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Theorem eqger 17842
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 17839 . . 3 Rel
32a1i 11 . 2 (𝑌 ∈ (SubGrp‘𝐺) → Rel )
4 subgrcl 17797 . . . . . 6 (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
5 eqger.x . . . . . . 7 𝑋 = (Base‘𝐺)
65subgss 17793 . . . . . 6 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌𝑋)
7 eqid 2806 . . . . . . 7 (invg𝐺) = (invg𝐺)
8 eqid 2806 . . . . . . 7 (+g𝐺) = (+g𝐺)
95, 7, 8, 1eqgval 17841 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝑥 𝑦 ↔ (𝑥𝑋𝑦𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌)))
104, 6, 9syl2anc 575 . . . . 5 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 𝑦 ↔ (𝑥𝑋𝑦𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌)))
1110biimpa 464 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → (𝑥𝑋𝑦𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌))
1211simp2d 1166 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → 𝑦𝑋)
1311simp1d 1165 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → 𝑥𝑋)
144adantr 468 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → 𝐺 ∈ Grp)
155, 7grpinvcl 17668 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → ((invg𝐺)‘𝑥) ∈ 𝑋)
1614, 13, 15syl2anc 575 . . . . . 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 1483 . . . . 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 575 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → ((invg𝐺)‘((invg𝐺)‘𝑥)) = 𝑥)
2120oveq2d 6886 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → (((invg𝐺)‘𝑦)(+g𝐺)((invg𝐺)‘((invg𝐺)‘𝑥))) = (((invg𝐺)‘𝑦)(+g𝐺)𝑥))
2218, 21eqtrd 2840 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → ((invg𝐺)‘(((invg𝐺)‘𝑥)(+g𝐺)𝑦)) = (((invg𝐺)‘𝑦)(+g𝐺)𝑥))
2311simp3d 1167 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌)
247subginvcl 17801 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌) → ((invg𝐺)‘(((invg𝐺)‘𝑥)(+g𝐺)𝑦)) ∈ 𝑌)
2523, 24syldan 581 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → ((invg𝐺)‘(((invg𝐺)‘𝑥)(+g𝐺)𝑦)) ∈ 𝑌)
2622, 25eqeltrrd 2886 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → (((invg𝐺)‘𝑦)(+g𝐺)𝑥) ∈ 𝑌)
276adantr 468 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → 𝑌𝑋)
285, 7, 8, 1eqgval 17841 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝑦 𝑥 ↔ (𝑦𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑥) ∈ 𝑌)))
2914, 27, 28syl2anc 575 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → (𝑦 𝑥 ↔ (𝑦𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑥) ∈ 𝑌)))
3012, 13, 26, 29mpbir3and 1435 . 2 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → 𝑦 𝑥)
3113adantrr 699 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝑥𝑋)
325, 7, 8, 1eqgval 17841 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝑦 𝑧 ↔ (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)))
334, 6, 32syl2anc 575 . . . . . 6 (𝑌 ∈ (SubGrp‘𝐺) → (𝑦 𝑧 ↔ (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)))
3433biimpa 464 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑦 𝑧) → (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌))
3534adantrl 698 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌))
3635simp2d 1166 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝑧𝑋)
374adantr 468 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝐺 ∈ Grp)
3837, 31, 15syl2anc 575 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((invg𝐺)‘𝑥) ∈ 𝑋)
3912adantrr 699 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝑦𝑋)
405, 7grpinvcl 17668 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑦𝑋) → ((invg𝐺)‘𝑦) ∈ 𝑋)
4137, 39, 40syl2anc 575 . . . . . . 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 1483 . . . . . 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 1484 . . . . 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 2806 . . . . . . . . . 10 (0g𝐺) = (0g𝐺)
475, 8, 46, 7grprinv 17670 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑦𝑋) → (𝑦(+g𝐺)((invg𝐺)‘𝑦)) = (0g𝐺))
4837, 39, 47syl2anc 575 . . . . . . . 8 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (𝑦(+g𝐺)((invg𝐺)‘𝑦)) = (0g𝐺))
4948oveq1d 6885 . . . . . . 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 1484 . . . . . . 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 575 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((0g𝐺)(+g𝐺)𝑧) = 𝑧)
5449, 51, 533eqtr3d 2848 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) = 𝑧)
5554oveq2d 6886 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (((invg𝐺)‘𝑥)(+g𝐺)(𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) = (((invg𝐺)‘𝑥)(+g𝐺)𝑧))
5645, 55eqtrd 2840 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((((invg𝐺)‘𝑥)(+g𝐺)𝑦)(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) = (((invg𝐺)‘𝑥)(+g𝐺)𝑧))
57 simpl 470 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝑌 ∈ (SubGrp‘𝐺))
5823adantrr 699 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌)
5935simp3d 1167 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)
608subgcl 17802 . . . . 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 1483 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((((invg𝐺)‘𝑥)(+g𝐺)𝑦)(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑌)
6256, 61eqeltrrd 2886 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (((invg𝐺)‘𝑥)(+g𝐺)𝑧) ∈ 𝑌)
636adantr 468 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝑌𝑋)
645, 7, 8, 1eqgval 17841 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝑥 𝑧 ↔ (𝑥𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑧) ∈ 𝑌)))
6537, 63, 64syl2anc 575 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (𝑥 𝑧 ↔ (𝑥𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑧) ∈ 𝑌)))
6631, 36, 62, 65mpbir3and 1435 . 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 571 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (((invg𝐺)‘𝑥)(+g𝐺)𝑥) = (0g𝐺))
6946subg0cl 17800 . . . . . . 7 (𝑌 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑌)
7069adantr 468 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (0g𝐺) ∈ 𝑌)
7168, 70eqeltrd 2885 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌)
7271ex 399 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥𝑋 → (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌))
7372pm4.71rd 554 . . 3 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥𝑋 ↔ ((((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌𝑥𝑋)))
745, 7, 8, 1eqgval 17841 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝑥 𝑥 ↔ (𝑥𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌)))
754, 6, 74syl2anc 575 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 𝑥 ↔ (𝑥𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌)))
76 df-3an 1102 . . . . 5 ((𝑥𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌) ↔ ((𝑥𝑋𝑥𝑋) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌))
77 anidm 556 . . . . . 6 ((𝑥𝑋𝑥𝑋) ↔ 𝑥𝑋)
7877anbi2ci 613 . . . . 5 (((𝑥𝑋𝑥𝑋) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌) ↔ ((((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌𝑥𝑋))
7976, 78bitri 266 . . . 4 ((𝑥𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌) ↔ ((((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌𝑥𝑋))
8075, 79syl6bb 278 . . 3 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 𝑥 ↔ ((((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌𝑥𝑋)))
8173, 80bitr4d 273 . 2 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥𝑋𝑥 𝑥))
823, 30, 66, 81iserd 8001 1 (𝑌 ∈ (SubGrp‘𝐺) → Er 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2156  wss 3769   class class class wbr 4844  Rel wrel 5316  cfv 6097  (class class class)co 6870   Er wer 7972  Basecbs 16064  +gcplusg 16149  0gc0g 16301  Grpcgrp 17623  invgcminusg 17624  SubGrpcsubg 17786   ~QG cqg 17788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175  ax-cnex 10273  ax-resscn 10274  ax-1cn 10275  ax-icn 10276  ax-addcl 10277  ax-addrcl 10278  ax-mulcl 10279  ax-mulrcl 10280  ax-mulcom 10281  ax-addass 10282  ax-mulass 10283  ax-distr 10284  ax-i2m1 10285  ax-1ne0 10286  ax-1rid 10287  ax-rnegex 10288  ax-rrecex 10289  ax-cnre 10290  ax-pre-lttri 10291  ax-pre-lttrn 10292  ax-pre-ltadd 10293  ax-pre-mulgt0 10294
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-nel 3082  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-riota 6831  df-ov 6873  df-oprab 6874  df-mpt2 6875  df-om 7292  df-1st 7394  df-2nd 7395  df-wrecs 7638  df-recs 7700  df-rdg 7738  df-er 7975  df-en 8189  df-dom 8190  df-sdom 8191  df-pnf 10357  df-mnf 10358  df-xr 10359  df-ltxr 10360  df-le 10361  df-sub 10549  df-neg 10550  df-nn 11302  df-2 11360  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-plusg 16162  df-0g 16303  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-grp 17626  df-minusg 17627  df-subg 17789  df-eqg 17791
This theorem is referenced by:  qusgrp  17847  qusadd  17849  lagsubg2  17853  lagsubg  17854  orbstafun  17941  orbstaval  17942  orbsta  17943  orbsta2  17944  sylow2blem1  18232  sylow2blem2  18233  sylow2blem3  18234  sylow3lem3  18241  sylow3lem4  18242  2idlcpbl  19439  qus1  19440  qusrhm  19442  quscrng  19445  zndvds  20101  cldsubg  22123  qustgpopn  22132  qustgphaus  22135  tgptsmscls  22162
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