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Theorem orbsta 17670
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 17668 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → Fun 𝐹)
6 simpr 477 . . . . . . . 8 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝐴𝑌)
76adantr 481 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑘𝑋) → 𝐴𝑌)
81gaf 17652 . . . . . . . . . 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 477 . . . . . . . 8 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑘𝑋) → 𝑘𝑋)
1210, 11, 7fovrnd 6762 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑘𝑋) → (𝑘 𝐴) ∈ 𝑌)
13 eqid 2621 . . . . . . . 8 (𝑘 𝐴) = (𝑘 𝐴)
14 oveq1 6614 . . . . . . . . . 10 ( = 𝑘 → ( 𝐴) = (𝑘 𝐴))
1514eqeq1d 2623 . . . . . . . . 9 ( = 𝑘 → (( 𝐴) = (𝑘 𝐴) ↔ (𝑘 𝐴) = (𝑘 𝐴)))
1615rspcev 3295 . . . . . . . 8 ((𝑘𝑋 ∧ (𝑘 𝐴) = (𝑘 𝐴)) → ∃𝑋 ( 𝐴) = (𝑘 𝐴))
1711, 13, 16sylancl 693 . . . . . . 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 17664 . . . . . . 7 (𝐴𝑂(𝑘 𝐴) ↔ (𝐴𝑌 ∧ (𝑘 𝐴) ∈ 𝑌 ∧ ∃𝑋 ( 𝐴) = (𝑘 𝐴)))
207, 12, 17, 19syl3anbrc 1244 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑘𝑋) → 𝐴𝑂(𝑘 𝐴))
21 ovex 6635 . . . . . . 7 (𝑘 𝐴) ∈ V
22 elecg 7733 . . . . . . 7 (((𝑘 𝐴) ∈ V ∧ 𝐴𝑌) → ((𝑘 𝐴) ∈ [𝐴]𝑂𝐴𝑂(𝑘 𝐴)))
2321, 7, 22sylancr 694 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑘𝑋) → ((𝑘 𝐴) ∈ [𝐴]𝑂𝐴𝑂(𝑘 𝐴)))
2420, 23mpbird 247 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑘𝑋) → (𝑘 𝐴) ∈ [𝐴]𝑂)
251, 2gastacl 17666 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝐻 ∈ (SubGrp‘𝐺))
261, 3eqger 17568 . . . . . 6 (𝐻 ∈ (SubGrp‘𝐺) → Er 𝑋)
2725, 26syl 17 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → Er 𝑋)
28 fvex 6160 . . . . . . 7 (Base‘𝐺) ∈ V
291, 28eqeltri 2694 . . . . . 6 𝑋 ∈ V
3029a1i 11 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝑋 ∈ V)
314, 24, 27, 30qliftf 7783 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → (Fun 𝐹𝐹:(𝑋 / )⟶[𝐴]𝑂))
325, 31mpbid 222 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝐹:(𝑋 / )⟶[𝐴]𝑂)
33 eqid 2621 . . . . 5 (𝑋 / ) = (𝑋 / )
34 fveq2 6150 . . . . . . . 8 ([𝑧] = 𝑎 → (𝐹‘[𝑧] ) = (𝐹𝑎))
3534eqeq1d 2623 . . . . . . 7 ([𝑧] = 𝑎 → ((𝐹‘[𝑧] ) = (𝐹𝑏) ↔ (𝐹𝑎) = (𝐹𝑏)))
36 eqeq1 2625 . . . . . . 7 ([𝑧] = 𝑎 → ([𝑧] = 𝑏𝑎 = 𝑏))
3735, 36imbi12d 334 . . . . . 6 ([𝑧] = 𝑎 → (((𝐹‘[𝑧] ) = (𝐹𝑏) → [𝑧] = 𝑏) ↔ ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏)))
3837ralbidv 2980 . . . . 5 ([𝑧] = 𝑎 → (∀𝑏 ∈ (𝑋 / )((𝐹‘[𝑧] ) = (𝐹𝑏) → [𝑧] = 𝑏) ↔ ∀𝑏 ∈ (𝑋 / )((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏)))
39 fveq2 6150 . . . . . . . . 9 ([𝑤] = 𝑏 → (𝐹‘[𝑤] ) = (𝐹𝑏))
4039eqeq2d 2631 . . . . . . . 8 ([𝑤] = 𝑏 → ((𝐹‘[𝑧] ) = (𝐹‘[𝑤] ) ↔ (𝐹‘[𝑧] ) = (𝐹𝑏)))
41 eqeq2 2632 . . . . . . . 8 ([𝑤] = 𝑏 → ([𝑧] = [𝑤] ↔ [𝑧] = 𝑏))
4240, 41imbi12d 334 . . . . . . 7 ([𝑤] = 𝑏 → (((𝐹‘[𝑧] ) = (𝐹‘[𝑤] ) → [𝑧] = [𝑤] ) ↔ ((𝐹‘[𝑧] ) = (𝐹𝑏) → [𝑧] = 𝑏)))
431, 2, 3, 4orbstaval 17669 . . . . . . . . . . . 12 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑧𝑋) → (𝐹‘[𝑧] ) = (𝑧 𝐴))
4443adantrr 752 . . . . . . . . . . 11 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → (𝐹‘[𝑧] ) = (𝑧 𝐴))
451, 2, 3, 4orbstaval 17669 . . . . . . . . . . . 12 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑤𝑋) → (𝐹‘[𝑤] ) = (𝑤 𝐴))
4645adantrl 751 . . . . . . . . . . 11 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → (𝐹‘[𝑤] ) = (𝑤 𝐴))
4744, 46eqeq12d 2636 . . . . . . . . . 10 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹‘[𝑧] ) = (𝐹‘[𝑤] ) ↔ (𝑧 𝐴) = (𝑤 𝐴)))
481, 2, 3gastacos 17667 . . . . . . . . . 10 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → (𝑧 𝑤 ↔ (𝑧 𝐴) = (𝑤 𝐴)))
4927adantr 481 . . . . . . . . . . 11 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → Er 𝑋)
50 simprl 793 . . . . . . . . . . 11 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → 𝑧𝑋)
5149, 50erth 7739 . . . . . . . . . 10 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → (𝑧 𝑤 ↔ [𝑧] = [𝑤] ))
5247, 48, 513bitr2d 296 . . . . . . . . 9 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹‘[𝑧] ) = (𝐹‘[𝑤] ) ↔ [𝑧] = [𝑤] ))
5352biimpd 219 . . . . . . . 8 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹‘[𝑧] ) = (𝐹‘[𝑤] ) → [𝑧] = [𝑤] ))
5453anassrs 679 . . . . . . 7 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑧𝑋) ∧ 𝑤𝑋) → ((𝐹‘[𝑧] ) = (𝐹‘[𝑤] ) → [𝑧] = [𝑤] ))
5533, 42, 54ectocld 7762 . . . . . 6 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑧𝑋) ∧ 𝑏 ∈ (𝑋 / )) → ((𝐹‘[𝑧] ) = (𝐹𝑏) → [𝑧] = 𝑏))
5655ralrimiva 2960 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑧𝑋) → ∀𝑏 ∈ (𝑋 / )((𝐹‘[𝑧] ) = (𝐹𝑏) → [𝑧] = 𝑏))
5733, 38, 56ectocld 7762 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑎 ∈ (𝑋 / )) → ∀𝑏 ∈ (𝑋 / )((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
5857ralrimiva 2960 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → ∀𝑎 ∈ (𝑋 / )∀𝑏 ∈ (𝑋 / )((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
59 dff13 6469 . . 3 (𝐹:(𝑋 / )–1-1→[𝐴]𝑂 ↔ (𝐹:(𝑋 / )⟶[𝐴]𝑂 ∧ ∀𝑎 ∈ (𝑋 / )∀𝑏 ∈ (𝑋 / )((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏)))
6032, 58, 59sylanbrc 697 . 2 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝐹:(𝑋 / )–1-1→[𝐴]𝑂)
61 vex 3189 . . . . . . . . 9 ∈ V
62 elecg 7733 . . . . . . . . 9 (( ∈ V ∧ 𝐴𝑌) → ( ∈ [𝐴]𝑂𝐴𝑂))
6361, 6, 62sylancr 694 . . . . . . . 8 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → ( ∈ [𝐴]𝑂𝐴𝑂))
6418gaorb 17664 . . . . . . . 8 (𝐴𝑂 ↔ (𝐴𝑌𝑌 ∧ ∃𝑤𝑋 (𝑤 𝐴) = ))
6563, 64syl6bb 276 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → ( ∈ [𝐴]𝑂 ↔ (𝐴𝑌𝑌 ∧ ∃𝑤𝑋 (𝑤 𝐴) = )))
6665biimpa 501 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ ∈ [𝐴]𝑂) → (𝐴𝑌𝑌 ∧ ∃𝑤𝑋 (𝑤 𝐴) = ))
6766simp3d 1073 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ ∈ [𝐴]𝑂) → ∃𝑤𝑋 (𝑤 𝐴) = )
68 ovex 6635 . . . . . . . . . . . 12 (𝐺 ~QG 𝐻) ∈ V
693, 68eqeltri 2694 . . . . . . . . . . 11 ∈ V
7069ecelqsi 7751 . . . . . . . . . 10 (𝑤𝑋 → [𝑤] ∈ (𝑋 / ))
7170adantl 482 . . . . . . . . 9 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑤𝑋) → [𝑤] ∈ (𝑋 / ))
7245eqcomd 2627 . . . . . . . . 9 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑤𝑋) → (𝑤 𝐴) = (𝐹‘[𝑤] ))
73 fveq2 6150 . . . . . . . . . . 11 (𝑧 = [𝑤] → (𝐹𝑧) = (𝐹‘[𝑤] ))
7473eqeq2d 2631 . . . . . . . . . 10 (𝑧 = [𝑤] → ((𝑤 𝐴) = (𝐹𝑧) ↔ (𝑤 𝐴) = (𝐹‘[𝑤] )))
7574rspcev 3295 . . . . . . . . 9 (([𝑤] ∈ (𝑋 / ) ∧ (𝑤 𝐴) = (𝐹‘[𝑤] )) → ∃𝑧 ∈ (𝑋 / )(𝑤 𝐴) = (𝐹𝑧))
7671, 72, 75syl2anc 692 . . . . . . . 8 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑤𝑋) → ∃𝑧 ∈ (𝑋 / )(𝑤 𝐴) = (𝐹𝑧))
77 eqeq1 2625 . . . . . . . . 9 ((𝑤 𝐴) = → ((𝑤 𝐴) = (𝐹𝑧) ↔ = (𝐹𝑧)))
7877rexbidv 3045 . . . . . . . 8 ((𝑤 𝐴) = → (∃𝑧 ∈ (𝑋 / )(𝑤 𝐴) = (𝐹𝑧) ↔ ∃𝑧 ∈ (𝑋 / ) = (𝐹𝑧)))
7976, 78syl5ibcom 235 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑤𝑋) → ((𝑤 𝐴) = → ∃𝑧 ∈ (𝑋 / ) = (𝐹𝑧)))
8079rexlimdva 3024 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → (∃𝑤𝑋 (𝑤 𝐴) = → ∃𝑧 ∈ (𝑋 / ) = (𝐹𝑧)))
8180imp 445 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ ∃𝑤𝑋 (𝑤 𝐴) = ) → ∃𝑧 ∈ (𝑋 / ) = (𝐹𝑧))
8267, 81syldan 487 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ ∈ [𝐴]𝑂) → ∃𝑧 ∈ (𝑋 / ) = (𝐹𝑧))
8382ralrimiva 2960 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → ∀ ∈ [ 𝐴]𝑂𝑧 ∈ (𝑋 / ) = (𝐹𝑧))
84 dffo3 6332 . . 3 (𝐹:(𝑋 / )–onto→[𝐴]𝑂 ↔ (𝐹:(𝑋 / )⟶[𝐴]𝑂 ∧ ∀ ∈ [ 𝐴]𝑂𝑧 ∈ (𝑋 / ) = (𝐹𝑧)))
8532, 83, 84sylanbrc 697 . 2 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝐹:(𝑋 / )–onto→[𝐴]𝑂)
86 df-f1o 5856 . 2 (𝐹:(𝑋 / )–1-1-onto→[𝐴]𝑂 ↔ (𝐹:(𝑋 / )–1-1→[𝐴]𝑂𝐹:(𝑋 / )–onto→[𝐴]𝑂))
8760, 85, 86sylanbrc 697 1 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝐹:(𝑋 / )–1-1-onto→[𝐴]𝑂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  {crab 2911  Vcvv 3186  wss 3556  {cpr 4152  cop 4156   class class class wbr 4615  {copab 4674  cmpt 4675   × cxp 5074  ran crn 5077  Fun wfun 5843  wf 5845  1-1wf1 5846  ontowfo 5847  1-1-ontowf1o 5848  cfv 5849  (class class class)co 6607   Er wer 7687  [cec 7688   / cqs 7689  Basecbs 15784  SubGrpcsubg 17512   ~QG cqg 17514   GrpAct cga 17646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-er 7690  df-ec 7692  df-qs 7696  df-map 7807  df-en 7903  df-dom 7904  df-sdom 7905  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-nn 10968  df-2 11026  df-ndx 15787  df-slot 15788  df-base 15789  df-sets 15790  df-ress 15791  df-plusg 15878  df-0g 16026  df-mgm 17166  df-sgrp 17208  df-mnd 17219  df-grp 17349  df-minusg 17350  df-subg 17515  df-eqg 17517  df-ga 17647
This theorem is referenced by:  orbsta2  17671
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