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Theorem eqger 19132
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 19129 . . 3 Rel
32a1i 11 . 2 (𝑌 ∈ (SubGrp‘𝐺) → Rel )
4 subgrcl 19085 . . . . . 6 (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
5 eqger.x . . . . . . 7 𝑋 = (Base‘𝐺)
65subgss 19081 . . . . . 6 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌𝑋)
7 eqid 2728 . . . . . . 7 (invg𝐺) = (invg𝐺)
8 eqid 2728 . . . . . . 7 (+g𝐺) = (+g𝐺)
95, 7, 8, 1eqgval 19131 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝑥 𝑦 ↔ (𝑥𝑋𝑦𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌)))
104, 6, 9syl2anc 583 . . . . 5 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 𝑦 ↔ (𝑥𝑋𝑦𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌)))
1110biimpa 476 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → (𝑥𝑋𝑦𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌))
1211simp2d 1141 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → 𝑦𝑋)
1311simp1d 1140 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → 𝑥𝑋)
144adantr 480 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → 𝐺 ∈ Grp)
155, 7, 14, 13grpinvcld 18944 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → ((invg𝐺)‘𝑥) ∈ 𝑋)
165, 8, 7grpinvadd 18973 . . . . . 6 ((𝐺 ∈ Grp ∧ ((invg𝐺)‘𝑥) ∈ 𝑋𝑦𝑋) → ((invg𝐺)‘(((invg𝐺)‘𝑥)(+g𝐺)𝑦)) = (((invg𝐺)‘𝑦)(+g𝐺)((invg𝐺)‘((invg𝐺)‘𝑥))))
1714, 15, 12, 16syl3anc 1369 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → ((invg𝐺)‘(((invg𝐺)‘𝑥)(+g𝐺)𝑦)) = (((invg𝐺)‘𝑦)(+g𝐺)((invg𝐺)‘((invg𝐺)‘𝑥))))
185, 7grpinvinv 18961 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → ((invg𝐺)‘((invg𝐺)‘𝑥)) = 𝑥)
1914, 13, 18syl2anc 583 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → ((invg𝐺)‘((invg𝐺)‘𝑥)) = 𝑥)
2019oveq2d 7436 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → (((invg𝐺)‘𝑦)(+g𝐺)((invg𝐺)‘((invg𝐺)‘𝑥))) = (((invg𝐺)‘𝑦)(+g𝐺)𝑥))
2117, 20eqtrd 2768 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → ((invg𝐺)‘(((invg𝐺)‘𝑥)(+g𝐺)𝑦)) = (((invg𝐺)‘𝑦)(+g𝐺)𝑥))
2211simp3d 1142 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌)
237subginvcl 19089 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌) → ((invg𝐺)‘(((invg𝐺)‘𝑥)(+g𝐺)𝑦)) ∈ 𝑌)
2422, 23syldan 590 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → ((invg𝐺)‘(((invg𝐺)‘𝑥)(+g𝐺)𝑦)) ∈ 𝑌)
2521, 24eqeltrrd 2830 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → (((invg𝐺)‘𝑦)(+g𝐺)𝑥) ∈ 𝑌)
266adantr 480 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → 𝑌𝑋)
275, 7, 8, 1eqgval 19131 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝑦 𝑥 ↔ (𝑦𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑥) ∈ 𝑌)))
2814, 26, 27syl2anc 583 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → (𝑦 𝑥 ↔ (𝑦𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑥) ∈ 𝑌)))
2912, 13, 25, 28mpbir3and 1340 . 2 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 𝑦) → 𝑦 𝑥)
3013adantrr 716 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝑥𝑋)
315, 7, 8, 1eqgval 19131 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝑦 𝑧 ↔ (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)))
324, 6, 31syl2anc 583 . . . . . 6 (𝑌 ∈ (SubGrp‘𝐺) → (𝑦 𝑧 ↔ (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)))
3332biimpa 476 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑦 𝑧) → (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌))
3433adantrl 715 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (𝑦𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌))
3534simp2d 1141 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝑧𝑋)
364adantr 480 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝐺 ∈ Grp)
375, 7, 36, 30grpinvcld 18944 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((invg𝐺)‘𝑥) ∈ 𝑋)
3812adantrr 716 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝑦𝑋)
395, 7, 36, 38grpinvcld 18944 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((invg𝐺)‘𝑦) ∈ 𝑋)
405, 8, 36, 39, 35grpcld 18903 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑋)
415, 8, 36, 37, 38, 40grpassd 18901 . . . . 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))))
42 eqid 2728 . . . . . . . . . 10 (0g𝐺) = (0g𝐺)
435, 8, 42, 7grprinv 18946 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑦𝑋) → (𝑦(+g𝐺)((invg𝐺)‘𝑦)) = (0g𝐺))
4436, 38, 43syl2anc 583 . . . . . . . 8 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (𝑦(+g𝐺)((invg𝐺)‘𝑦)) = (0g𝐺))
4544oveq1d 7435 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((𝑦(+g𝐺)((invg𝐺)‘𝑦))(+g𝐺)𝑧) = ((0g𝐺)(+g𝐺)𝑧))
465, 8, 36, 38, 39, 35grpassd 18901 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((𝑦(+g𝐺)((invg𝐺)‘𝑦))(+g𝐺)𝑧) = (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)))
475, 8, 42, 36, 35grplidd 18925 . . . . . . 7 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((0g𝐺)(+g𝐺)𝑧) = 𝑧)
4845, 46, 473eqtr3d 2776 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) = 𝑧)
4948oveq2d 7436 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (((invg𝐺)‘𝑥)(+g𝐺)(𝑦(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧))) = (((invg𝐺)‘𝑥)(+g𝐺)𝑧))
5041, 49eqtrd 2768 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((((invg𝐺)‘𝑥)(+g𝐺)𝑦)(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) = (((invg𝐺)‘𝑥)(+g𝐺)𝑧))
51 simpl 482 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝑌 ∈ (SubGrp‘𝐺))
5222adantrr 716 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌)
5334simp3d 1142 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌)
548subgcl 19090 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑌 ∧ (((invg𝐺)‘𝑦)(+g𝐺)𝑧) ∈ 𝑌) → ((((invg𝐺)‘𝑥)(+g𝐺)𝑦)(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑌)
5551, 52, 53, 54syl3anc 1369 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → ((((invg𝐺)‘𝑥)(+g𝐺)𝑦)(+g𝐺)(((invg𝐺)‘𝑦)(+g𝐺)𝑧)) ∈ 𝑌)
5650, 55eqeltrrd 2830 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (((invg𝐺)‘𝑥)(+g𝐺)𝑧) ∈ 𝑌)
576adantr 480 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝑌𝑋)
585, 7, 8, 1eqgval 19131 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝑥 𝑧 ↔ (𝑥𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑧) ∈ 𝑌)))
5936, 57, 58syl2anc 583 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → (𝑥 𝑧 ↔ (𝑥𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑧) ∈ 𝑌)))
6030, 35, 56, 59mpbir3and 1340 . 2 ((𝑌 ∈ (SubGrp‘𝐺) ∧ (𝑥 𝑦𝑦 𝑧)) → 𝑥 𝑧)
615, 8, 42, 7grplinv 18945 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → (((invg𝐺)‘𝑥)(+g𝐺)𝑥) = (0g𝐺))
624, 61sylan 579 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (((invg𝐺)‘𝑥)(+g𝐺)𝑥) = (0g𝐺))
6342subg0cl 19088 . . . . . . 7 (𝑌 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑌)
6463adantr 480 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (0g𝐺) ∈ 𝑌)
6562, 64eqeltrd 2829 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌)
6665ex 412 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥𝑋 → (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌))
6766pm4.71rd 562 . . 3 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥𝑋 ↔ ((((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌𝑥𝑋)))
685, 7, 8, 1eqgval 19131 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝑥 𝑥 ↔ (𝑥𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌)))
694, 6, 68syl2anc 583 . . . 4 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 𝑥 ↔ (𝑥𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌)))
70 df-3an 1087 . . . . 5 ((𝑥𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌) ↔ ((𝑥𝑋𝑥𝑋) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌))
71 anidm 564 . . . . . 6 ((𝑥𝑋𝑥𝑋) ↔ 𝑥𝑋)
7271anbi2ci 624 . . . . 5 (((𝑥𝑋𝑥𝑋) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌) ↔ ((((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌𝑥𝑋))
7370, 72bitri 275 . . . 4 ((𝑥𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌) ↔ ((((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌𝑥𝑋))
7469, 73bitrdi 287 . . 3 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥 𝑥 ↔ ((((invg𝐺)‘𝑥)(+g𝐺)𝑥) ∈ 𝑌𝑥𝑋)))
7567, 74bitr4d 282 . 2 (𝑌 ∈ (SubGrp‘𝐺) → (𝑥𝑋𝑥 𝑥))
763, 29, 60, 75iserd 8750 1 (𝑌 ∈ (SubGrp‘𝐺) → Er 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  wss 3947   class class class wbr 5148  Rel wrel 5683  cfv 6548  (class class class)co 7420   Er wer 8721  Basecbs 17179  +gcplusg 17232  0gc0g 17420  Grpcgrp 18889  invgcminusg 18890  SubGrpcsubg 19074   ~QG cqg 19076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-er 8724  df-en 8964  df-dom 8965  df-sdom 8966  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-2 12305  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17180  df-ress 17209  df-plusg 17245  df-0g 17422  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-grp 18892  df-minusg 18893  df-subg 19077  df-eqg 19079
This theorem is referenced by:  eqg0el  19137  qusgrp  19140  qusadd  19142  lagsubg2  19148  lagsubg  19149  qus0subgadd  19153  ghmquskerlem1  19233  ghmquskerlem2  19235  ghmquskerlem3  19236  ghmqusker  19237  orbstafun  19261  orbstaval  19262  orbsta  19263  orbsta2  19264  sylow2blem1  19574  sylow2blem2  19575  sylow2blem3  19576  sylow3lem3  19583  sylow3lem4  19584  qusecsub  19789  2idlcpblrng  21164  qus2idrng  21166  qus1  21167  qusrhm  21169  qusmul2  21170  qusmulrng  21173  rngqipring1  21205  zndvds  21482  cldsubg  24014  qustgpopn  24023  qustgphaus  24026  tgptsmscls  24053  qusker  33061  qusvscpbl  33063  quslmod  33070  qusxpid  33075  qustrivr  33077  qusmul  33114  nsgqusf1olem3  33125  lmhmqusker  33127  ghmqusnsglem1  33129  ghmqusnsglem2  33130  ghmqusnsg  33131  rhmquskerlem  33140  rhmqusnsg  33143  qsidomlem1  33168  qsidomlem2  33169  qsnzr  33171  qsdrngilem  33205  qsdrnglem2  33207
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