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Theorem lactghmga 17647
 Description: The converse of galactghm 17646. The uncurrying of a homomorphism into (SymGrp‘𝑌) is a group action. Thus, group actions and group homomorphisms into a symmetric group are essentially equivalent notions. (Contributed by Mario Carneiro, 15-Jan-2015.)
Hypotheses
Ref Expression
lactghmga.x 𝑋 = (Base‘𝐺)
lactghmga.h 𝐻 = (SymGrp‘𝑌)
lactghmga.f = (𝑥𝑋, 𝑦𝑌 ↦ ((𝐹𝑥)‘𝑦))
Assertion
Ref Expression
lactghmga (𝐹 ∈ (𝐺 GrpHom 𝐻) → ∈ (𝐺 GrpAct 𝑌))
Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐺,𝑦   𝑥,𝐻,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦
Allowed substitution hints:   (𝑥,𝑦)

Proof of Theorem lactghmga
Dummy variables 𝑣 𝑢 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghmgrp1 17485 . . 3 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
2 ghmgrp2 17486 . . . 4 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐻 ∈ Grp)
3 grpn0 17277 . . . 4 (𝐻 ∈ Grp → 𝐻 ≠ ∅)
4 lactghmga.h . . . . . 6 𝐻 = (SymGrp‘𝑌)
5 fvprc 6097 . . . . . 6 𝑌 ∈ V → (SymGrp‘𝑌) = ∅)
64, 5syl5eq 2656 . . . . 5 𝑌 ∈ V → 𝐻 = ∅)
76necon1ai 2809 . . . 4 (𝐻 ≠ ∅ → 𝑌 ∈ V)
82, 3, 73syl 18 . . 3 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝑌 ∈ V)
91, 8jca 553 . 2 (𝐹 ∈ (𝐺 GrpHom 𝐻) → (𝐺 ∈ Grp ∧ 𝑌 ∈ V))
10 lactghmga.x . . . . . . . . . . 11 𝑋 = (Base‘𝐺)
11 eqid 2610 . . . . . . . . . . 11 (Base‘𝐻) = (Base‘𝐻)
1210, 11ghmf 17487 . . . . . . . . . 10 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐹:𝑋⟶(Base‘𝐻))
1312ffvelrnda 6267 . . . . . . . . 9 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥𝑋) → (𝐹𝑥) ∈ (Base‘𝐻))
148adantr 480 . . . . . . . . . 10 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥𝑋) → 𝑌 ∈ V)
154, 11elsymgbas 17625 . . . . . . . . . 10 (𝑌 ∈ V → ((𝐹𝑥) ∈ (Base‘𝐻) ↔ (𝐹𝑥):𝑌1-1-onto𝑌))
1614, 15syl 17 . . . . . . . . 9 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥𝑋) → ((𝐹𝑥) ∈ (Base‘𝐻) ↔ (𝐹𝑥):𝑌1-1-onto𝑌))
1713, 16mpbid 221 . . . . . . . 8 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥𝑋) → (𝐹𝑥):𝑌1-1-onto𝑌)
18 f1of 6050 . . . . . . . 8 ((𝐹𝑥):𝑌1-1-onto𝑌 → (𝐹𝑥):𝑌𝑌)
1917, 18syl 17 . . . . . . 7 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥𝑋) → (𝐹𝑥):𝑌𝑌)
2019ffvelrnda 6267 . . . . . 6 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥𝑋) ∧ 𝑦𝑌) → ((𝐹𝑥)‘𝑦) ∈ 𝑌)
2120ralrimiva 2949 . . . . 5 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥𝑋) → ∀𝑦𝑌 ((𝐹𝑥)‘𝑦) ∈ 𝑌)
2221ralrimiva 2949 . . . 4 (𝐹 ∈ (𝐺 GrpHom 𝐻) → ∀𝑥𝑋𝑦𝑌 ((𝐹𝑥)‘𝑦) ∈ 𝑌)
23 lactghmga.f . . . . 5 = (𝑥𝑋, 𝑦𝑌 ↦ ((𝐹𝑥)‘𝑦))
2423fmpt2 7126 . . . 4 (∀𝑥𝑋𝑦𝑌 ((𝐹𝑥)‘𝑦) ∈ 𝑌 :(𝑋 × 𝑌)⟶𝑌)
2522, 24sylib 207 . . 3 (𝐹 ∈ (𝐺 GrpHom 𝐻) → :(𝑋 × 𝑌)⟶𝑌)
26 eqid 2610 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
2710, 26grpidcl 17273 . . . . . . . 8 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝑋)
281, 27syl 17 . . . . . . 7 (𝐹 ∈ (𝐺 GrpHom 𝐻) → (0g𝐺) ∈ 𝑋)
29 fveq2 6103 . . . . . . . . 9 (𝑥 = (0g𝐺) → (𝐹𝑥) = (𝐹‘(0g𝐺)))
3029fveq1d 6105 . . . . . . . 8 (𝑥 = (0g𝐺) → ((𝐹𝑥)‘𝑦) = ((𝐹‘(0g𝐺))‘𝑦))
31 fveq2 6103 . . . . . . . 8 (𝑦 = 𝑧 → ((𝐹‘(0g𝐺))‘𝑦) = ((𝐹‘(0g𝐺))‘𝑧))
32 fvex 6113 . . . . . . . 8 ((𝐹‘(0g𝐺))‘𝑧) ∈ V
3330, 31, 23, 32ovmpt2 6694 . . . . . . 7 (((0g𝐺) ∈ 𝑋𝑧𝑌) → ((0g𝐺) 𝑧) = ((𝐹‘(0g𝐺))‘𝑧))
3428, 33sylan 487 . . . . . 6 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → ((0g𝐺) 𝑧) = ((𝐹‘(0g𝐺))‘𝑧))
35 eqid 2610 . . . . . . . . . 10 (0g𝐻) = (0g𝐻)
3626, 35ghmid 17489 . . . . . . . . 9 (𝐹 ∈ (𝐺 GrpHom 𝐻) → (𝐹‘(0g𝐺)) = (0g𝐻))
3736adantr 480 . . . . . . . 8 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → (𝐹‘(0g𝐺)) = (0g𝐻))
388adantr 480 . . . . . . . . 9 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → 𝑌 ∈ V)
394symgid 17644 . . . . . . . . 9 (𝑌 ∈ V → ( I ↾ 𝑌) = (0g𝐻))
4038, 39syl 17 . . . . . . . 8 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → ( I ↾ 𝑌) = (0g𝐻))
4137, 40eqtr4d 2647 . . . . . . 7 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → (𝐹‘(0g𝐺)) = ( I ↾ 𝑌))
4241fveq1d 6105 . . . . . 6 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → ((𝐹‘(0g𝐺))‘𝑧) = (( I ↾ 𝑌)‘𝑧))
43 fvresi 6344 . . . . . . 7 (𝑧𝑌 → (( I ↾ 𝑌)‘𝑧) = 𝑧)
4443adantl 481 . . . . . 6 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → (( I ↾ 𝑌)‘𝑧) = 𝑧)
4534, 42, 443eqtrd 2648 . . . . 5 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → ((0g𝐺) 𝑧) = 𝑧)
4612ad2antrr 758 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → 𝐹:𝑋⟶(Base‘𝐻))
47 simprr 792 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → 𝑣𝑋)
4846, 47ffvelrnd 6268 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝐹𝑣) ∈ (Base‘𝐻))
498ad2antrr 758 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → 𝑌 ∈ V)
504, 11elsymgbas 17625 . . . . . . . . . . . 12 (𝑌 ∈ V → ((𝐹𝑣) ∈ (Base‘𝐻) ↔ (𝐹𝑣):𝑌1-1-onto𝑌))
5149, 50syl 17 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → ((𝐹𝑣) ∈ (Base‘𝐻) ↔ (𝐹𝑣):𝑌1-1-onto𝑌))
5248, 51mpbid 221 . . . . . . . . . 10 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝐹𝑣):𝑌1-1-onto𝑌)
53 f1of 6050 . . . . . . . . . 10 ((𝐹𝑣):𝑌1-1-onto𝑌 → (𝐹𝑣):𝑌𝑌)
5452, 53syl 17 . . . . . . . . 9 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝐹𝑣):𝑌𝑌)
55 simplr 788 . . . . . . . . 9 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → 𝑧𝑌)
56 fvco3 6185 . . . . . . . . 9 (((𝐹𝑣):𝑌𝑌𝑧𝑌) → (((𝐹𝑢) ∘ (𝐹𝑣))‘𝑧) = ((𝐹𝑢)‘((𝐹𝑣)‘𝑧)))
5754, 55, 56syl2anc 691 . . . . . . . 8 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (((𝐹𝑢) ∘ (𝐹𝑣))‘𝑧) = ((𝐹𝑢)‘((𝐹𝑣)‘𝑧)))
58 simpll 786 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
59 simprl 790 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → 𝑢𝑋)
60 eqid 2610 . . . . . . . . . . . 12 (+g𝐺) = (+g𝐺)
61 eqid 2610 . . . . . . . . . . . 12 (+g𝐻) = (+g𝐻)
6210, 60, 61ghmlin 17488 . . . . . . . . . . 11 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑢𝑋𝑣𝑋) → (𝐹‘(𝑢(+g𝐺)𝑣)) = ((𝐹𝑢)(+g𝐻)(𝐹𝑣)))
6358, 59, 47, 62syl3anc 1318 . . . . . . . . . 10 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝐹‘(𝑢(+g𝐺)𝑣)) = ((𝐹𝑢)(+g𝐻)(𝐹𝑣)))
6446, 59ffvelrnd 6268 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝐹𝑢) ∈ (Base‘𝐻))
654, 11, 61symgov 17633 . . . . . . . . . . 11 (((𝐹𝑢) ∈ (Base‘𝐻) ∧ (𝐹𝑣) ∈ (Base‘𝐻)) → ((𝐹𝑢)(+g𝐻)(𝐹𝑣)) = ((𝐹𝑢) ∘ (𝐹𝑣)))
6664, 48, 65syl2anc 691 . . . . . . . . . 10 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → ((𝐹𝑢)(+g𝐻)(𝐹𝑣)) = ((𝐹𝑢) ∘ (𝐹𝑣)))
6763, 66eqtrd 2644 . . . . . . . . 9 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝐹‘(𝑢(+g𝐺)𝑣)) = ((𝐹𝑢) ∘ (𝐹𝑣)))
6867fveq1d 6105 . . . . . . . 8 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑧) = (((𝐹𝑢) ∘ (𝐹𝑣))‘𝑧))
6954, 55ffvelrnd 6268 . . . . . . . . 9 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → ((𝐹𝑣)‘𝑧) ∈ 𝑌)
70 fveq2 6103 . . . . . . . . . . 11 (𝑥 = 𝑢 → (𝐹𝑥) = (𝐹𝑢))
7170fveq1d 6105 . . . . . . . . . 10 (𝑥 = 𝑢 → ((𝐹𝑥)‘𝑦) = ((𝐹𝑢)‘𝑦))
72 fveq2 6103 . . . . . . . . . 10 (𝑦 = ((𝐹𝑣)‘𝑧) → ((𝐹𝑢)‘𝑦) = ((𝐹𝑢)‘((𝐹𝑣)‘𝑧)))
73 fvex 6113 . . . . . . . . . 10 ((𝐹𝑢)‘((𝐹𝑣)‘𝑧)) ∈ V
7471, 72, 23, 73ovmpt2 6694 . . . . . . . . 9 ((𝑢𝑋 ∧ ((𝐹𝑣)‘𝑧) ∈ 𝑌) → (𝑢 ((𝐹𝑣)‘𝑧)) = ((𝐹𝑢)‘((𝐹𝑣)‘𝑧)))
7559, 69, 74syl2anc 691 . . . . . . . 8 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝑢 ((𝐹𝑣)‘𝑧)) = ((𝐹𝑢)‘((𝐹𝑣)‘𝑧)))
7657, 68, 753eqtr4d 2654 . . . . . . 7 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑧) = (𝑢 ((𝐹𝑣)‘𝑧)))
771ad2antrr 758 . . . . . . . . 9 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → 𝐺 ∈ Grp)
7810, 60grpcl 17253 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑢𝑋𝑣𝑋) → (𝑢(+g𝐺)𝑣) ∈ 𝑋)
7977, 59, 47, 78syl3anc 1318 . . . . . . . 8 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝑢(+g𝐺)𝑣) ∈ 𝑋)
80 fveq2 6103 . . . . . . . . . 10 (𝑥 = (𝑢(+g𝐺)𝑣) → (𝐹𝑥) = (𝐹‘(𝑢(+g𝐺)𝑣)))
8180fveq1d 6105 . . . . . . . . 9 (𝑥 = (𝑢(+g𝐺)𝑣) → ((𝐹𝑥)‘𝑦) = ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑦))
82 fveq2 6103 . . . . . . . . 9 (𝑦 = 𝑧 → ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑦) = ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑧))
83 fvex 6113 . . . . . . . . 9 ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑧) ∈ V
8481, 82, 23, 83ovmpt2 6694 . . . . . . . 8 (((𝑢(+g𝐺)𝑣) ∈ 𝑋𝑧𝑌) → ((𝑢(+g𝐺)𝑣) 𝑧) = ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑧))
8579, 55, 84syl2anc 691 . . . . . . 7 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → ((𝑢(+g𝐺)𝑣) 𝑧) = ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑧))
86 fveq2 6103 . . . . . . . . . . 11 (𝑥 = 𝑣 → (𝐹𝑥) = (𝐹𝑣))
8786fveq1d 6105 . . . . . . . . . 10 (𝑥 = 𝑣 → ((𝐹𝑥)‘𝑦) = ((𝐹𝑣)‘𝑦))
88 fveq2 6103 . . . . . . . . . 10 (𝑦 = 𝑧 → ((𝐹𝑣)‘𝑦) = ((𝐹𝑣)‘𝑧))
89 fvex 6113 . . . . . . . . . 10 ((𝐹𝑣)‘𝑧) ∈ V
9087, 88, 23, 89ovmpt2 6694 . . . . . . . . 9 ((𝑣𝑋𝑧𝑌) → (𝑣 𝑧) = ((𝐹𝑣)‘𝑧))
9147, 55, 90syl2anc 691 . . . . . . . 8 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝑣 𝑧) = ((𝐹𝑣)‘𝑧))
9291oveq2d 6565 . . . . . . 7 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝑢 (𝑣 𝑧)) = (𝑢 ((𝐹𝑣)‘𝑧)))
9376, 85, 923eqtr4d 2654 . . . . . 6 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧)))
9493ralrimivva 2954 . . . . 5 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → ∀𝑢𝑋𝑣𝑋 ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧)))
9545, 94jca 553 . . . 4 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → (((0g𝐺) 𝑧) = 𝑧 ∧ ∀𝑢𝑋𝑣𝑋 ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧))))
9695ralrimiva 2949 . . 3 (𝐹 ∈ (𝐺 GrpHom 𝐻) → ∀𝑧𝑌 (((0g𝐺) 𝑧) = 𝑧 ∧ ∀𝑢𝑋𝑣𝑋 ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧))))
9725, 96jca 553 . 2 (𝐹 ∈ (𝐺 GrpHom 𝐻) → ( :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑧𝑌 (((0g𝐺) 𝑧) = 𝑧 ∧ ∀𝑢𝑋𝑣𝑋 ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧)))))
9810, 60, 26isga 17547 . 2 ( ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑧𝑌 (((0g𝐺) 𝑧) = 𝑧 ∧ ∀𝑢𝑋𝑣𝑋 ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧))))))
999, 97, 98sylanbrc 695 1 (𝐹 ∈ (𝐺 GrpHom 𝐻) → ∈ (𝐺 GrpAct 𝑌))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173  ∅c0 3874   I cid 4948   × cxp 5036   ↾ cres 5040   ∘ ccom 5042  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Basecbs 15695  +gcplusg 15768  0gc0g 15923  Grpcgrp 17245   GrpHom cghm 17480   GrpAct cga 17545  SymGrpcsymg 17620 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-plusg 15781  df-tset 15787  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-ghm 17481  df-ga 17546  df-symg 17621 This theorem is referenced by:  symgga  17649
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