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Theorem eqger 17576
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 17573 . . 3 Rel
32a1i 11 . 2 (𝑌 ∈ (SubGrp‘𝐺) → Rel )
4 subgrcl 17531 . . . . . 6 (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
5 eqger.x . . . . . . 7 𝑋 = (Base‘𝐺)
65subgss 17527 . . . . . 6 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌𝑋)
7 eqid 2621 . . . . . . 7 (invg𝐺) = (invg𝐺)
8 eqid 2621 . . . . . . 7 (+g𝐺) = (+g𝐺)
95, 7, 8, 1eqgval 17575 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝑥 𝑦 ↔ (𝑥𝑋𝑦𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌)))
104, 6, 9syl2anc 692 . . . . 5 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 𝑦 ↔ (𝑥𝑋𝑦𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌)))
1110biimpa 501 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → (𝑥𝑋𝑦𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌))
1211simp2d 1072 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → 𝑦𝑋)
1311simp1d 1071 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → 𝑥𝑋)
144adantr 481 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → 𝐺 ∈ Grp)
155, 7grpinvcl 17399 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → ((invg𝐺)‘𝑥) ∈ 𝑋)
1614, 13, 15syl2anc 692 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → ((invg𝐺)‘𝑥) ∈ 𝑋)
175, 8, 7grpinvadd 17425 . . . . . 6 ((𝐺 ∈ Grp ∧ ((invg𝐺)‘𝑥) ∈ 𝑋𝑦𝑋) → ((invg𝐺)‘(((invg𝐺)‘𝑥)(+g𝐺)𝑦)) = (((invg𝐺)‘𝑦)(+g𝐺)((invg𝐺)‘((invg𝐺)‘𝑥))))
1814, 16, 12, 17syl3anc 1323 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → ((invg𝐺)‘(((invg𝐺)‘𝑥)(+g𝐺)𝑦)) = (((invg𝐺)‘𝑦)(+g𝐺)((invg𝐺)‘((invg𝐺)‘𝑥))))
195, 7grpinvinv 17414 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → ((invg𝐺)‘((invg𝐺)‘𝑥)) = 𝑥)
2014, 13, 19syl2anc 692 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → ((invg𝐺)‘((invg𝐺)‘𝑥)) = 𝑥)
2120oveq2d 6626 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → (((invg𝐺)‘𝑦)(+g𝐺)((invg𝐺)‘((invg𝐺)‘𝑥))) = (((invg𝐺)‘𝑦)(+g𝐺)𝑥))
2218, 21eqtrd 2655 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → ((invg𝐺)‘(((invg𝐺)‘𝑥)(+g𝐺)𝑦)) = (((invg𝐺)‘𝑦)(+g𝐺)𝑥))
2311simp3d 1073 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌)
247subginvcl 17535 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌) → ((invg𝐺)‘(((invg𝐺)‘𝑥)(+g𝐺)𝑦)) ∈ 𝑌)
2523, 24syldan 487 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → ((invg𝐺)‘(((invg𝐺)‘𝑥)(+g𝐺)𝑦)) ∈ 𝑌)
2622, 25eqeltrrd 2699 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → (((invg𝐺)‘𝑦)(+g𝐺)𝑥) ∈ 𝑌)
276adantr 481 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → 𝑌𝑋)
285, 7, 8, 1eqgval 17575 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝑦 𝑥 ↔ (𝑦𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑥) ∈ 𝑌)))
2914, 27, 28syl2anc 692 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → (𝑦 𝑥 ↔ (𝑦𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑥) ∈ 𝑌)))
3012, 13, 26, 29mpbir3and 1243 . 2 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → 𝑦 𝑥)
3113adantrr 752 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝑥𝑋)
325, 7, 8, 1eqgval 17575 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝑦 𝑧 ↔ (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)))
334, 6, 32syl2anc 692 . . . . . 6 (𝑌 ∈ (SubGrp‘𝐺) → (𝑦 𝑧 ↔ (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)))
3433biimpa 501 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑦 𝑧) → (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌))
3534adantrl 751 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌))
3635simp2d 1072 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝑧𝑋)
374adantr 481 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝐺 ∈ Grp)
3837, 31, 15syl2anc 692 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((invg𝐺)‘𝑥) ∈ 𝑋)
3912adantrr 752 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝑦𝑋)
405, 7grpinvcl 17399 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑦𝑋) → ((invg𝐺)‘𝑦) ∈ 𝑋)
4137, 39, 40syl2anc 692 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((invg𝐺)‘𝑦) ∈ 𝑋)
425, 8grpcl 17362 . . . . . . 7 ((𝐺 ∈ Grp ∧ ((invg𝐺)‘𝑦) ∈ 𝑋𝑧𝑋) → (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑋)
4337, 41, 36, 42syl3anc 1323 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑋)
445, 8grpass 17363 . . . . . 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 1325 . . . . 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 2621 . . . . . . . . . 10 (0g𝐺) = (0g𝐺)
475, 8, 46, 7grprinv 17401 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑦𝑋) → (𝑦(+g𝐺)((invg𝐺)‘𝑦)) = (0g𝐺))
4837, 39, 47syl2anc 692 . . . . . . . 8 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (𝑦(+g𝐺)((invg𝐺)‘𝑦)) = (0g𝐺))
4948oveq1d 6625 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((𝑦(+g𝐺)((invg𝐺)‘𝑦))(+g𝐺)𝑧) = ((0g𝐺)(+g𝐺)𝑧))
505, 8grpass 17363 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑦𝑋 ∧ ((invg𝐺)‘𝑦) ∈ 𝑋𝑧𝑋)) → ((𝑦(+g𝐺)((invg𝐺)‘𝑦))(+g𝐺)𝑧) = (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)))
5137, 39, 41, 36, 50syl13anc 1325 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((𝑦(+g𝐺)((invg𝐺)‘𝑦))(+g𝐺)𝑧) = (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)))
525, 8, 46grplid 17384 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑧𝑋) → ((0g𝐺)(+g𝐺)𝑧) = 𝑧)
5337, 36, 52syl2anc 692 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((0g𝐺)(+g𝐺)𝑧) = 𝑧)
5449, 51, 533eqtr3d 2663 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) = 𝑧)
5554oveq2d 6626 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (((invg𝐺)‘𝑥)(+g𝐺)(𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) = (((invg𝐺)‘𝑥)(+g𝐺)𝑧))
5645, 55eqtrd 2655 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((((invg𝐺)‘𝑥)(+g𝐺)𝑦)(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) = (((invg𝐺)‘𝑥)(+g𝐺)𝑧))
57 simpl 473 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝑌 ∈ (SubGrp‘𝐺))
5823adantrr 752 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌)
5935simp3d 1073 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)
608subgcl 17536 . . . . 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 1323 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((((invg𝐺)‘𝑥)(+g𝐺)𝑦)(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑌)
6256, 61eqeltrrd 2699 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (((invg𝐺)‘𝑥)(+g𝐺)𝑧) ∈ 𝑌)
636adantr 481 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝑌𝑋)
645, 7, 8, 1eqgval 17575 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝑥 𝑧 ↔ (𝑥𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑧) ∈ 𝑌)))
6537, 63, 64syl2anc 692 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (𝑥 𝑧 ↔ (𝑥𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑧) ∈ 𝑌)))
6631, 36, 62, 65mpbir3and 1243 . 2 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝑥 𝑧)
675, 8, 46, 7grplinv 17400 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → (((invg𝐺)‘𝑥)(+g𝐺)𝑥) = (0g𝐺))
684, 67sylan 488 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (((invg𝐺)‘𝑥)(+g𝐺)𝑥) = (0g𝐺))
6946subg0cl 17534 . . . . . . 7 (𝑌 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑌)
7069adantr 481 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (0g𝐺) ∈ 𝑌)
7168, 70eqeltrd 2698 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌)
7271ex 450 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥𝑋 → (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌))
7372pm4.71rd 666 . . 3 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥𝑋 ↔ ((((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌𝑥𝑋)))
745, 7, 8, 1eqgval 17575 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝑥 𝑥 ↔ (𝑥𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌)))
754, 6, 74syl2anc 692 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 𝑥 ↔ (𝑥𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌)))
76 df-3an 1038 . . . . 5 ((𝑥𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌) ↔ ((𝑥𝑋𝑥𝑋) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌))
77 anidm 675 . . . . . 6 ((𝑥𝑋𝑥𝑋) ↔ 𝑥𝑋)
7877anbi2ci 731 . . . . 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 7720 1 (𝑌 ∈ (SubGrp‘𝐺) → Er 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wss 3559   class class class wbr 4618  Rel wrel 5084  cfv 5852  (class class class)co 6610   Er wer 7691  Basecbs 15792  +gcplusg 15873  0gc0g 16032  Grpcgrp 17354  invgcminusg 17355  SubGrpcsubg 17520   ~QG cqg 17522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-plusg 15886  df-0g 16034  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-grp 17357  df-minusg 17358  df-subg 17523  df-eqg 17525
This theorem is referenced by:  qusgrp  17581  qusadd  17583  lagsubg2  17587  lagsubg  17588  orbstafun  17676  orbstaval  17677  orbsta  17678  orbsta2  17679  sylow2blem1  17967  sylow2blem2  17968  sylow2blem3  17969  sylow3lem3  17976  sylow3lem4  17977  2idlcpbl  19166  qus1  19167  qusrhm  19169  quscrng  19172  zndvds  19830  cldsubg  21837  qustgpopn  21846  qustgphaus  21849  tgptsmscls  21876
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