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Theorem orbsta 19343
Description: The Orbit-Stabilizer theorem. The mapping 𝐹 is a bijection from the cosets of the stabilizer subgroup of 𝐴 to the orbit of 𝐴. (Contributed by Mario Carneiro, 15-Jan-2015.)
Hypotheses
Ref Expression
gasta.1 𝑋 = (Base‘𝐺)
gasta.2 𝐻 = {𝑢𝑋 ∣ (𝑢 𝐴) = 𝐴}
orbsta.r = (𝐺 ~QG 𝐻)
orbsta.f 𝐹 = ran (𝑘𝑋 ↦ ⟨[𝑘] , (𝑘 𝐴)⟩)
orbsta.o 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}
Assertion
Ref Expression
orbsta (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝐹:(𝑋 / )–1-1-onto→[𝐴]𝑂)
Distinct variable groups:   𝑔,𝑘,𝑥,𝑦,   𝑢,𝑔, ,𝑘,𝑥,𝑦   𝑥,𝐻,𝑦   𝐴,𝑔,𝑘,𝑢,𝑥,𝑦   𝑔,𝐺,𝑘,𝑢,𝑥,𝑦   𝑔,𝑋,𝑘,𝑢,𝑥,𝑦   𝑘,𝑂   𝑔,𝑌,𝑘,𝑥,𝑦
Allowed substitution hints:   (𝑢)   𝐹(𝑥,𝑦,𝑢,𝑔,𝑘)   𝐻(𝑢,𝑔,𝑘)   𝑂(𝑥,𝑦,𝑢,𝑔)   𝑌(𝑢)

Proof of Theorem orbsta
Dummy variables 𝑎 𝑏 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gasta.1 . . . . 5 𝑋 = (Base‘𝐺)
2 gasta.2 . . . . 5 𝐻 = {𝑢𝑋 ∣ (𝑢 𝐴) = 𝐴}
3 orbsta.r . . . . 5 = (𝐺 ~QG 𝐻)
4 orbsta.f . . . . 5 𝐹 = ran (𝑘𝑋 ↦ ⟨[𝑘] , (𝑘 𝐴)⟩)
51, 2, 3, 4orbstafun 19341 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → Fun 𝐹)
6 simpr 484 . . . . . . . 8 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝐴𝑌)
76adantr 480 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑘𝑋) → 𝐴𝑌)
81gaf 19325 . . . . . . . . . 10 ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)⟶𝑌)
98adantr 480 . . . . . . . . 9 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → :(𝑋 × 𝑌)⟶𝑌)
109adantr 480 . . . . . . . 8 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑘𝑋) → :(𝑋 × 𝑌)⟶𝑌)
11 simpr 484 . . . . . . . 8 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑘𝑋) → 𝑘𝑋)
1210, 11, 7fovcdmd 7604 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑘𝑋) → (𝑘 𝐴) ∈ 𝑌)
13 eqid 2734 . . . . . . . 8 (𝑘 𝐴) = (𝑘 𝐴)
14 oveq1 7437 . . . . . . . . . 10 ( = 𝑘 → ( 𝐴) = (𝑘 𝐴))
1514eqeq1d 2736 . . . . . . . . 9 ( = 𝑘 → (( 𝐴) = (𝑘 𝐴) ↔ (𝑘 𝐴) = (𝑘 𝐴)))
1615rspcev 3621 . . . . . . . 8 ((𝑘𝑋 ∧ (𝑘 𝐴) = (𝑘 𝐴)) → ∃𝑋 ( 𝐴) = (𝑘 𝐴))
1711, 13, 16sylancl 586 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑘𝑋) → ∃𝑋 ( 𝐴) = (𝑘 𝐴))
18 orbsta.o . . . . . . . 8 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}
1918gaorb 19337 . . . . . . 7 (𝐴𝑂(𝑘 𝐴) ↔ (𝐴𝑌 ∧ (𝑘 𝐴) ∈ 𝑌 ∧ ∃𝑋 ( 𝐴) = (𝑘 𝐴)))
207, 12, 17, 19syl3anbrc 1342 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑘𝑋) → 𝐴𝑂(𝑘 𝐴))
21 ovex 7463 . . . . . . 7 (𝑘 𝐴) ∈ V
22 elecg 8787 . . . . . . 7 (((𝑘 𝐴) ∈ V ∧ 𝐴𝑌) → ((𝑘 𝐴) ∈ [𝐴]𝑂𝐴𝑂(𝑘 𝐴)))
2321, 7, 22sylancr 587 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑘𝑋) → ((𝑘 𝐴) ∈ [𝐴]𝑂𝐴𝑂(𝑘 𝐴)))
2420, 23mpbird 257 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑘𝑋) → (𝑘 𝐴) ∈ [𝐴]𝑂)
251, 2gastacl 19339 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝐻 ∈ (SubGrp‘𝐺))
261, 3eqger 19208 . . . . . 6 (𝐻 ∈ (SubGrp‘𝐺) → Er 𝑋)
2725, 26syl 17 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → Er 𝑋)
281fvexi 6920 . . . . . 6 𝑋 ∈ V
2928a1i 11 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝑋 ∈ V)
304, 24, 27, 29qliftf 8843 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → (Fun 𝐹𝐹:(𝑋 / )⟶[𝐴]𝑂))
315, 30mpbid 232 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝐹:(𝑋 / )⟶[𝐴]𝑂)
32 eqid 2734 . . . . 5 (𝑋 / ) = (𝑋 / )
33 fveqeq2 6915 . . . . . . 7 ([𝑧] = 𝑎 → ((𝐹‘[𝑧] ) = (𝐹𝑏) ↔ (𝐹𝑎) = (𝐹𝑏)))
34 eqeq1 2738 . . . . . . 7 ([𝑧] = 𝑎 → ([𝑧] = 𝑏𝑎 = 𝑏))
3533, 34imbi12d 344 . . . . . 6 ([𝑧] = 𝑎 → (((𝐹‘[𝑧] ) = (𝐹𝑏) → [𝑧] = 𝑏) ↔ ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏)))
3635ralbidv 3175 . . . . 5 ([𝑧] = 𝑎 → (∀𝑏 ∈ (𝑋 / )((𝐹‘[𝑧] ) = (𝐹𝑏) → [𝑧] = 𝑏) ↔ ∀𝑏 ∈ (𝑋 / )((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏)))
37 fveq2 6906 . . . . . . . . 9 ([𝑤] = 𝑏 → (𝐹‘[𝑤] ) = (𝐹𝑏))
3837eqeq2d 2745 . . . . . . . 8 ([𝑤] = 𝑏 → ((𝐹‘[𝑧] ) = (𝐹‘[𝑤] ) ↔ (𝐹‘[𝑧] ) = (𝐹𝑏)))
39 eqeq2 2746 . . . . . . . 8 ([𝑤] = 𝑏 → ([𝑧] = [𝑤] ↔ [𝑧] = 𝑏))
4038, 39imbi12d 344 . . . . . . 7 ([𝑤] = 𝑏 → (((𝐹‘[𝑧] ) = (𝐹‘[𝑤] ) → [𝑧] = [𝑤] ) ↔ ((𝐹‘[𝑧] ) = (𝐹𝑏) → [𝑧] = 𝑏)))
411, 2, 3, 4orbstaval 19342 . . . . . . . . . . . 12 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑧𝑋) → (𝐹‘[𝑧] ) = (𝑧 𝐴))
4241adantrr 717 . . . . . . . . . . 11 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → (𝐹‘[𝑧] ) = (𝑧 𝐴))
431, 2, 3, 4orbstaval 19342 . . . . . . . . . . . 12 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑤𝑋) → (𝐹‘[𝑤] ) = (𝑤 𝐴))
4443adantrl 716 . . . . . . . . . . 11 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → (𝐹‘[𝑤] ) = (𝑤 𝐴))
4542, 44eqeq12d 2750 . . . . . . . . . 10 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹‘[𝑧] ) = (𝐹‘[𝑤] ) ↔ (𝑧 𝐴) = (𝑤 𝐴)))
461, 2, 3gastacos 19340 . . . . . . . . . 10 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → (𝑧 𝑤 ↔ (𝑧 𝐴) = (𝑤 𝐴)))
4727adantr 480 . . . . . . . . . . 11 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → Er 𝑋)
48 simprl 771 . . . . . . . . . . 11 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → 𝑧𝑋)
4947, 48erth 8794 . . . . . . . . . 10 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → (𝑧 𝑤 ↔ [𝑧] = [𝑤] ))
5045, 46, 493bitr2d 307 . . . . . . . . 9 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹‘[𝑧] ) = (𝐹‘[𝑤] ) ↔ [𝑧] = [𝑤] ))
5150biimpd 229 . . . . . . . 8 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹‘[𝑧] ) = (𝐹‘[𝑤] ) → [𝑧] = [𝑤] ))
5251anassrs 467 . . . . . . 7 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑧𝑋) ∧ 𝑤𝑋) → ((𝐹‘[𝑧] ) = (𝐹‘[𝑤] ) → [𝑧] = [𝑤] ))
5332, 40, 52ectocld 8822 . . . . . 6 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑧𝑋) ∧ 𝑏 ∈ (𝑋 / )) → ((𝐹‘[𝑧] ) = (𝐹𝑏) → [𝑧] = 𝑏))
5453ralrimiva 3143 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑧𝑋) → ∀𝑏 ∈ (𝑋 / )((𝐹‘[𝑧] ) = (𝐹𝑏) → [𝑧] = 𝑏))
5532, 36, 54ectocld 8822 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑎 ∈ (𝑋 / )) → ∀𝑏 ∈ (𝑋 / )((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
5655ralrimiva 3143 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → ∀𝑎 ∈ (𝑋 / )∀𝑏 ∈ (𝑋 / )((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
57 dff13 7274 . . 3 (𝐹:(𝑋 / )–1-1→[𝐴]𝑂 ↔ (𝐹:(𝑋 / )⟶[𝐴]𝑂 ∧ ∀𝑎 ∈ (𝑋 / )∀𝑏 ∈ (𝑋 / )((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏)))
5831, 56, 57sylanbrc 583 . 2 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝐹:(𝑋 / )–1-1→[𝐴]𝑂)
59 vex 3481 . . . . . . . . 9 ∈ V
60 elecg 8787 . . . . . . . . 9 (( ∈ V ∧ 𝐴𝑌) → ( ∈ [𝐴]𝑂𝐴𝑂))
6159, 6, 60sylancr 587 . . . . . . . 8 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → ( ∈ [𝐴]𝑂𝐴𝑂))
6218gaorb 19337 . . . . . . . 8 (𝐴𝑂 ↔ (𝐴𝑌𝑌 ∧ ∃𝑤𝑋 (𝑤 𝐴) = ))
6361, 62bitrdi 287 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → ( ∈ [𝐴]𝑂 ↔ (𝐴𝑌𝑌 ∧ ∃𝑤𝑋 (𝑤 𝐴) = )))
6463biimpa 476 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ ∈ [𝐴]𝑂) → (𝐴𝑌𝑌 ∧ ∃𝑤𝑋 (𝑤 𝐴) = ))
6564simp3d 1143 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ ∈ [𝐴]𝑂) → ∃𝑤𝑋 (𝑤 𝐴) = )
663ovexi 7464 . . . . . . . . . 10 ∈ V
6766ecelqsi 8811 . . . . . . . . 9 (𝑤𝑋 → [𝑤] ∈ (𝑋 / ))
6843eqcomd 2740 . . . . . . . . 9 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑤𝑋) → (𝑤 𝐴) = (𝐹‘[𝑤] ))
69 fveq2 6906 . . . . . . . . . 10 (𝑧 = [𝑤] → (𝐹𝑧) = (𝐹‘[𝑤] ))
7069rspceeqv 3644 . . . . . . . . 9 (([𝑤] ∈ (𝑋 / ) ∧ (𝑤 𝐴) = (𝐹‘[𝑤] )) → ∃𝑧 ∈ (𝑋 / )(𝑤 𝐴) = (𝐹𝑧))
7167, 68, 70syl2an2 686 . . . . . . . 8 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑤𝑋) → ∃𝑧 ∈ (𝑋 / )(𝑤 𝐴) = (𝐹𝑧))
72 eqeq1 2738 . . . . . . . . 9 ((𝑤 𝐴) = → ((𝑤 𝐴) = (𝐹𝑧) ↔ = (𝐹𝑧)))
7372rexbidv 3176 . . . . . . . 8 ((𝑤 𝐴) = → (∃𝑧 ∈ (𝑋 / )(𝑤 𝐴) = (𝐹𝑧) ↔ ∃𝑧 ∈ (𝑋 / ) = (𝐹𝑧)))
7471, 73syl5ibcom 245 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑤𝑋) → ((𝑤 𝐴) = → ∃𝑧 ∈ (𝑋 / ) = (𝐹𝑧)))
7574rexlimdva 3152 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → (∃𝑤𝑋 (𝑤 𝐴) = → ∃𝑧 ∈ (𝑋 / ) = (𝐹𝑧)))
7675imp 406 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ ∃𝑤𝑋 (𝑤 𝐴) = ) → ∃𝑧 ∈ (𝑋 / ) = (𝐹𝑧))
7765, 76syldan 591 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ ∈ [𝐴]𝑂) → ∃𝑧 ∈ (𝑋 / ) = (𝐹𝑧))
7877ralrimiva 3143 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → ∀ ∈ [ 𝐴]𝑂𝑧 ∈ (𝑋 / ) = (𝐹𝑧))
79 dffo3 7121 . . 3 (𝐹:(𝑋 / )–onto→[𝐴]𝑂 ↔ (𝐹:(𝑋 / )⟶[𝐴]𝑂 ∧ ∀ ∈ [ 𝐴]𝑂𝑧 ∈ (𝑋 / ) = (𝐹𝑧)))
8031, 78, 79sylanbrc 583 . 2 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝐹:(𝑋 / )–onto→[𝐴]𝑂)
81 df-f1o 6569 . 2 (𝐹:(𝑋 / )–1-1-onto→[𝐴]𝑂 ↔ (𝐹:(𝑋 / )–1-1→[𝐴]𝑂𝐹:(𝑋 / )–onto→[𝐴]𝑂))
8258, 80, 81sylanbrc 583 1 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝐹:(𝑋 / )–1-1-onto→[𝐴]𝑂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058  wrex 3067  {crab 3432  Vcvv 3477  wss 3962  {cpr 4632  cop 4636   class class class wbr 5147  {copab 5209  cmpt 5230   × cxp 5686  ran crn 5689  Fun wfun 6556  wf 6558  1-1wf1 6559  ontowfo 6560  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430   Er wer 8740  [cec 8741   / cqs 8742  Basecbs 17244  SubGrpcsubg 19150   ~QG cqg 19152   GrpAct cga 19319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-ec 8745  df-qs 8749  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-0g 17487  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-grp 18966  df-minusg 18967  df-subg 19153  df-eqg 19155  df-ga 19320
This theorem is referenced by:  orbsta2  19344
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