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Theorem orbsta 19093
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 19091 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → Fun 𝐹)
6 simpr 485 . . . . . . . 8 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝐴𝑌)
76adantr 481 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑘𝑋) → 𝐴𝑌)
81gaf 19075 . . . . . . . . . 10 ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)⟶𝑌)
98adantr 481 . . . . . . . . 9 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → :(𝑋 × 𝑌)⟶𝑌)
109adantr 481 . . . . . . . 8 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑘𝑋) → :(𝑋 × 𝑌)⟶𝑌)
11 simpr 485 . . . . . . . 8 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑘𝑋) → 𝑘𝑋)
1210, 11, 7fovcdmd 7526 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑘𝑋) → (𝑘 𝐴) ∈ 𝑌)
13 eqid 2736 . . . . . . . 8 (𝑘 𝐴) = (𝑘 𝐴)
14 oveq1 7364 . . . . . . . . . 10 ( = 𝑘 → ( 𝐴) = (𝑘 𝐴))
1514eqeq1d 2738 . . . . . . . . 9 ( = 𝑘 → (( 𝐴) = (𝑘 𝐴) ↔ (𝑘 𝐴) = (𝑘 𝐴)))
1615rspcev 3581 . . . . . . . 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 19087 . . . . . . 7 (𝐴𝑂(𝑘 𝐴) ↔ (𝐴𝑌 ∧ (𝑘 𝐴) ∈ 𝑌 ∧ ∃𝑋 ( 𝐴) = (𝑘 𝐴)))
207, 12, 17, 19syl3anbrc 1343 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑘𝑋) → 𝐴𝑂(𝑘 𝐴))
21 ovex 7390 . . . . . . 7 (𝑘 𝐴) ∈ V
22 elecg 8691 . . . . . . 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 256 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑘𝑋) → (𝑘 𝐴) ∈ [𝐴]𝑂)
251, 2gastacl 19089 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝐻 ∈ (SubGrp‘𝐺))
261, 3eqger 18980 . . . . . 6 (𝐻 ∈ (SubGrp‘𝐺) → Er 𝑋)
2725, 26syl 17 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → Er 𝑋)
281fvexi 6856 . . . . . 6 𝑋 ∈ V
2928a1i 11 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝑋 ∈ V)
304, 24, 27, 29qliftf 8744 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → (Fun 𝐹𝐹:(𝑋 / )⟶[𝐴]𝑂))
315, 30mpbid 231 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝐹:(𝑋 / )⟶[𝐴]𝑂)
32 eqid 2736 . . . . 5 (𝑋 / ) = (𝑋 / )
33 fveqeq2 6851 . . . . . . 7 ([𝑧] = 𝑎 → ((𝐹‘[𝑧] ) = (𝐹𝑏) ↔ (𝐹𝑎) = (𝐹𝑏)))
34 eqeq1 2740 . . . . . . 7 ([𝑧] = 𝑎 → ([𝑧] = 𝑏𝑎 = 𝑏))
3533, 34imbi12d 344 . . . . . 6 ([𝑧] = 𝑎 → (((𝐹‘[𝑧] ) = (𝐹𝑏) → [𝑧] = 𝑏) ↔ ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏)))
3635ralbidv 3174 . . . . 5 ([𝑧] = 𝑎 → (∀𝑏 ∈ (𝑋 / )((𝐹‘[𝑧] ) = (𝐹𝑏) → [𝑧] = 𝑏) ↔ ∀𝑏 ∈ (𝑋 / )((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏)))
37 fveq2 6842 . . . . . . . . 9 ([𝑤] = 𝑏 → (𝐹‘[𝑤] ) = (𝐹𝑏))
3837eqeq2d 2747 . . . . . . . 8 ([𝑤] = 𝑏 → ((𝐹‘[𝑧] ) = (𝐹‘[𝑤] ) ↔ (𝐹‘[𝑧] ) = (𝐹𝑏)))
39 eqeq2 2748 . . . . . . . 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 19092 . . . . . . . . . . . 12 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑧𝑋) → (𝐹‘[𝑧] ) = (𝑧 𝐴))
4241adantrr 715 . . . . . . . . . . 11 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → (𝐹‘[𝑧] ) = (𝑧 𝐴))
431, 2, 3, 4orbstaval 19092 . . . . . . . . . . . 12 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑤𝑋) → (𝐹‘[𝑤] ) = (𝑤 𝐴))
4443adantrl 714 . . . . . . . . . . 11 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → (𝐹‘[𝑤] ) = (𝑤 𝐴))
4542, 44eqeq12d 2752 . . . . . . . . . 10 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹‘[𝑧] ) = (𝐹‘[𝑤] ) ↔ (𝑧 𝐴) = (𝑤 𝐴)))
461, 2, 3gastacos 19090 . . . . . . . . . 10 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → (𝑧 𝑤 ↔ (𝑧 𝐴) = (𝑤 𝐴)))
4727adantr 481 . . . . . . . . . . 11 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → Er 𝑋)
48 simprl 769 . . . . . . . . . . 11 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → 𝑧𝑋)
4947, 48erth 8697 . . . . . . . . . 10 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → (𝑧 𝑤 ↔ [𝑧] = [𝑤] ))
5045, 46, 493bitr2d 306 . . . . . . . . 9 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹‘[𝑧] ) = (𝐹‘[𝑤] ) ↔ [𝑧] = [𝑤] ))
5150biimpd 228 . . . . . . . 8 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹‘[𝑧] ) = (𝐹‘[𝑤] ) → [𝑧] = [𝑤] ))
5251anassrs 468 . . . . . . 7 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑧𝑋) ∧ 𝑤𝑋) → ((𝐹‘[𝑧] ) = (𝐹‘[𝑤] ) → [𝑧] = [𝑤] ))
5332, 40, 52ectocld 8723 . . . . . 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 8723 . . . 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 7202 . . 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 3449 . . . . . . . . 9 ∈ V
60 elecg 8691 . . . . . . . . 9 (( ∈ V ∧ 𝐴𝑌) → ( ∈ [𝐴]𝑂𝐴𝑂))
6159, 6, 60sylancr 587 . . . . . . . 8 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → ( ∈ [𝐴]𝑂𝐴𝑂))
6218gaorb 19087 . . . . . . . 8 (𝐴𝑂 ↔ (𝐴𝑌𝑌 ∧ ∃𝑤𝑋 (𝑤 𝐴) = ))
6361, 62bitrdi 286 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → ( ∈ [𝐴]𝑂 ↔ (𝐴𝑌𝑌 ∧ ∃𝑤𝑋 (𝑤 𝐴) = )))
6463biimpa 477 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ ∈ [𝐴]𝑂) → (𝐴𝑌𝑌 ∧ ∃𝑤𝑋 (𝑤 𝐴) = ))
6564simp3d 1144 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ ∈ [𝐴]𝑂) → ∃𝑤𝑋 (𝑤 𝐴) = )
663ovexi 7391 . . . . . . . . . 10 ∈ V
6766ecelqsi 8712 . . . . . . . . 9 (𝑤𝑋 → [𝑤] ∈ (𝑋 / ))
6843eqcomd 2742 . . . . . . . . 9 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑤𝑋) → (𝑤 𝐴) = (𝐹‘[𝑤] ))
69 fveq2 6842 . . . . . . . . . 10 (𝑧 = [𝑤] → (𝐹𝑧) = (𝐹‘[𝑤] ))
7069rspceeqv 3595 . . . . . . . . 9 (([𝑤] ∈ (𝑋 / ) ∧ (𝑤 𝐴) = (𝐹‘[𝑤] )) → ∃𝑧 ∈ (𝑋 / )(𝑤 𝐴) = (𝐹𝑧))
7167, 68, 70syl2an2 684 . . . . . . . 8 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑤𝑋) → ∃𝑧 ∈ (𝑋 / )(𝑤 𝐴) = (𝐹𝑧))
72 eqeq1 2740 . . . . . . . . 9 ((𝑤 𝐴) = → ((𝑤 𝐴) = (𝐹𝑧) ↔ = (𝐹𝑧)))
7372rexbidv 3175 . . . . . . . 8 ((𝑤 𝐴) = → (∃𝑧 ∈ (𝑋 / )(𝑤 𝐴) = (𝐹𝑧) ↔ ∃𝑧 ∈ (𝑋 / ) = (𝐹𝑧)))
7471, 73syl5ibcom 244 . . . . . . 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 407 . . . . 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 7052 . . 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 6503 . 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  wss 3910  {cpr 4588  cop 4592   class class class wbr 5105  {copab 5167  cmpt 5188   × cxp 5631  ran crn 5634  Fun wfun 6490  wf 6492  1-1wf1 6493  ontowfo 6494  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357   Er wer 8645  [cec 8646   / cqs 8647  Basecbs 17083  SubGrpcsubg 18922   ~QG cqg 18924   GrpAct cga 19069
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-ec 8650  df-qs 8654  df-map 8767  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  df-ga 19070
This theorem is referenced by:  orbsta2  19094
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