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Theorem gasubg 18362
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 18353 . . 3 ( ∈ (𝐺 GrpAct 𝑌) → 𝑌 ∈ V)
2 gasubg.1 . . . 4 𝐻 = (𝐺s 𝑆)
32subggrp 18212 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
41, 3anim12ci 613 . 2 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝐻 ∈ Grp ∧ 𝑌 ∈ V))
5 eqid 2826 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
65gaf 18355 . . . . . 6 ( ∈ (𝐺 GrpAct 𝑌) → :((Base‘𝐺) × 𝑌)⟶𝑌)
76adantr 481 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → :((Base‘𝐺) × 𝑌)⟶𝑌)
8 simpr 485 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ∈ (SubGrp‘𝐺))
95subgss 18210 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
108, 9syl 17 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
11 xpss1 5573 . . . . . 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 6542 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ( ↾ (𝑆 × 𝑌)):(𝑆 × 𝑌)⟶𝑌)
142subgbas 18213 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
158, 14syl 17 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 = (Base‘𝐻))
1615xpeq1d 5583 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑆 × 𝑌) = ((Base‘𝐻) × 𝑌))
1716feq2d 6497 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (( ↾ (𝑆 × 𝑌)):(𝑆 × 𝑌)⟶𝑌 ↔ ( ↾ (𝑆 × 𝑌)):((Base‘𝐻) × 𝑌)⟶𝑌))
1813, 17mpbid 233 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ( ↾ (𝑆 × 𝑌)):((Base‘𝐻) × 𝑌)⟶𝑌)
198adantr 481 . . . . . . . 8 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → 𝑆 ∈ (SubGrp‘𝐺))
20 eqid 2826 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
2120subg0cl 18217 . . . . . . . 8 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑆)
2219, 21syl 17 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → (0g𝐺) ∈ 𝑆)
23 simpr 485 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → 𝑥𝑌)
24 ovres 7304 . . . . . . 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 18215 . . . . . . . 8 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) = (0g𝐻))
2719, 26syl 17 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → (0g𝐺) = (0g𝐻))
2827oveq1d 7163 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → ((0g𝐺)( ↾ (𝑆 × 𝑌))𝑥) = ((0g𝐻)( ↾ (𝑆 × 𝑌))𝑥))
2920gagrpid 18354 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥𝑌) → ((0g𝐺) 𝑥) = 𝑥)
3029adantlr 711 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → ((0g𝐺) 𝑥) = 𝑥)
3125, 28, 303eqtr3d 2869 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → ((0g𝐻)( ↾ (𝑆 × 𝑌))𝑥) = 𝑥)
32 eqimss2 4028 . . . . . . . . . . 11 (𝑆 = (Base‘𝐻) → (Base‘𝐻) ⊆ 𝑆)
3315, 32syl 17 . . . . . . . . . 10 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (Base‘𝐻) ⊆ 𝑆)
3433adantr 481 . . . . . . . . 9 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) → (Base‘𝐻) ⊆ 𝑆)
3534sselda 3971 . . . . . . . 8 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ 𝑦 ∈ (Base‘𝐻)) → 𝑦𝑆)
3634sselda 3971 . . . . . . . 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 767 . . . . . . . . . 10 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → 𝑦𝑆)
397ad2antrr 722 . . . . . . . . . . 11 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → :((Base‘𝐺) × 𝑌)⟶𝑌)
409ad3antlr 727 . . . . . . . . . . . 12 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → 𝑆 ⊆ (Base‘𝐺))
41 simprr 769 . . . . . . . . . . . 12 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → 𝑧𝑆)
4240, 41sseldd 3972 . . . . . . . . . . 11 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → 𝑧 ∈ (Base‘𝐺))
4323adantr 481 . . . . . . . . . . 11 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → 𝑥𝑌)
4439, 42, 43fovrnd 7310 . . . . . . . . . 10 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (𝑧 𝑥) ∈ 𝑌)
45 ovres 7304 . . . . . . . . . 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 7304 . . . . . . . . . . 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 7164 . . . . . . . . 9 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (𝑦( ↾ (𝑆 × 𝑌))(𝑧( ↾ (𝑆 × 𝑌))𝑥)) = (𝑦( ↾ (𝑆 × 𝑌))(𝑧 𝑥)))
50 simplll 771 . . . . . . . . . 10 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → ∈ (𝐺 GrpAct 𝑌))
5140, 38sseldd 3972 . . . . . . . . . 10 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → 𝑦 ∈ (Base‘𝐺))
52 eqid 2826 . . . . . . . . . . 11 (+g𝐺) = (+g𝐺)
535, 52gaass 18357 . . . . . . . . . 10 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺) ∧ 𝑥𝑌)) → ((𝑦(+g𝐺)𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))
5450, 51, 42, 43, 53syl13anc 1366 . . . . . . . . 9 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → ((𝑦(+g𝐺)𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))
5546, 49, 543eqtr4d 2871 . . . . . . . 8 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (𝑦( ↾ (𝑆 × 𝑌))(𝑧( ↾ (𝑆 × 𝑌))𝑥)) = ((𝑦(+g𝐺)𝑧) 𝑥))
5652subgcl 18219 . . . . . . . . . . 11 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑦𝑆𝑧𝑆) → (𝑦(+g𝐺)𝑧) ∈ 𝑆)
57563expb 1114 . . . . . . . . . 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 7304 . . . . . . . . 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 16602 . . . . . . . . . . 11 (𝑆 ∈ (SubGrp‘𝐺) → (+g𝐺) = (+g𝐻))
6261ad3antlr 727 . . . . . . . . . 10 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (+g𝐺) = (+g𝐻))
6362oveqd 7165 . . . . . . . . 9 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → (𝑦(+g𝐺)𝑧) = (𝑦(+g𝐻)𝑧))
6463oveq1d 7163 . . . . . . . 8 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑌) ∧ (𝑦𝑆𝑧𝑆)) → ((𝑦(+g𝐺)𝑧)( ↾ (𝑆 × 𝑌))𝑥) = ((𝑦(+g𝐻)𝑧)( ↾ (𝑆 × 𝑌))𝑥))
6555, 60, 643eqtr2rd 2868 . . . . . . 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 3196 . . . . 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 512 . . . 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 3187 . . 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 512 . 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 2826 . . 3 (Base‘𝐻) = (Base‘𝐻)
72 eqid 2826 . . 3 (+g𝐻) = (+g𝐻)
73 eqid 2826 . . 3 (0g𝐻) = (0g𝐻)
7471, 72, 73isga 18351 . 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 396   = wceq 1530  wcel 2107  wral 3143  Vcvv 3500  wss 3940   × cxp 5552  cres 5556  wf 6348  cfv 6352  (class class class)co 7148  Basecbs 16473  s cress 16474  +gcplusg 16555  0gc0g 16703  Grpcgrp 18033  SubGrpcsubg 18203   GrpAct cga 18349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-er 8279  df-map 8398  df-en 8499  df-dom 8500  df-sdom 8501  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-ndx 16476  df-slot 16477  df-base 16479  df-sets 16480  df-ress 16481  df-plusg 16568  df-0g 16705  df-mgm 17842  df-sgrp 17890  df-mnd 17901  df-grp 18036  df-subg 18206  df-ga 18350
This theorem is referenced by:  sylow3lem5  18676
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