+g𝐺)𝑧) = (𝑦(+g𝐻)𝑧))
6463oveq1d 7427 . . . . . . . 8 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → ((𝑦(+g𝐺)𝑧)( ↾ (𝑆 × 𝑌))𝑥) = ((𝑦(+g𝐻)𝑧)( ↾ (𝑆 × 𝑌))𝑥))
6555, 60, 643eqtr2rd 2778 . . . . . . 7 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → ((𝑦(+g𝐻)𝑧)( ↾ (𝑆 × 𝑌))𝑥) = (𝑦( ↾ (𝑆 × 𝑌))(𝑧( ↾ (𝑆 × 𝑌))𝑥)))
6637, 65syldan 590 . . . . . 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 3199 . . . . 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 3145 . . 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 2731 . . 3 (Base‘𝐻) = (Base‘𝐻)
72 eqid 2731 . . 3 (+g𝐻) = (+g𝐻)
73 eqid 2731 . . 3 (0g𝐻) = (0g𝐻)
7471, 72, 73isga 19203 . 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 582 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 2105  wral 3060  Vcvv 3473  wss 3948   × cxp 5674  cres 5678  wf 6539  cfv 6543  (class class class)co 7412  Basecbs 17151  s cress 17180  +gcplusg 17204  0gc0g 17392  Grpcgrp 18861  SubGrpcsubg 19043   GrpAct cga 19201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-er 8709  df-map 8828  df-en 8946  df-dom 8947  df-sdom 8948  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-ress 17181  df-plusg 17217  df-0g 17394  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-grp 18864  df-subg 19046  df-ga 19202
This theorem is referenced by:  sylow3lem5  19547
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