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Theorem lactghmga 18928
Description: The converse of galactghm 18927. 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 18751 . 2 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
2 ghmgrp2 18752 . . 3 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐻 ∈ Grp)
3 grpn0 18526 . . 3 (𝐻 ∈ Grp → 𝐻 ≠ ∅)
4 lactghmga.h . . . . 5 𝐻 = (SymGrp‘𝑌)
5 fvprc 6748 . . . . 5 𝑌 ∈ V → (SymGrp‘𝑌) = ∅)
64, 5eqtrid 2790 . . . 4 𝑌 ∈ V → 𝐻 = ∅)
76necon1ai 2970 . . 3 (𝐻 ≠ ∅ → 𝑌 ∈ V)
82, 3, 73syl 18 . 2 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝑌 ∈ V)
9 lactghmga.x . . . . . . . . . . 11 𝑋 = (Base‘𝐺)
10 eqid 2738 . . . . . . . . . . 11 (Base‘𝐻) = (Base‘𝐻)
119, 10ghmf 18753 . . . . . . . . . 10 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐹:𝑋⟶(Base‘𝐻))
1211ffvelrnda 6943 . . . . . . . . 9 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥𝑋) → (𝐹𝑥) ∈ (Base‘𝐻))
138adantr 480 . . . . . . . . . 10 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥𝑋) → 𝑌 ∈ V)
144, 10elsymgbas 18896 . . . . . . . . . 10 (𝑌 ∈ V → ((𝐹𝑥) ∈ (Base‘𝐻) ↔ (𝐹𝑥):𝑌1-1-onto𝑌))
1513, 14syl 17 . . . . . . . . 9 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥𝑋) → ((𝐹𝑥) ∈ (Base‘𝐻) ↔ (𝐹𝑥):𝑌1-1-onto𝑌))
1612, 15mpbid 231 . . . . . . . 8 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥𝑋) → (𝐹𝑥):𝑌1-1-onto𝑌)
17 f1of 6700 . . . . . . . 8 ((𝐹𝑥):𝑌1-1-onto𝑌 → (𝐹𝑥):𝑌𝑌)
1816, 17syl 17 . . . . . . 7 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥𝑋) → (𝐹𝑥):𝑌𝑌)
1918ffvelrnda 6943 . . . . . 6 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥𝑋) ∧ 𝑦𝑌) → ((𝐹𝑥)‘𝑦) ∈ 𝑌)
2019ralrimiva 3107 . . . . 5 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥𝑋) → ∀𝑦𝑌 ((𝐹𝑥)‘𝑦) ∈ 𝑌)
2120ralrimiva 3107 . . . 4 (𝐹 ∈ (𝐺 GrpHom 𝐻) → ∀𝑥𝑋𝑦𝑌 ((𝐹𝑥)‘𝑦) ∈ 𝑌)
22 lactghmga.f . . . . 5 = (𝑥𝑋, 𝑦𝑌 ↦ ((𝐹𝑥)‘𝑦))
2322fmpo 7881 . . . 4 (∀𝑥𝑋𝑦𝑌 ((𝐹𝑥)‘𝑦) ∈ 𝑌 :(𝑋 × 𝑌)⟶𝑌)
2421, 23sylib 217 . . 3 (𝐹 ∈ (𝐺 GrpHom 𝐻) → :(𝑋 × 𝑌)⟶𝑌)
25 eqid 2738 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
269, 25grpidcl 18522 . . . . . . . 8 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝑋)
271, 26syl 17 . . . . . . 7 (𝐹 ∈ (𝐺 GrpHom 𝐻) → (0g𝐺) ∈ 𝑋)
28 fveq2 6756 . . . . . . . . 9 (𝑥 = (0g𝐺) → (𝐹𝑥) = (𝐹‘(0g𝐺)))
2928fveq1d 6758 . . . . . . . 8 (𝑥 = (0g𝐺) → ((𝐹𝑥)‘𝑦) = ((𝐹‘(0g𝐺))‘𝑦))
30 fveq2 6756 . . . . . . . 8 (𝑦 = 𝑧 → ((𝐹‘(0g𝐺))‘𝑦) = ((𝐹‘(0g𝐺))‘𝑧))
31 fvex 6769 . . . . . . . 8 ((𝐹‘(0g𝐺))‘𝑧) ∈ V
3229, 30, 22, 31ovmpo 7411 . . . . . . 7 (((0g𝐺) ∈ 𝑋𝑧𝑌) → ((0g𝐺) 𝑧) = ((𝐹‘(0g𝐺))‘𝑧))
3327, 32sylan 579 . . . . . 6 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → ((0g𝐺) 𝑧) = ((𝐹‘(0g𝐺))‘𝑧))
34 eqid 2738 . . . . . . . . . 10 (0g𝐻) = (0g𝐻)
3525, 34ghmid 18755 . . . . . . . . 9 (𝐹 ∈ (𝐺 GrpHom 𝐻) → (𝐹‘(0g𝐺)) = (0g𝐻))
3635adantr 480 . . . . . . . 8 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → (𝐹‘(0g𝐺)) = (0g𝐻))
378adantr 480 . . . . . . . . 9 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → 𝑌 ∈ V)
384symgid 18924 . . . . . . . . 9 (𝑌 ∈ V → ( I ↾ 𝑌) = (0g𝐻))
3937, 38syl 17 . . . . . . . 8 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → ( I ↾ 𝑌) = (0g𝐻))
4036, 39eqtr4d 2781 . . . . . . 7 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → (𝐹‘(0g𝐺)) = ( I ↾ 𝑌))
4140fveq1d 6758 . . . . . 6 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → ((𝐹‘(0g𝐺))‘𝑧) = (( I ↾ 𝑌)‘𝑧))
42 fvresi 7027 . . . . . . 7 (𝑧𝑌 → (( I ↾ 𝑌)‘𝑧) = 𝑧)
4342adantl 481 . . . . . 6 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → (( I ↾ 𝑌)‘𝑧) = 𝑧)
4433, 41, 433eqtrd 2782 . . . . 5 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → ((0g𝐺) 𝑧) = 𝑧)
4511ad2antrr 722 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → 𝐹:𝑋⟶(Base‘𝐻))
46 simprr 769 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → 𝑣𝑋)
4745, 46ffvelrnd 6944 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝐹𝑣) ∈ (Base‘𝐻))
488ad2antrr 722 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → 𝑌 ∈ V)
494, 10elsymgbas 18896 . . . . . . . . . . . 12 (𝑌 ∈ V → ((𝐹𝑣) ∈ (Base‘𝐻) ↔ (𝐹𝑣):𝑌1-1-onto𝑌))
5048, 49syl 17 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → ((𝐹𝑣) ∈ (Base‘𝐻) ↔ (𝐹𝑣):𝑌1-1-onto𝑌))
5147, 50mpbid 231 . . . . . . . . . 10 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝐹𝑣):𝑌1-1-onto𝑌)
52 f1of 6700 . . . . . . . . . 10 ((𝐹𝑣):𝑌1-1-onto𝑌 → (𝐹𝑣):𝑌𝑌)
5351, 52syl 17 . . . . . . . . 9 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝐹𝑣):𝑌𝑌)
54 simplr 765 . . . . . . . . 9 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → 𝑧𝑌)
55 fvco3 6849 . . . . . . . . 9 (((𝐹𝑣):𝑌𝑌𝑧𝑌) → (((𝐹𝑢) ∘ (𝐹𝑣))‘𝑧) = ((𝐹𝑢)‘((𝐹𝑣)‘𝑧)))
5653, 54, 55syl2anc 583 . . . . . . . 8 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (((𝐹𝑢) ∘ (𝐹𝑣))‘𝑧) = ((𝐹𝑢)‘((𝐹𝑣)‘𝑧)))
57 simpll 763 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
58 simprl 767 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → 𝑢𝑋)
59 eqid 2738 . . . . . . . . . . . 12 (+g𝐺) = (+g𝐺)
60 eqid 2738 . . . . . . . . . . . 12 (+g𝐻) = (+g𝐻)
619, 59, 60ghmlin 18754 . . . . . . . . . . 11 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑢𝑋𝑣𝑋) → (𝐹‘(𝑢(+g𝐺)𝑣)) = ((𝐹𝑢)(+g𝐻)(𝐹𝑣)))
6257, 58, 46, 61syl3anc 1369 . . . . . . . . . 10 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝐹‘(𝑢(+g𝐺)𝑣)) = ((𝐹𝑢)(+g𝐻)(𝐹𝑣)))
6345, 58ffvelrnd 6944 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝐹𝑢) ∈ (Base‘𝐻))
644, 10, 60symgov 18906 . . . . . . . . . . 11 (((𝐹𝑢) ∈ (Base‘𝐻) ∧ (𝐹𝑣) ∈ (Base‘𝐻)) → ((𝐹𝑢)(+g𝐻)(𝐹𝑣)) = ((𝐹𝑢) ∘ (𝐹𝑣)))
6563, 47, 64syl2anc 583 . . . . . . . . . 10 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → ((𝐹𝑢)(+g𝐻)(𝐹𝑣)) = ((𝐹𝑢) ∘ (𝐹𝑣)))
6662, 65eqtrd 2778 . . . . . . . . 9 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝐹‘(𝑢(+g𝐺)𝑣)) = ((𝐹𝑢) ∘ (𝐹𝑣)))
6766fveq1d 6758 . . . . . . . 8 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑧) = (((𝐹𝑢) ∘ (𝐹𝑣))‘𝑧))
6853, 54ffvelrnd 6944 . . . . . . . . 9 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → ((𝐹𝑣)‘𝑧) ∈ 𝑌)
69 fveq2 6756 . . . . . . . . . . 11 (𝑥 = 𝑢 → (𝐹𝑥) = (𝐹𝑢))
7069fveq1d 6758 . . . . . . . . . 10 (𝑥 = 𝑢 → ((𝐹𝑥)‘𝑦) = ((𝐹𝑢)‘𝑦))
71 fveq2 6756 . . . . . . . . . 10 (𝑦 = ((𝐹𝑣)‘𝑧) → ((𝐹𝑢)‘𝑦) = ((𝐹𝑢)‘((𝐹𝑣)‘𝑧)))
72 fvex 6769 . . . . . . . . . 10 ((𝐹𝑢)‘((𝐹𝑣)‘𝑧)) ∈ V
7370, 71, 22, 72ovmpo 7411 . . . . . . . . 9 ((𝑢𝑋 ∧ ((𝐹𝑣)‘𝑧) ∈ 𝑌) → (𝑢 ((𝐹𝑣)‘𝑧)) = ((𝐹𝑢)‘((𝐹𝑣)‘𝑧)))
7458, 68, 73syl2anc 583 . . . . . . . 8 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝑢 ((𝐹𝑣)‘𝑧)) = ((𝐹𝑢)‘((𝐹𝑣)‘𝑧)))
7556, 67, 743eqtr4d 2788 . . . . . . 7 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑧) = (𝑢 ((𝐹𝑣)‘𝑧)))
761ad2antrr 722 . . . . . . . . 9 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → 𝐺 ∈ Grp)
779, 59grpcl 18500 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑢𝑋𝑣𝑋) → (𝑢(+g𝐺)𝑣) ∈ 𝑋)
7876, 58, 46, 77syl3anc 1369 . . . . . . . 8 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝑢(+g𝐺)𝑣) ∈ 𝑋)
79 fveq2 6756 . . . . . . . . . 10 (𝑥 = (𝑢(+g𝐺)𝑣) → (𝐹𝑥) = (𝐹‘(𝑢(+g𝐺)𝑣)))
8079fveq1d 6758 . . . . . . . . 9 (𝑥 = (𝑢(+g𝐺)𝑣) → ((𝐹𝑥)‘𝑦) = ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑦))
81 fveq2 6756 . . . . . . . . 9 (𝑦 = 𝑧 → ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑦) = ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑧))
82 fvex 6769 . . . . . . . . 9 ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑧) ∈ V
8380, 81, 22, 82ovmpo 7411 . . . . . . . 8 (((𝑢(+g𝐺)𝑣) ∈ 𝑋𝑧𝑌) → ((𝑢(+g𝐺)𝑣) 𝑧) = ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑧))
8478, 54, 83syl2anc 583 . . . . . . 7 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → ((𝑢(+g𝐺)𝑣) 𝑧) = ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑧))
85 fveq2 6756 . . . . . . . . . . 11 (𝑥 = 𝑣 → (𝐹𝑥) = (𝐹𝑣))
8685fveq1d 6758 . . . . . . . . . 10 (𝑥 = 𝑣 → ((𝐹𝑥)‘𝑦) = ((𝐹𝑣)‘𝑦))
87 fveq2 6756 . . . . . . . . . 10 (𝑦 = 𝑧 → ((𝐹𝑣)‘𝑦) = ((𝐹𝑣)‘𝑧))
88 fvex 6769 . . . . . . . . . 10 ((𝐹𝑣)‘𝑧) ∈ V
8986, 87, 22, 88ovmpo 7411 . . . . . . . . 9 ((𝑣𝑋𝑧𝑌) → (𝑣 𝑧) = ((𝐹𝑣)‘𝑧))
9046, 54, 89syl2anc 583 . . . . . . . 8 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝑣 𝑧) = ((𝐹𝑣)‘𝑧))
9190oveq2d 7271 . . . . . . 7 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝑢 (𝑣 𝑧)) = (𝑢 ((𝐹𝑣)‘𝑧)))
9275, 84, 913eqtr4d 2788 . . . . . 6 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧)))
9392ralrimivva 3114 . . . . 5 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → ∀𝑢𝑋𝑣𝑋 ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧)))
9444, 93jca 511 . . . 4 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → (((0g𝐺) 𝑧) = 𝑧 ∧ ∀𝑢𝑋𝑣𝑋 ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧))))
9594ralrimiva 3107 . . 3 (𝐹 ∈ (𝐺 GrpHom 𝐻) → ∀𝑧𝑌 (((0g𝐺) 𝑧) = 𝑧 ∧ ∀𝑢𝑋𝑣𝑋 ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧))))
9624, 95jca 511 . 2 (𝐹 ∈ (𝐺 GrpHom 𝐻) → ( :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑧𝑌 (((0g𝐺) 𝑧) = 𝑧 ∧ ∀𝑢𝑋𝑣𝑋 ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧)))))
979, 59, 25isga 18812 . 2 ( ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑧𝑌 (((0g𝐺) 𝑧) = 𝑧 ∧ ∀𝑢𝑋𝑣𝑋 ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧))))))
981, 8, 96, 97syl21anbrc 1342 1 (𝐹 ∈ (𝐺 GrpHom 𝐻) → ∈ (𝐺 GrpAct 𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395   = wceq 1539  wcel 2108  wne 2942  wral 3063  Vcvv 3422  c0 4253   I cid 5479   × cxp 5578  cres 5582  ccom 5584  wf 6414  1-1-ontowf1o 6417  cfv 6418  (class class class)co 7255  cmpo 7257  Basecbs 16840  +gcplusg 16888  0gc0g 17067  Grpcgrp 18492   GrpHom cghm 18746   GrpAct cga 18810  SymGrpcsymg 18889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-tset 16907  df-0g 17069  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-submnd 18346  df-efmnd 18423  df-grp 18495  df-ghm 18747  df-ga 18811  df-symg 18890
This theorem is referenced by:  symgga  18930
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