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Theorem eqger 18980
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 18977 . . 3 Rel
32a1i 11 . 2 (𝑌 ∈ (SubGrp‘𝐺) → Rel )
4 subgrcl 18933 . . . . . 6 (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
5 eqger.x . . . . . . 7 𝑋 = (Base‘𝐺)
65subgss 18929 . . . . . 6 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌𝑋)
7 eqid 2736 . . . . . . 7 (invg𝐺) = (invg𝐺)
8 eqid 2736 . . . . . . 7 (+g𝐺) = (+g𝐺)
95, 7, 8, 1eqgval 18979 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝑥 𝑦 ↔ (𝑥𝑋𝑦𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌)))
104, 6, 9syl2anc 584 . . . . 5 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 𝑦 ↔ (𝑥𝑋𝑦𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌)))
1110biimpa 477 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → (𝑥𝑋𝑦𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌))
1211simp2d 1143 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → 𝑦𝑋)
1311simp1d 1142 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → 𝑥𝑋)
144adantr 481 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → 𝐺 ∈ Grp)
155, 7grpinvcl 18798 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → ((invg𝐺)‘𝑥) ∈ 𝑋)
1614, 13, 15syl2anc 584 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → ((invg𝐺)‘𝑥) ∈ 𝑋)
175, 8, 7grpinvadd 18825 . . . . . 6 ((𝐺 ∈ Grp ∧ ((invg𝐺)‘𝑥) ∈ 𝑋𝑦𝑋) → ((invg𝐺)‘(((invg𝐺)‘𝑥)(+g𝐺)𝑦)) = (((invg𝐺)‘𝑦)(+g𝐺)((invg𝐺)‘((invg𝐺)‘𝑥))))
1814, 16, 12, 17syl3anc 1371 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → ((invg𝐺)‘(((invg𝐺)‘𝑥)(+g𝐺)𝑦)) = (((invg𝐺)‘𝑦)(+g𝐺)((invg𝐺)‘((invg𝐺)‘𝑥))))
195, 7grpinvinv 18814 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → ((invg𝐺)‘((invg𝐺)‘𝑥)) = 𝑥)
2014, 13, 19syl2anc 584 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → ((invg𝐺)‘((invg𝐺)‘𝑥)) = 𝑥)
2120oveq2d 7373 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → (((invg𝐺)‘𝑦)(+g𝐺)((invg𝐺)‘((invg𝐺)‘𝑥))) = (((invg𝐺)‘𝑦)(+g𝐺)𝑥))
2218, 21eqtrd 2776 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → ((invg𝐺)‘(((invg𝐺)‘𝑥)(+g𝐺)𝑦)) = (((invg𝐺)‘𝑦)(+g𝐺)𝑥))
2311simp3d 1144 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌)
247subginvcl 18937 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌) → ((invg𝐺)‘(((invg𝐺)‘𝑥)(+g𝐺)𝑦)) ∈ 𝑌)
2523, 24syldan 591 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → ((invg𝐺)‘(((invg𝐺)‘𝑥)(+g𝐺)𝑦)) ∈ 𝑌)
2622, 25eqeltrrd 2839 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → (((invg𝐺)‘𝑦)(+g𝐺)𝑥) ∈ 𝑌)
276adantr 481 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → 𝑌𝑋)
285, 7, 8, 1eqgval 18979 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝑦 𝑥 ↔ (𝑦𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑥) ∈ 𝑌)))
2914, 27, 28syl2anc 584 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → (𝑦 𝑥 ↔ (𝑦𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑥) ∈ 𝑌)))
3012, 13, 26, 29mpbir3and 1342 . 2 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → 𝑦 𝑥)
3113adantrr 715 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝑥𝑋)
325, 7, 8, 1eqgval 18979 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝑦 𝑧 ↔ (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)))
334, 6, 32syl2anc 584 . . . . . 6 (𝑌 ∈ (SubGrp‘𝐺) → (𝑦 𝑧 ↔ (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)))
3433biimpa 477 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑦 𝑧) → (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌))
3534adantrl 714 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌))
3635simp2d 1143 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝑧𝑋)
374adantr 481 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝐺 ∈ Grp)
3837, 31, 15syl2anc 584 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((invg𝐺)‘𝑥) ∈ 𝑋)
3912adantrr 715 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝑦𝑋)
405, 7grpinvcl 18798 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑦𝑋) → ((invg𝐺)‘𝑦) ∈ 𝑋)
4137, 39, 40syl2anc 584 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((invg𝐺)‘𝑦) ∈ 𝑋)
425, 8grpcl 18756 . . . . . . 7 ((𝐺 ∈ Grp ∧ ((invg𝐺)‘𝑦) ∈ 𝑋𝑧𝑋) → (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑋)
4337, 41, 36, 42syl3anc 1371 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑋)
445, 8grpass 18757 . . . . . 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 1372 . . . . 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 2736 . . . . . . . . . 10 (0g𝐺) = (0g𝐺)
475, 8, 46, 7grprinv 18801 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑦𝑋) → (𝑦(+g𝐺)((invg𝐺)‘𝑦)) = (0g𝐺))
4837, 39, 47syl2anc 584 . . . . . . . 8 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (𝑦(+g𝐺)((invg𝐺)‘𝑦)) = (0g𝐺))
4948oveq1d 7372 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((𝑦(+g𝐺)((invg𝐺)‘𝑦))(+g𝐺)𝑧) = ((0g𝐺)(+g𝐺)𝑧))
505, 8grpass 18757 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑦𝑋 ∧ ((invg𝐺)‘𝑦) ∈ 𝑋𝑧𝑋)) → ((𝑦(+g𝐺)((invg𝐺)‘𝑦))(+g𝐺)𝑧) = (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)))
5137, 39, 41, 36, 50syl13anc 1372 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((𝑦(+g𝐺)((invg𝐺)‘𝑦))(+g𝐺)𝑧) = (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)))
525, 8, 46grplid 18780 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑧𝑋) → ((0g𝐺)(+g𝐺)𝑧) = 𝑧)
5337, 36, 52syl2anc 584 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((0g𝐺)(+g𝐺)𝑧) = 𝑧)
5449, 51, 533eqtr3d 2784 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) = 𝑧)
5554oveq2d 7373 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (((invg𝐺)‘𝑥)(+g𝐺)(𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) = (((invg𝐺)‘𝑥)(+g𝐺)𝑧))
5645, 55eqtrd 2776 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((((invg𝐺)‘𝑥)(+g𝐺)𝑦)(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) = (((invg𝐺)‘𝑥)(+g𝐺)𝑧))
57 simpl 483 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝑌 ∈ (SubGrp‘𝐺))
5823adantrr 715 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌)
5935simp3d 1144 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)
608subgcl 18938 . . . . 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 1371 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((((invg𝐺)‘𝑥)(+g𝐺)𝑦)(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑌)
6256, 61eqeltrrd 2839 . . 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 18979 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝑥 𝑧 ↔ (𝑥𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑧) ∈ 𝑌)))
6537, 63, 64syl2anc 584 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (𝑥 𝑧 ↔ (𝑥𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑧) ∈ 𝑌)))
6631, 36, 62, 65mpbir3and 1342 . 2 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝑥 𝑧)
675, 8, 46, 7grplinv 18800 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → (((invg𝐺)‘𝑥)(+g𝐺)𝑥) = (0g𝐺))
684, 67sylan 580 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (((invg𝐺)‘𝑥)(+g𝐺)𝑥) = (0g𝐺))
6946subg0cl 18936 . . . . . . 7 (𝑌 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑌)
7069adantr 481 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (0g𝐺) ∈ 𝑌)
7168, 70eqeltrd 2838 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌)
7271ex 413 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥𝑋 → (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌))
7372pm4.71rd 563 . . 3 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥𝑋 ↔ ((((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌𝑥𝑋)))
745, 7, 8, 1eqgval 18979 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝑥 𝑥 ↔ (𝑥𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌)))
754, 6, 74syl2anc 584 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 𝑥 ↔ (𝑥𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌)))
76 df-3an 1089 . . . . 5 ((𝑥𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌) ↔ ((𝑥𝑋𝑥𝑋) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌))
77 anidm 565 . . . . . 6 ((𝑥𝑋𝑥𝑋) ↔ 𝑥𝑋)
7877anbi2ci 625 . . . . 5 (((𝑥𝑋𝑥𝑋) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌) ↔ ((((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌𝑥𝑋))
7976, 78bitri 274 . . . 4 ((𝑥𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌) ↔ ((((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌𝑥𝑋))
8075, 79bitrdi 286 . . 3 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 𝑥 ↔ ((((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌𝑥𝑋)))
8173, 80bitr4d 281 . 2 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥𝑋𝑥 𝑥))
823, 30, 66, 81iserd 8674 1 (𝑌 ∈ (SubGrp‘𝐺) → Er 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wss 3910   class class class wbr 5105  Rel wrel 5638  cfv 6496  (class class class)co 7357   Er wer 8645  Basecbs 17083  +gcplusg 17133  0gc0g 17321  Grpcgrp 18748  invgcminusg 18749  SubGrpcsubg 18922   ~QG cqg 18924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-subg 18925  df-eqg 18927
This theorem is referenced by:  qusgrp  18985  qusadd  18987  lagsubg2  18991  lagsubg  18992  orbstafun  19091  orbstaval  19092  orbsta  19093  orbsta2  19094  sylow2blem1  19402  sylow2blem2  19403  sylow2blem3  19404  sylow3lem3  19411  sylow3lem4  19412  2idlcpbl  20704  qus1  20705  qusrhm  20707  quscrng  20710  zndvds  20956  cldsubg  23462  qustgpopn  23471  qustgphaus  23474  tgptsmscls  23501  qusker  32141  qusvscpbl  32143  quslmod  32146  eqg0el  32149  qusxpid  32151  qustrivr  32153  nsgqusf1olem3  32193  ghmquskerlem1  32195  ghmquskerlem2  32197  ghmqusker  32198  qsidomlem1  32225  qsidomlem2  32226
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