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Theorem lactghmga 18022
Description: The converse of galactghm 18021. 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 17861 . . 3 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
2 ghmgrp2 17862 . . . 4 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐻 ∈ Grp)
3 grpn0 17653 . . . 4 (𝐻 ∈ Grp → 𝐻 ≠ ∅)
4 lactghmga.h . . . . . 6 𝐻 = (SymGrp‘𝑌)
5 fvprc 6344 . . . . . 6 𝑌 ∈ V → (SymGrp‘𝑌) = ∅)
64, 5syl5eq 2804 . . . . 5 𝑌 ∈ V → 𝐻 = ∅)
76necon1ai 2957 . . . 4 (𝐻 ≠ ∅ → 𝑌 ∈ V)
82, 3, 73syl 18 . . 3 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝑌 ∈ V)
91, 8jca 555 . 2 (𝐹 ∈ (𝐺 GrpHom 𝐻) → (𝐺 ∈ Grp ∧ 𝑌 ∈ V))
10 lactghmga.x . . . . . . . . . . 11 𝑋 = (Base‘𝐺)
11 eqid 2758 . . . . . . . . . . 11 (Base‘𝐻) = (Base‘𝐻)
1210, 11ghmf 17863 . . . . . . . . . 10 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐹:𝑋⟶(Base‘𝐻))
1312ffvelrnda 6520 . . . . . . . . 9 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥𝑋) → (𝐹𝑥) ∈ (Base‘𝐻))
148adantr 472 . . . . . . . . . 10 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥𝑋) → 𝑌 ∈ V)
154, 11elsymgbas 18000 . . . . . . . . . 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 222 . . . . . . . 8 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥𝑋) → (𝐹𝑥):𝑌1-1-onto𝑌)
18 f1of 6296 . . . . . . . 8 ((𝐹𝑥):𝑌1-1-onto𝑌 → (𝐹𝑥):𝑌𝑌)
1917, 18syl 17 . . . . . . 7 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥𝑋) → (𝐹𝑥):𝑌𝑌)
2019ffvelrnda 6520 . . . . . 6 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥𝑋) ∧ 𝑦𝑌) → ((𝐹𝑥)‘𝑦) ∈ 𝑌)
2120ralrimiva 3102 . . . . 5 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥𝑋) → ∀𝑦𝑌 ((𝐹𝑥)‘𝑦) ∈ 𝑌)
2221ralrimiva 3102 . . . 4 (𝐹 ∈ (𝐺 GrpHom 𝐻) → ∀𝑥𝑋𝑦𝑌 ((𝐹𝑥)‘𝑦) ∈ 𝑌)
23 lactghmga.f . . . . 5 = (𝑥𝑋, 𝑦𝑌 ↦ ((𝐹𝑥)‘𝑦))
2423fmpt2 7403 . . . 4 (∀𝑥𝑋𝑦𝑌 ((𝐹𝑥)‘𝑦) ∈ 𝑌 :(𝑋 × 𝑌)⟶𝑌)
2522, 24sylib 208 . . 3 (𝐹 ∈ (𝐺 GrpHom 𝐻) → :(𝑋 × 𝑌)⟶𝑌)
26 eqid 2758 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
2710, 26grpidcl 17649 . . . . . . . 8 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝑋)
281, 27syl 17 . . . . . . 7 (𝐹 ∈ (𝐺 GrpHom 𝐻) → (0g𝐺) ∈ 𝑋)
29 fveq2 6350 . . . . . . . . 9 (𝑥 = (0g𝐺) → (𝐹𝑥) = (𝐹‘(0g𝐺)))
3029fveq1d 6352 . . . . . . . 8 (𝑥 = (0g𝐺) → ((𝐹𝑥)‘𝑦) = ((𝐹‘(0g𝐺))‘𝑦))
31 fveq2 6350 . . . . . . . 8 (𝑦 = 𝑧 → ((𝐹‘(0g𝐺))‘𝑦) = ((𝐹‘(0g𝐺))‘𝑧))
32 fvex 6360 . . . . . . . 8 ((𝐹‘(0g𝐺))‘𝑧) ∈ V
3330, 31, 23, 32ovmpt2 6959 . . . . . . 7 (((0g𝐺) ∈ 𝑋𝑧𝑌) → ((0g𝐺) 𝑧) = ((𝐹‘(0g𝐺))‘𝑧))
3428, 33sylan 489 . . . . . 6 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → ((0g𝐺) 𝑧) = ((𝐹‘(0g𝐺))‘𝑧))
35 eqid 2758 . . . . . . . . . 10 (0g𝐻) = (0g𝐻)
3626, 35ghmid 17865 . . . . . . . . 9 (𝐹 ∈ (𝐺 GrpHom 𝐻) → (𝐹‘(0g𝐺)) = (0g𝐻))
3736adantr 472 . . . . . . . 8 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → (𝐹‘(0g𝐺)) = (0g𝐻))
388adantr 472 . . . . . . . . 9 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → 𝑌 ∈ V)
394symgid 18019 . . . . . . . . 9 (𝑌 ∈ V → ( I ↾ 𝑌) = (0g𝐻))
4038, 39syl 17 . . . . . . . 8 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → ( I ↾ 𝑌) = (0g𝐻))
4137, 40eqtr4d 2795 . . . . . . 7 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → (𝐹‘(0g𝐺)) = ( I ↾ 𝑌))
4241fveq1d 6352 . . . . . 6 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → ((𝐹‘(0g𝐺))‘𝑧) = (( I ↾ 𝑌)‘𝑧))
43 fvresi 6601 . . . . . . 7 (𝑧𝑌 → (( I ↾ 𝑌)‘𝑧) = 𝑧)
4443adantl 473 . . . . . 6 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → (( I ↾ 𝑌)‘𝑧) = 𝑧)
4534, 42, 443eqtrd 2796 . . . . 5 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → ((0g𝐺) 𝑧) = 𝑧)
4612ad2antrr 764 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → 𝐹:𝑋⟶(Base‘𝐻))
47 simprr 813 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → 𝑣𝑋)
4846, 47ffvelrnd 6521 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝐹𝑣) ∈ (Base‘𝐻))
498ad2antrr 764 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → 𝑌 ∈ V)
504, 11elsymgbas 18000 . . . . . . . . . . . 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 222 . . . . . . . . . 10 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝐹𝑣):𝑌1-1-onto𝑌)
53 f1of 6296 . . . . . . . . . 10 ((𝐹𝑣):𝑌1-1-onto𝑌 → (𝐹𝑣):𝑌𝑌)
5452, 53syl 17 . . . . . . . . 9 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝐹𝑣):𝑌𝑌)
55 simplr 809 . . . . . . . . 9 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → 𝑧𝑌)
56 fvco3 6435 . . . . . . . . 9 (((𝐹𝑣):𝑌𝑌𝑧𝑌) → (((𝐹𝑢) ∘ (𝐹𝑣))‘𝑧) = ((𝐹𝑢)‘((𝐹𝑣)‘𝑧)))
5754, 55, 56syl2anc 696 . . . . . . . 8 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (((𝐹𝑢) ∘ (𝐹𝑣))‘𝑧) = ((𝐹𝑢)‘((𝐹𝑣)‘𝑧)))
58 simpll 807 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
59 simprl 811 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → 𝑢𝑋)
60 eqid 2758 . . . . . . . . . . . 12 (+g𝐺) = (+g𝐺)
61 eqid 2758 . . . . . . . . . . . 12 (+g𝐻) = (+g𝐻)
6210, 60, 61ghmlin 17864 . . . . . . . . . . 11 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑢𝑋𝑣𝑋) → (𝐹‘(𝑢(+g𝐺)𝑣)) = ((𝐹𝑢)(+g𝐻)(𝐹𝑣)))
6358, 59, 47, 62syl3anc 1477 . . . . . . . . . 10 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝐹‘(𝑢(+g𝐺)𝑣)) = ((𝐹𝑢)(+g𝐻)(𝐹𝑣)))
6446, 59ffvelrnd 6521 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝐹𝑢) ∈ (Base‘𝐻))
654, 11, 61symgov 18008 . . . . . . . . . . 11 (((𝐹𝑢) ∈ (Base‘𝐻) ∧ (𝐹𝑣) ∈ (Base‘𝐻)) → ((𝐹𝑢)(+g𝐻)(𝐹𝑣)) = ((𝐹𝑢) ∘ (𝐹𝑣)))
6664, 48, 65syl2anc 696 . . . . . . . . . 10 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → ((𝐹𝑢)(+g𝐻)(𝐹𝑣)) = ((𝐹𝑢) ∘ (𝐹𝑣)))
6763, 66eqtrd 2792 . . . . . . . . 9 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝐹‘(𝑢(+g𝐺)𝑣)) = ((𝐹𝑢) ∘ (𝐹𝑣)))
6867fveq1d 6352 . . . . . . . 8 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑧) = (((𝐹𝑢) ∘ (𝐹𝑣))‘𝑧))
6954, 55ffvelrnd 6521 . . . . . . . . 9 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → ((𝐹𝑣)‘𝑧) ∈ 𝑌)
70 fveq2 6350 . . . . . . . . . . 11 (𝑥 = 𝑢 → (𝐹𝑥) = (𝐹𝑢))
7170fveq1d 6352 . . . . . . . . . 10 (𝑥 = 𝑢 → ((𝐹𝑥)‘𝑦) = ((𝐹𝑢)‘𝑦))
72 fveq2 6350 . . . . . . . . . 10 (𝑦 = ((𝐹𝑣)‘𝑧) → ((𝐹𝑢)‘𝑦) = ((𝐹𝑢)‘((𝐹𝑣)‘𝑧)))
73 fvex 6360 . . . . . . . . . 10 ((𝐹𝑢)‘((𝐹𝑣)‘𝑧)) ∈ V
7471, 72, 23, 73ovmpt2 6959 . . . . . . . . 9 ((𝑢𝑋 ∧ ((𝐹𝑣)‘𝑧) ∈ 𝑌) → (𝑢 ((𝐹𝑣)‘𝑧)) = ((𝐹𝑢)‘((𝐹𝑣)‘𝑧)))
7559, 69, 74syl2anc 696 . . . . . . . 8 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝑢 ((𝐹𝑣)‘𝑧)) = ((𝐹𝑢)‘((𝐹𝑣)‘𝑧)))
7657, 68, 753eqtr4d 2802 . . . . . . 7 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑧) = (𝑢 ((𝐹𝑣)‘𝑧)))
771ad2antrr 764 . . . . . . . . 9 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → 𝐺 ∈ Grp)
7810, 60grpcl 17629 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑢𝑋𝑣𝑋) → (𝑢(+g𝐺)𝑣) ∈ 𝑋)
7977, 59, 47, 78syl3anc 1477 . . . . . . . 8 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝑢(+g𝐺)𝑣) ∈ 𝑋)
80 fveq2 6350 . . . . . . . . . 10 (𝑥 = (𝑢(+g𝐺)𝑣) → (𝐹𝑥) = (𝐹‘(𝑢(+g𝐺)𝑣)))
8180fveq1d 6352 . . . . . . . . 9 (𝑥 = (𝑢(+g𝐺)𝑣) → ((𝐹𝑥)‘𝑦) = ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑦))
82 fveq2 6350 . . . . . . . . 9 (𝑦 = 𝑧 → ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑦) = ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑧))
83 fvex 6360 . . . . . . . . 9 ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑧) ∈ V
8481, 82, 23, 83ovmpt2 6959 . . . . . . . 8 (((𝑢(+g𝐺)𝑣) ∈ 𝑋𝑧𝑌) → ((𝑢(+g𝐺)𝑣) 𝑧) = ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑧))
8579, 55, 84syl2anc 696 . . . . . . 7 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → ((𝑢(+g𝐺)𝑣) 𝑧) = ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑧))
86 fveq2 6350 . . . . . . . . . . 11 (𝑥 = 𝑣 → (𝐹𝑥) = (𝐹𝑣))
8786fveq1d 6352 . . . . . . . . . 10 (𝑥 = 𝑣 → ((𝐹𝑥)‘𝑦) = ((𝐹𝑣)‘𝑦))
88 fveq2 6350 . . . . . . . . . 10 (𝑦 = 𝑧 → ((𝐹𝑣)‘𝑦) = ((𝐹𝑣)‘𝑧))
89 fvex 6360 . . . . . . . . . 10 ((𝐹𝑣)‘𝑧) ∈ V
9087, 88, 23, 89ovmpt2 6959 . . . . . . . . 9 ((𝑣𝑋𝑧𝑌) → (𝑣 𝑧) = ((𝐹𝑣)‘𝑧))
9147, 55, 90syl2anc 696 . . . . . . . 8 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝑣 𝑧) = ((𝐹𝑣)‘𝑧))
9291oveq2d 6827 . . . . . . 7 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝑢 (𝑣 𝑧)) = (𝑢 ((𝐹𝑣)‘𝑧)))
9376, 85, 923eqtr4d 2802 . . . . . 6 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧)))
9493ralrimivva 3107 . . . . 5 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → ∀𝑢𝑋𝑣𝑋 ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧)))
9545, 94jca 555 . . . 4 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → (((0g𝐺) 𝑧) = 𝑧 ∧ ∀𝑢𝑋𝑣𝑋 ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧))))
9695ralrimiva 3102 . . 3 (𝐹 ∈ (𝐺 GrpHom 𝐻) → ∀𝑧𝑌 (((0g𝐺) 𝑧) = 𝑧 ∧ ∀𝑢𝑋𝑣𝑋 ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧))))
9725, 96jca 555 . 2 (𝐹 ∈ (𝐺 GrpHom 𝐻) → ( :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑧𝑌 (((0g𝐺) 𝑧) = 𝑧 ∧ ∀𝑢𝑋𝑣𝑋 ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧)))))
9810, 60, 26isga 17922 . 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 701 1 (𝐹 ∈ (𝐺 GrpHom 𝐻) → ∈ (𝐺 GrpAct 𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1630  wcel 2137  wne 2930  wral 3048  Vcvv 3338  c0 4056   I cid 5171   × cxp 5262  cres 5266  ccom 5268  wf 6043  1-1-ontowf1o 6046  cfv 6047  (class class class)co 6811  cmpt2 6813  Basecbs 16057  +gcplusg 16141  0gc0g 16300  Grpcgrp 17621   GrpHom cghm 17856   GrpAct cga 17920  SymGrpcsymg 17995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-cnex 10182  ax-resscn 10183  ax-1cn 10184  ax-icn 10185  ax-addcl 10186  ax-addrcl 10187  ax-mulcl 10188  ax-mulrcl 10189  ax-mulcom 10190  ax-addass 10191  ax-mulass 10192  ax-distr 10193  ax-i2m1 10194  ax-1ne0 10195  ax-1rid 10196  ax-rnegex 10197  ax-rrecex 10198  ax-cnre 10199  ax-pre-lttri 10200  ax-pre-lttrn 10201  ax-pre-ltadd 10202  ax-pre-mulgt0 10203
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-nel 3034  df-ral 3053  df-rex 3054  df-reu 3055  df-rmo 3056  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-riota 6772  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-om 7229  df-1st 7331  df-2nd 7332  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-1o 7727  df-oadd 7731  df-er 7909  df-map 8023  df-en 8120  df-dom 8121  df-sdom 8122  df-fin 8123  df-pnf 10266  df-mnf 10267  df-xr 10268  df-ltxr 10269  df-le 10270  df-sub 10458  df-neg 10459  df-nn 11211  df-2 11269  df-3 11270  df-4 11271  df-5 11272  df-6 11273  df-7 11274  df-8 11275  df-9 11276  df-n0 11483  df-z 11568  df-uz 11878  df-fz 12518  df-struct 16059  df-ndx 16060  df-slot 16061  df-base 16063  df-plusg 16154  df-tset 16160  df-0g 16302  df-mgm 17441  df-sgrp 17483  df-mnd 17494  df-grp 17624  df-ghm 17857  df-ga 17921  df-symg 17996
This theorem is referenced by:  symgga  18024
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