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Theorem gaorber 18203
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 5538 . . 3 Rel
32a1i 11 . 2 ( ∈ (𝐺 GrpAct 𝑌) → Rel )
4 simpr 477 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) → 𝑢 𝑣)
51gaorb 18202 . . . . 5 (𝑢 𝑣 ↔ (𝑢𝑌𝑣𝑌 ∧ ∃𝑋 ( 𝑢) = 𝑣))
64, 5sylib 210 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) → (𝑢𝑌𝑣𝑌 ∧ ∃𝑋 ( 𝑢) = 𝑣))
76simp2d 1123 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) → 𝑣𝑌)
86simp1d 1122 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) → 𝑢𝑌)
96simp3d 1124 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) → ∃𝑋 ( 𝑢) = 𝑣)
10 simpll 754 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) ∧ 𝑋) → ∈ (𝐺 GrpAct 𝑌))
11 simpr 477 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) ∧ 𝑋) → 𝑋)
128adantr 473 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) ∧ 𝑋) → 𝑢𝑌)
137adantr 473 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) ∧ 𝑋) → 𝑣𝑌)
14 gaorber.2 . . . . . . . 8 𝑋 = (Base‘𝐺)
15 eqid 2772 . . . . . . . 8 (invg𝐺) = (invg𝐺)
1614, 15gacan 18200 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑋𝑢𝑌𝑣𝑌)) → (( 𝑢) = 𝑣 ↔ (((invg𝐺)‘) 𝑣) = 𝑢))
1710, 11, 12, 13, 16syl13anc 1352 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) ∧ 𝑋) → (( 𝑢) = 𝑣 ↔ (((invg𝐺)‘) 𝑣) = 𝑢))
18 gagrp 18187 . . . . . . . . 9 ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
1918adantr 473 . . . . . . . 8 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) → 𝐺 ∈ Grp)
2014, 15grpinvcl 17932 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑋) → ((invg𝐺)‘) ∈ 𝑋)
2119, 20sylan 572 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) ∧ 𝑋) → ((invg𝐺)‘) ∈ 𝑋)
22 oveq1 6977 . . . . . . . . . 10 (𝑘 = ((invg𝐺)‘) → (𝑘 𝑣) = (((invg𝐺)‘) 𝑣))
2322eqeq1d 2774 . . . . . . . . 9 (𝑘 = ((invg𝐺)‘) → ((𝑘 𝑣) = 𝑢 ↔ (((invg𝐺)‘) 𝑣) = 𝑢))
2423rspcev 3529 . . . . . . . 8 ((((invg𝐺)‘) ∈ 𝑋 ∧ (((invg𝐺)‘) 𝑣) = 𝑢) → ∃𝑘𝑋 (𝑘 𝑣) = 𝑢)
2524ex 405 . . . . . . 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 232 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) ∧ 𝑋) → (( 𝑢) = 𝑣 → ∃𝑘𝑋 (𝑘 𝑣) = 𝑢))
2827rexlimdva 3223 . . . 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 18202 . . 3 (𝑣 𝑢 ↔ (𝑣𝑌𝑢𝑌 ∧ ∃𝑘𝑋 (𝑘 𝑣) = 𝑢))
317, 8, 29, 30syl3anbrc 1323 . 2 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢 𝑣) → 𝑣 𝑢)
328adantrr 704 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) → 𝑢𝑌)
33 simprr 760 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) → 𝑣 𝑤)
341gaorb 18202 . . . . 5 (𝑣 𝑤 ↔ (𝑣𝑌𝑤𝑌 ∧ ∃𝑘𝑋 (𝑘 𝑣) = 𝑤))
3533, 34sylib 210 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) → (𝑣𝑌𝑤𝑌 ∧ ∃𝑘𝑋 (𝑘 𝑣) = 𝑤))
3635simp2d 1123 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) → 𝑤𝑌)
379adantrr 704 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) → ∃𝑋 ( 𝑢) = 𝑣)
3835simp3d 1124 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) → ∃𝑘𝑋 (𝑘 𝑣) = 𝑤)
39 reeanv 3302 . . . . 5 (∃𝑋𝑘𝑋 (( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤) ↔ (∃𝑋 ( 𝑢) = 𝑣 ∧ ∃𝑘𝑋 (𝑘 𝑣) = 𝑤))
4018ad2antrr 713 . . . . . . . . 9 ((( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) ∧ ((𝑋𝑘𝑋) ∧ (( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤))) → 𝐺 ∈ Grp)
41 simprlr 767 . . . . . . . . 9 ((( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) ∧ ((𝑋𝑘𝑋) ∧ (( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤))) → 𝑘𝑋)
42 simprll 766 . . . . . . . . 9 ((( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) ∧ ((𝑋𝑘𝑋) ∧ (( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤))) → 𝑋)
43 eqid 2772 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
4414, 43grpcl 17893 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑘𝑋𝑋) → (𝑘(+g𝐺)) ∈ 𝑋)
4540, 41, 42, 44syl3anc 1351 . . . . . . . 8 ((( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) ∧ ((𝑋𝑘𝑋) ∧ (( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤))) → (𝑘(+g𝐺)) ∈ 𝑋)
46 simpll 754 . . . . . . . . . 10 ((( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) ∧ ((𝑋𝑘𝑋) ∧ (( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤))) → ∈ (𝐺 GrpAct 𝑌))
4732adantr 473 . . . . . . . . . 10 ((( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) ∧ ((𝑋𝑘𝑋) ∧ (( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤))) → 𝑢𝑌)
4814, 43gaass 18192 . . . . . . . . . 10 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑘𝑋𝑋𝑢𝑌)) → ((𝑘(+g𝐺)) 𝑢) = (𝑘 ( 𝑢)))
4946, 41, 42, 47, 48syl13anc 1352 . . . . . . . . 9 ((( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) ∧ ((𝑋𝑘𝑋) ∧ (( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤))) → ((𝑘(+g𝐺)) 𝑢) = (𝑘 ( 𝑢)))
50 simprrl 768 . . . . . . . . . 10 ((( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) ∧ ((𝑋𝑘𝑋) ∧ (( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤))) → ( 𝑢) = 𝑣)
5150oveq2d 6986 . . . . . . . . 9 ((( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) ∧ ((𝑋𝑘𝑋) ∧ (( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤))) → (𝑘 ( 𝑢)) = (𝑘 𝑣))
52 simprrr 769 . . . . . . . . 9 ((( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) ∧ ((𝑋𝑘𝑋) ∧ (( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤))) → (𝑘 𝑣) = 𝑤)
5349, 51, 523eqtrd 2812 . . . . . . . 8 ((( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) ∧ ((𝑋𝑘𝑋) ∧ (( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤))) → ((𝑘(+g𝐺)) 𝑢) = 𝑤)
54 oveq1 6977 . . . . . . . . . 10 (𝑓 = (𝑘(+g𝐺)) → (𝑓 𝑢) = ((𝑘(+g𝐺)) 𝑢))
5554eqeq1d 2774 . . . . . . . . 9 (𝑓 = (𝑘(+g𝐺)) → ((𝑓 𝑢) = 𝑤 ↔ ((𝑘(+g𝐺)) 𝑢) = 𝑤))
5655rspcev 3529 . . . . . . . 8 (((𝑘(+g𝐺)) ∈ 𝑋 ∧ ((𝑘(+g𝐺)) 𝑢) = 𝑤) → ∃𝑓𝑋 (𝑓 𝑢) = 𝑤)
5745, 53, 56syl2anc 576 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) ∧ ((𝑋𝑘𝑋) ∧ (( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤))) → ∃𝑓𝑋 (𝑓 𝑢) = 𝑤)
5857expr 449 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) ∧ (𝑋𝑘𝑋)) → ((( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤) → ∃𝑓𝑋 (𝑓 𝑢) = 𝑤))
5958rexlimdvva 3233 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) → (∃𝑋𝑘𝑋 (( 𝑢) = 𝑣 ∧ (𝑘 𝑣) = 𝑤) → ∃𝑓𝑋 (𝑓 𝑢) = 𝑤))
6039, 59syl5bir 235 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) → ((∃𝑋 ( 𝑢) = 𝑣 ∧ ∃𝑘𝑋 (𝑘 𝑣) = 𝑤) → ∃𝑓𝑋 (𝑓 𝑢) = 𝑤))
6137, 38, 60mp2and 686 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) → ∃𝑓𝑋 (𝑓 𝑢) = 𝑤)
621gaorb 18202 . . 3 (𝑢 𝑤 ↔ (𝑢𝑌𝑤𝑌 ∧ ∃𝑓𝑋 (𝑓 𝑢) = 𝑤))
6332, 36, 61, 62syl3anbrc 1323 . 2 (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝑢 𝑣𝑣 𝑤)) → 𝑢 𝑤)
6418adantr 473 . . . . . . . 8 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢𝑌) → 𝐺 ∈ Grp)
65 eqid 2772 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
6614, 65grpidcl 17913 . . . . . . . 8 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝑋)
6764, 66syl 17 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢𝑌) → (0g𝐺) ∈ 𝑋)
6865gagrpid 18189 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢𝑌) → ((0g𝐺) 𝑢) = 𝑢)
69 oveq1 6977 . . . . . . . . 9 ( = (0g𝐺) → ( 𝑢) = ((0g𝐺) 𝑢))
7069eqeq1d 2774 . . . . . . . 8 ( = (0g𝐺) → (( 𝑢) = 𝑢 ↔ ((0g𝐺) 𝑢) = 𝑢))
7170rspcev 3529 . . . . . . 7 (((0g𝐺) ∈ 𝑋 ∧ ((0g𝐺) 𝑢) = 𝑢) → ∃𝑋 ( 𝑢) = 𝑢)
7267, 68, 71syl2anc 576 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑢𝑌) → ∃𝑋 ( 𝑢) = 𝑢)
7372ex 405 . . . . 5 ( ∈ (𝐺 GrpAct 𝑌) → (𝑢𝑌 → ∃𝑋 ( 𝑢) = 𝑢))
7473pm4.71rd 555 . . . 4 ( ∈ (𝐺 GrpAct 𝑌) → (𝑢𝑌 ↔ (∃𝑋 ( 𝑢) = 𝑢𝑢𝑌)))
75 df-3an 1070 . . . . 5 ((𝑢𝑌𝑢𝑌 ∧ ∃𝑋 ( 𝑢) = 𝑢) ↔ ((𝑢𝑌𝑢𝑌) ∧ ∃𝑋 ( 𝑢) = 𝑢))
76 anidm 557 . . . . . 6 ((𝑢𝑌𝑢𝑌) ↔ 𝑢𝑌)
7776anbi2ci 615 . . . . 5 (((𝑢𝑌𝑢𝑌) ∧ ∃𝑋 ( 𝑢) = 𝑢) ↔ (∃𝑋 ( 𝑢) = 𝑢𝑢𝑌))
7875, 77bitri 267 . . . 4 ((𝑢𝑌𝑢𝑌 ∧ ∃𝑋 ( 𝑢) = 𝑢) ↔ (∃𝑋 ( 𝑢) = 𝑢𝑢𝑌))
7974, 78syl6bbr 281 . . 3 ( ∈ (𝐺 GrpAct 𝑌) → (𝑢𝑌 ↔ (𝑢𝑌𝑢𝑌 ∧ ∃𝑋 ( 𝑢) = 𝑢)))
801gaorb 18202 . . 3 (𝑢 𝑢 ↔ (𝑢𝑌𝑢𝑌 ∧ ∃𝑋 ( 𝑢) = 𝑢))
8179, 80syl6bbr 281 . 2 ( ∈ (𝐺 GrpAct 𝑌) → (𝑢𝑌𝑢 𝑢))
823, 31, 63, 81iserd 8109 1 ( ∈ (𝐺 GrpAct 𝑌) → Er 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1068   = wceq 1507  wcel 2050  wrex 3083  wss 3823  {cpr 4437   class class class wbr 4923  {copab 4985  Rel wrel 5406  cfv 6182  (class class class)co 6970   Er wer 8080  Basecbs 16333  +gcplusg 16415  0gc0g 16563  Grpcgrp 17885  invgcminusg 17886   GrpAct cga 18184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5306  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-fv 6190  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-er 8083  df-map 8202  df-0g 16565  df-mgm 17704  df-sgrp 17746  df-mnd 17757  df-grp 17888  df-minusg 17889  df-ga 18185
This theorem is referenced by:  sylow1lem3  18480  sylow1lem5  18482  sylow2alem1  18497  sylow2alem2  18498  sylow2a  18499  sylow3lem3  18509
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