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Theorem gaorber 18429
Description: The orbit equivalence relation is an equivalence relation on the target set of the group action. (Contributed by NM, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
Hypotheses
Ref Expression
gaorb.1 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}
gaorber.2 𝑋 = (Base‘𝐺)
Assertion
Ref Expression
gaorber ( ∈ (𝐺 GrpAct 𝑌) → Er 𝑌)
Distinct variable groups:   𝑥,𝑔,𝑦,   𝑔,𝑋,𝑥,𝑦   𝑥,𝑌,𝑦
Allowed substitution hints:   (𝑥,𝑦,𝑔)   𝐺(𝑥,𝑦,𝑔)   𝑌(𝑔)

Proof of Theorem gaorber
Dummy variables 𝑓 𝑘 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gaorb.1 . . . 4 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}
21relopabi 5671 . . 3 Rel
32a1i 11 . 2 ( ∈ (𝐺 GrpAct 𝑌) → Rel )
4 simpr 488 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) → 𝑢 𝑣)
51gaorb 18428 . . . . 5 (𝑢 𝑣 ↔ (𝑢𝑌𝑣𝑌 ∧ ∃𝑋 ( 𝑢) = 𝑣))
64, 5sylib 221 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) → (𝑢𝑌𝑣𝑌 ∧ ∃𝑋 ( 𝑢) = 𝑣))
76simp2d 1140 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) → 𝑣𝑌)
86simp1d 1139 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) → 𝑢𝑌)
96simp3d 1141 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) → ∃𝑋 ( 𝑢) = 𝑣)
10 simpll 766 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) ∧ 𝑋) → ∈ (𝐺 GrpAct 𝑌))
11 simpr 488 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) ∧ 𝑋) → 𝑋)
128adantr 484 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) ∧ 𝑋) → 𝑢𝑌)
137adantr 484 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) ∧ 𝑋) → 𝑣𝑌)
14 gaorber.2 . . . . . . . 8 𝑋 = (Base‘𝐺)
15 eqid 2822 . . . . . . . 8 (invg𝐺) = (invg𝐺)
1614, 15gacan 18426 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑋𝑢𝑌𝑣𝑌)) → (( 𝑢) = 𝑣 ↔ (((invg𝐺)‘) 𝑣) = 𝑢))
1710, 11, 12, 13, 16syl13anc 1369 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) ∧ 𝑋) → (( 𝑢) = 𝑣 ↔ (((invg𝐺)‘) 𝑣) = 𝑢))
18 gagrp 18413 . . . . . . . . 9 ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
1918adantr 484 . . . . . . . 8 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) → 𝐺 ∈ Grp)
2014, 15grpinvcl 18142 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑋) → ((invg𝐺)‘) ∈ 𝑋)
2119, 20sylan 583 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) ∧ 𝑋) → ((invg𝐺)‘) ∈ 𝑋)
22 oveq1 7147 . . . . . . . . . 10 (𝑘 = ((invg𝐺)‘) → (𝑘 𝑣) = (((invg𝐺)‘) 𝑣))
2322eqeq1d 2824 . . . . . . . . 9 (𝑘 = ((invg𝐺)‘) → ((𝑘 𝑣) = 𝑢 ↔ (((invg𝐺)‘) 𝑣) = 𝑢))
2423rspcev 3598 . . . . . . . 8 ((((invg𝐺)‘) ∈ 𝑋 ∧ (((invg𝐺)‘) 𝑣) = 𝑢) → ∃𝑘𝑋 (𝑘 𝑣) = 𝑢)
2524ex 416 . . . . . . 7 (((invg𝐺)‘) ∈ 𝑋 → ((((invg𝐺)‘) 𝑣) = 𝑢 → ∃𝑘𝑋 (𝑘 𝑣) = 𝑢))
2621, 25syl 17 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) ∧ 𝑋) → ((((invg𝐺)‘) 𝑣) = 𝑢 → ∃𝑘𝑋 (𝑘 𝑣) = 𝑢))
2717, 26sylbid 243 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) ∧ 𝑋) → (( 𝑢) = 𝑣 → ∃𝑘𝑋 (𝑘 𝑣) = 𝑢))
2827rexlimdva 3270 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) → (∃𝑋 ( 𝑢) = 𝑣 → ∃𝑘𝑋 (𝑘 𝑣) = 𝑢))
299, 28mpd 15 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) → ∃𝑘𝑋 (𝑘 𝑣) = 𝑢)
301gaorb 18428 . . 3 (𝑣 𝑢 ↔ (𝑣𝑌𝑢𝑌 ∧ ∃𝑘𝑋 (𝑘 𝑣) = 𝑢))
317, 8, 29, 30syl3anbrc 1340 . 2 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) → 𝑣 𝑢)
328adantrr 716 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) → 𝑢𝑌)
33 simprr 772 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) → 𝑣 𝑤)
341gaorb 18428 . . . . 5 (𝑣 𝑤 ↔ (𝑣𝑌𝑤𝑌 ∧ ∃𝑘𝑋 (𝑘 𝑣) = 𝑤))
3533, 34sylib 221 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) → (𝑣𝑌𝑤𝑌 ∧ ∃𝑘𝑋 (𝑘 𝑣) = 𝑤))
3635simp2d 1140 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) → 𝑤𝑌)
379adantrr 716 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) → ∃𝑋 ( 𝑢) = 𝑣)
3835simp3d 1141 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) → ∃𝑘𝑋 (𝑘 𝑣) = 𝑤)
39 reeanv 3348 . . . . 5 (∃𝑋𝑘𝑋 (( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤) ↔ (∃𝑋 ( 𝑢) = 𝑣 ∧ ∃𝑘𝑋 (𝑘 𝑣) = 𝑤))
4018ad2antrr 725 . . . . . . . . 9 ((( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) ∧ ((𝑋𝑘𝑋) ∧ (( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤))) → 𝐺 ∈ Grp)
41 simprlr 779 . . . . . . . . 9 ((( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) ∧ ((𝑋𝑘𝑋) ∧ (( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤))) → 𝑘𝑋)
42 simprll 778 . . . . . . . . 9 ((( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) ∧ ((𝑋𝑘𝑋) ∧ (( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤))) → 𝑋)
43 eqid 2822 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
4414, 43grpcl 18102 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑘𝑋𝑋) → (𝑘(+g𝐺)) ∈ 𝑋)
4540, 41, 42, 44syl3anc 1368 . . . . . . . 8 ((( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) ∧ ((𝑋𝑘𝑋) ∧ (( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤))) → (𝑘(+g𝐺)) ∈ 𝑋)
46 simpll 766 . . . . . . . . . 10 ((( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) ∧ ((𝑋𝑘𝑋) ∧ (( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤))) → ∈ (𝐺 GrpAct 𝑌))
4732adantr 484 . . . . . . . . . 10 ((( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) ∧ ((𝑋𝑘𝑋) ∧ (( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤))) → 𝑢𝑌)
4814, 43gaass 18418 . . . . . . . . . 10 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑘𝑋𝑋𝑢𝑌)) → ((𝑘(+g𝐺)) 𝑢) = (𝑘 ( 𝑢)))
4946, 41, 42, 47, 48syl13anc 1369 . . . . . . . . 9 ((( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) ∧ ((𝑋𝑘𝑋) ∧ (( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤))) → ((𝑘(+g𝐺)) 𝑢) = (𝑘 ( 𝑢)))
50 simprrl 780 . . . . . . . . . 10 ((( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) ∧ ((𝑋𝑘𝑋) ∧ (( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤))) → ( 𝑢) = 𝑣)
5150oveq2d 7156 . . . . . . . . 9 ((( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) ∧ ((𝑋𝑘𝑋) ∧ (( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤))) → (𝑘 ( 𝑢)) = (𝑘 𝑣))
52 simprrr 781 . . . . . . . . 9 ((( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) ∧ ((𝑋𝑘𝑋) ∧ (( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤))) → (𝑘 𝑣) = 𝑤)
5349, 51, 523eqtrd 2861 . . . . . . . 8 ((( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) ∧ ((𝑋𝑘𝑋) ∧ (( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤))) → ((𝑘(+g𝐺)) 𝑢) = 𝑤)
54 oveq1 7147 . . . . . . . . . 10 (𝑓 = (𝑘(+g𝐺)) → (𝑓 𝑢) = ((𝑘(+g𝐺)) 𝑢))
5554eqeq1d 2824 . . . . . . . . 9 (𝑓 = (𝑘(+g𝐺)) → ((𝑓 𝑢) = 𝑤 ↔ ((𝑘(+g𝐺)) 𝑢) = 𝑤))
5655rspcev 3598 . . . . . . . 8 (((𝑘(+g𝐺)) ∈ 𝑋 ∧ ((𝑘(+g𝐺)) 𝑢) = 𝑤) → ∃𝑓𝑋 (𝑓 𝑢) = 𝑤)
5745, 53, 56syl2anc 587 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) ∧ ((𝑋𝑘𝑋) ∧ (( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤))) → ∃𝑓𝑋 (𝑓 𝑢) = 𝑤)
5857expr 460 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) ∧ (𝑋𝑘𝑋)) → ((( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤) → ∃𝑓𝑋 (𝑓 𝑢) = 𝑤))
5958rexlimdvva 3280 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) → (∃𝑋𝑘𝑋 (( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤) → ∃𝑓𝑋 (𝑓 𝑢) = 𝑤))
6039, 59syl5bir 246 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) → ((∃𝑋 ( 𝑢) = 𝑣 ∧ ∃𝑘𝑋 (𝑘 𝑣) = 𝑤) → ∃𝑓𝑋 (𝑓 𝑢) = 𝑤))
6137, 38, 60mp2and 698 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) → ∃𝑓𝑋 (𝑓 𝑢) = 𝑤)
621gaorb 18428 . . 3 (𝑢 𝑤 ↔ (𝑢𝑌𝑤𝑌 ∧ ∃𝑓𝑋 (𝑓 𝑢) = 𝑤))
6332, 36, 61, 62syl3anbrc 1340 . 2 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) → 𝑢 𝑤)
6418adantr 484 . . . . . . . 8 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢𝑌) → 𝐺 ∈ Grp)
65 eqid 2822 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
6614, 65grpidcl 18122 . . . . . . . 8 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝑋)
6764, 66syl 17 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢𝑌) → (0g𝐺) ∈ 𝑋)
6865gagrpid 18415 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢𝑌) → ((0g𝐺) 𝑢) = 𝑢)
69 oveq1 7147 . . . . . . . . 9 ( = (0g𝐺) → ( 𝑢) = ((0g𝐺) 𝑢))
7069eqeq1d 2824 . . . . . . . 8 ( = (0g𝐺) → (( 𝑢) = 𝑢 ↔ ((0g𝐺) 𝑢) = 𝑢))
7170rspcev 3598 . . . . . . 7 (((0g𝐺) ∈ 𝑋 ∧ ((0g𝐺) 𝑢) = 𝑢) → ∃𝑋 ( 𝑢) = 𝑢)
7267, 68, 71syl2anc 587 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢𝑌) → ∃𝑋 ( 𝑢) = 𝑢)
7372ex 416 . . . . 5 ( ∈ (𝐺 GrpAct 𝑌) → (𝑢𝑌 → ∃𝑋 ( 𝑢) = 𝑢))
7473pm4.71rd 566 . . . 4 ( ∈ (𝐺 GrpAct 𝑌) → (𝑢𝑌 ↔ (∃𝑋 ( 𝑢) = 𝑢𝑢𝑌)))
75 df-3an 1086 . . . . 5 ((𝑢𝑌𝑢𝑌 ∧ ∃𝑋 ( 𝑢) = 𝑢) ↔ ((𝑢𝑌𝑢𝑌) ∧ ∃𝑋 ( 𝑢) = 𝑢))
76 anidm 568 . . . . . 6 ((𝑢𝑌𝑢𝑌) ↔ 𝑢𝑌)
7776anbi2ci 627 . . . . 5 (((𝑢𝑌𝑢𝑌) ∧ ∃𝑋 ( 𝑢) = 𝑢) ↔ (∃𝑋 ( 𝑢) = 𝑢𝑢𝑌))
7875, 77bitri 278 . . . 4 ((𝑢𝑌𝑢𝑌 ∧ ∃𝑋 ( 𝑢) = 𝑢) ↔ (∃𝑋 ( 𝑢) = 𝑢𝑢𝑌))
7974, 78syl6bbr 292 . . 3 ( ∈ (𝐺 GrpAct 𝑌) → (𝑢𝑌 ↔ (𝑢𝑌𝑢𝑌 ∧ ∃𝑋 ( 𝑢) = 𝑢)))
801gaorb 18428 . . 3 (𝑢 𝑢 ↔ (𝑢𝑌𝑢𝑌 ∧ ∃𝑋 ( 𝑢) = 𝑢))
8179, 80syl6bbr 292 . 2 ( ∈ (𝐺 GrpAct 𝑌) → (𝑢𝑌𝑢 𝑢))
823, 31, 63, 81iserd 8302 1 ( ∈ (𝐺 GrpAct 𝑌) → Er 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2114  wrex 3131  wss 3908  {cpr 4541   class class class wbr 5042  {copab 5104  Rel wrel 5537  cfv 6334  (class class class)co 7140   Er wer 8273  Basecbs 16474  +gcplusg 16556  0gc0g 16704  Grpcgrp 18094  invgcminusg 18095   GrpAct cga 18410
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-er 8276  df-map 8395  df-0g 16706  df-mgm 17843  df-sgrp 17892  df-mnd 17903  df-grp 18097  df-minusg 18098  df-ga 18411
This theorem is referenced by:  sylow1lem3  18716  sylow1lem5  18718  sylow2alem1  18733  sylow2alem2  18734  sylow2a  18735  sylow3lem3  18745
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