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Theorem eqger 17845
Description: The subgroup coset equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 13-Jan-2015.)
Hypotheses
Ref Expression
eqger.x 𝑋 = (Base‘𝐺)
eqger.r = (𝐺 ~QG 𝑌)
Assertion
Ref Expression
eqger (𝑌 ∈ (SubGrp‘𝐺) → Er 𝑋)

Proof of Theorem eqger
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqger.r . . . 4 = (𝐺 ~QG 𝑌)
21releqg 17842 . . 3 Rel
32a1i 11 . 2 (𝑌 ∈ (SubGrp‘𝐺) → Rel )
4 subgrcl 17800 . . . . . 6 (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
5 eqger.x . . . . . . 7 𝑋 = (Base‘𝐺)
65subgss 17796 . . . . . 6 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌𝑋)
7 eqid 2760 . . . . . . 7 (invg𝐺) = (invg𝐺)
8 eqid 2760 . . . . . . 7 (+g𝐺) = (+g𝐺)
95, 7, 8, 1eqgval 17844 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝑥 𝑦 ↔ (𝑥𝑋𝑦𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌)))
104, 6, 9syl2anc 696 . . . . 5 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 𝑦 ↔ (𝑥𝑋𝑦𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌)))
1110biimpa 502 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → (𝑥𝑋𝑦𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌))
1211simp2d 1138 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → 𝑦𝑋)
1311simp1d 1137 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → 𝑥𝑋)
144adantr 472 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → 𝐺 ∈ Grp)
155, 7grpinvcl 17668 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → ((invg𝐺)‘𝑥) ∈ 𝑋)
1614, 13, 15syl2anc 696 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → ((invg𝐺)‘𝑥) ∈ 𝑋)
175, 8, 7grpinvadd 17694 . . . . . 6 ((𝐺 ∈ Grp ∧ ((invg𝐺)‘𝑥) ∈ 𝑋𝑦𝑋) → ((invg𝐺)‘(((invg𝐺)‘𝑥)(+g𝐺)𝑦)) = (((invg𝐺)‘𝑦)(+g𝐺)((invg𝐺)‘((invg𝐺)‘𝑥))))
1814, 16, 12, 17syl3anc 1477 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → ((invg𝐺)‘(((invg𝐺)‘𝑥)(+g𝐺)𝑦)) = (((invg𝐺)‘𝑦)(+g𝐺)((invg𝐺)‘((invg𝐺)‘𝑥))))
195, 7grpinvinv 17683 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → ((invg𝐺)‘((invg𝐺)‘𝑥)) = 𝑥)
2014, 13, 19syl2anc 696 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → ((invg𝐺)‘((invg𝐺)‘𝑥)) = 𝑥)
2120oveq2d 6829 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → (((invg𝐺)‘𝑦)(+g𝐺)((invg𝐺)‘((invg𝐺)‘𝑥))) = (((invg𝐺)‘𝑦)(+g𝐺)𝑥))
2218, 21eqtrd 2794 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → ((invg𝐺)‘(((invg𝐺)‘𝑥)(+g𝐺)𝑦)) = (((invg𝐺)‘𝑦)(+g𝐺)𝑥))
2311simp3d 1139 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌)
247subginvcl 17804 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌) → ((invg𝐺)‘(((invg𝐺)‘𝑥)(+g𝐺)𝑦)) ∈ 𝑌)
2523, 24syldan 488 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → ((invg𝐺)‘(((invg𝐺)‘𝑥)(+g𝐺)𝑦)) ∈ 𝑌)
2622, 25eqeltrrd 2840 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → (((invg𝐺)‘𝑦)(+g𝐺)𝑥) ∈ 𝑌)
276adantr 472 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → 𝑌𝑋)
285, 7, 8, 1eqgval 17844 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝑦 𝑥 ↔ (𝑦𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑥) ∈ 𝑌)))
2914, 27, 28syl2anc 696 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → (𝑦 𝑥 ↔ (𝑦𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑥) ∈ 𝑌)))
3012, 13, 26, 29mpbir3and 1428 . 2 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → 𝑦 𝑥)
3113adantrr 755 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝑥𝑋)
325, 7, 8, 1eqgval 17844 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝑦 𝑧 ↔ (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)))
334, 6, 32syl2anc 696 . . . . . 6 (𝑌 ∈ (SubGrp‘𝐺) → (𝑦 𝑧 ↔ (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)))
3433biimpa 502 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑦 𝑧) → (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌))
3534adantrl 754 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌))
3635simp2d 1138 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝑧𝑋)
374adantr 472 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝐺 ∈ Grp)
3837, 31, 15syl2anc 696 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((invg𝐺)‘𝑥) ∈ 𝑋)
3912adantrr 755 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝑦𝑋)
405, 7grpinvcl 17668 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑦𝑋) → ((invg𝐺)‘𝑦) ∈ 𝑋)
4137, 39, 40syl2anc 696 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((invg𝐺)‘𝑦) ∈ 𝑋)
425, 8grpcl 17631 . . . . . . 7 ((𝐺 ∈ Grp ∧ ((invg𝐺)‘𝑦) ∈ 𝑋𝑧𝑋) → (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑋)
4337, 41, 36, 42syl3anc 1477 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑋)
445, 8grpass 17632 . . . . . 6 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝑥) ∈ 𝑋𝑦𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑋)) → ((((invg𝐺)‘𝑥)(+g𝐺)𝑦)(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) = (((invg𝐺)‘𝑥)(+g𝐺)(𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))))
4537, 38, 39, 43, 44syl13anc 1479 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((((invg𝐺)‘𝑥)(+g𝐺)𝑦)(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) = (((invg𝐺)‘𝑥)(+g𝐺)(𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))))
46 eqid 2760 . . . . . . . . . 10 (0g𝐺) = (0g𝐺)
475, 8, 46, 7grprinv 17670 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑦𝑋) → (𝑦(+g𝐺)((invg𝐺)‘𝑦)) = (0g𝐺))
4837, 39, 47syl2anc 696 . . . . . . . 8 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (𝑦(+g𝐺)((invg𝐺)‘𝑦)) = (0g𝐺))
4948oveq1d 6828 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((𝑦(+g𝐺)((invg𝐺)‘𝑦))(+g𝐺)𝑧) = ((0g𝐺)(+g𝐺)𝑧))
505, 8grpass 17632 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑦𝑋 ∧ ((invg𝐺)‘𝑦) ∈ 𝑋𝑧𝑋)) → ((𝑦(+g𝐺)((invg𝐺)‘𝑦))(+g𝐺)𝑧) = (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)))
5137, 39, 41, 36, 50syl13anc 1479 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((𝑦(+g𝐺)((invg𝐺)‘𝑦))(+g𝐺)𝑧) = (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)))
525, 8, 46grplid 17653 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑧𝑋) → ((0g𝐺)(+g𝐺)𝑧) = 𝑧)
5337, 36, 52syl2anc 696 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((0g𝐺)(+g𝐺)𝑧) = 𝑧)
5449, 51, 533eqtr3d 2802 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) = 𝑧)
5554oveq2d 6829 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (((invg𝐺)‘𝑥)(+g𝐺)(𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) = (((invg𝐺)‘𝑥)(+g𝐺)𝑧))
5645, 55eqtrd 2794 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((((invg𝐺)‘𝑥)(+g𝐺)𝑦)(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) = (((invg𝐺)‘𝑥)(+g𝐺)𝑧))
57 simpl 474 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝑌 ∈ (SubGrp‘𝐺))
5823adantrr 755 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌)
5935simp3d 1139 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)
608subgcl 17805 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → ((((invg𝐺)‘𝑥)(+g𝐺)𝑦)(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑌)
6157, 58, 59, 60syl3anc 1477 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((((invg𝐺)‘𝑥)(+g𝐺)𝑦)(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑌)
6256, 61eqeltrrd 2840 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (((invg𝐺)‘𝑥)(+g𝐺)𝑧) ∈ 𝑌)
636adantr 472 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝑌𝑋)
645, 7, 8, 1eqgval 17844 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝑥 𝑧 ↔ (𝑥𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑧) ∈ 𝑌)))
6537, 63, 64syl2anc 696 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (𝑥 𝑧 ↔ (𝑥𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑧) ∈ 𝑌)))
6631, 36, 62, 65mpbir3and 1428 . 2 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝑥 𝑧)
675, 8, 46, 7grplinv 17669 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → (((invg𝐺)‘𝑥)(+g𝐺)𝑥) = (0g𝐺))
684, 67sylan 489 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (((invg𝐺)‘𝑥)(+g𝐺)𝑥) = (0g𝐺))
6946subg0cl 17803 . . . . . . 7 (𝑌 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑌)
7069adantr 472 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (0g𝐺) ∈ 𝑌)
7168, 70eqeltrd 2839 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌)
7271ex 449 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥𝑋 → (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌))
7372pm4.71rd 670 . . 3 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥𝑋 ↔ ((((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌𝑥𝑋)))
745, 7, 8, 1eqgval 17844 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝑥 𝑥 ↔ (𝑥𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌)))
754, 6, 74syl2anc 696 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 𝑥 ↔ (𝑥𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌)))
76 df-3an 1074 . . . . 5 ((𝑥𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌) ↔ ((𝑥𝑋𝑥𝑋) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌))
77 anidm 679 . . . . . 6 ((𝑥𝑋𝑥𝑋) ↔ 𝑥𝑋)
7877anbi2ci 734 . . . . 5 (((𝑥𝑋𝑥𝑋) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌) ↔ ((((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌𝑥𝑋))
7976, 78bitri 264 . . . 4 ((𝑥𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌) ↔ ((((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌𝑥𝑋))
8075, 79syl6bb 276 . . 3 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 𝑥 ↔ ((((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌𝑥𝑋)))
8173, 80bitr4d 271 . 2 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥𝑋𝑥 𝑥))
823, 30, 66, 81iserd 7937 1 (𝑌 ∈ (SubGrp‘𝐺) → Er 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wss 3715   class class class wbr 4804  Rel wrel 5271  cfv 6049  (class class class)co 6813   Er wer 7908  Basecbs 16059  +gcplusg 16143  0gc0g 16302  Grpcgrp 17623  invgcminusg 17624  SubGrpcsubg 17789   ~QG cqg 17791
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-rep 4923  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-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-er 7911  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-minusg 17627  df-subg 17792  df-eqg 17794
This theorem is referenced by:  qusgrp  17850  qusadd  17852  lagsubg2  17856  lagsubg  17857  orbstafun  17944  orbstaval  17945  orbsta  17946  orbsta2  17947  sylow2blem1  18235  sylow2blem2  18236  sylow2blem3  18237  sylow3lem3  18244  sylow3lem4  18245  2idlcpbl  19436  qus1  19437  qusrhm  19439  quscrng  19442  zndvds  20100  cldsubg  22115  qustgpopn  22124  qustgphaus  22127  tgptsmscls  22154
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