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Theorem lactghmga 19187
Description: The converse of galactghm 19186. 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 19010 . 2 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
2 ghmgrp2 19011 . . 3 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐻 ∈ Grp)
3 grpn0 18782 . . 3 (𝐻 ∈ Grp → 𝐻 ≠ ∅)
4 lactghmga.h . . . . 5 𝐻 = (SymGrp‘𝑌)
5 fvprc 6834 . . . . 5 𝑌 ∈ V → (SymGrp‘𝑌) = ∅)
64, 5eqtrid 2788 . . . 4 𝑌 ∈ V → 𝐻 = ∅)
76necon1ai 2971 . . 3 (𝐻 ≠ ∅ → 𝑌 ∈ V)
82, 3, 73syl 18 . 2 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝑌 ∈ V)
9 lactghmga.x . . . . . . . . . . 11 𝑋 = (Base‘𝐺)
10 eqid 2736 . . . . . . . . . . 11 (Base‘𝐻) = (Base‘𝐻)
119, 10ghmf 19012 . . . . . . . . . 10 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐹:𝑋⟶(Base‘𝐻))
1211ffvelcdmda 7035 . . . . . . . . 9 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥𝑋) → (𝐹𝑥) ∈ (Base‘𝐻))
138adantr 481 . . . . . . . . . 10 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥𝑋) → 𝑌 ∈ V)
144, 10elsymgbas 19155 . . . . . . . . . 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 6784 . . . . . . . 8 ((𝐹𝑥):𝑌1-1-onto𝑌 → (𝐹𝑥):𝑌𝑌)
1816, 17syl 17 . . . . . . 7 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥𝑋) → (𝐹𝑥):𝑌𝑌)
1918ffvelcdmda 7035 . . . . . 6 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥𝑋) ∧ 𝑦𝑌) → ((𝐹𝑥)‘𝑦) ∈ 𝑌)
2019ralrimiva 3143 . . . . 5 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥𝑋) → ∀𝑦𝑌 ((𝐹𝑥)‘𝑦) ∈ 𝑌)
2120ralrimiva 3143 . . . 4 (𝐹 ∈ (𝐺 GrpHom 𝐻) → ∀𝑥𝑋𝑦𝑌 ((𝐹𝑥)‘𝑦) ∈ 𝑌)
22 lactghmga.f . . . . 5 = (𝑥𝑋, 𝑦𝑌 ↦ ((𝐹𝑥)‘𝑦))
2322fmpo 8000 . . . 4 (∀𝑥𝑋𝑦𝑌 ((𝐹𝑥)‘𝑦) ∈ 𝑌 :(𝑋 × 𝑌)⟶𝑌)
2421, 23sylib 217 . . 3 (𝐹 ∈ (𝐺 GrpHom 𝐻) → :(𝑋 × 𝑌)⟶𝑌)
25 eqid 2736 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
269, 25grpidcl 18778 . . . . . . . 8 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝑋)
271, 26syl 17 . . . . . . 7 (𝐹 ∈ (𝐺 GrpHom 𝐻) → (0g𝐺) ∈ 𝑋)
28 fveq2 6842 . . . . . . . . 9 (𝑥 = (0g𝐺) → (𝐹𝑥) = (𝐹‘(0g𝐺)))
2928fveq1d 6844 . . . . . . . 8 (𝑥 = (0g𝐺) → ((𝐹𝑥)‘𝑦) = ((𝐹‘(0g𝐺))‘𝑦))
30 fveq2 6842 . . . . . . . 8 (𝑦 = 𝑧 → ((𝐹‘(0g𝐺))‘𝑦) = ((𝐹‘(0g𝐺))‘𝑧))
31 fvex 6855 . . . . . . . 8 ((𝐹‘(0g𝐺))‘𝑧) ∈ V
3229, 30, 22, 31ovmpo 7515 . . . . . . 7 (((0g𝐺) ∈ 𝑋𝑧𝑌) → ((0g𝐺) 𝑧) = ((𝐹‘(0g𝐺))‘𝑧))
3327, 32sylan 580 . . . . . 6 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → ((0g𝐺) 𝑧) = ((𝐹‘(0g𝐺))‘𝑧))
34 eqid 2736 . . . . . . . . . 10 (0g𝐻) = (0g𝐻)
3525, 34ghmid 19014 . . . . . . . . 9 (𝐹 ∈ (𝐺 GrpHom 𝐻) → (𝐹‘(0g𝐺)) = (0g𝐻))
3635adantr 481 . . . . . . . 8 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → (𝐹‘(0g𝐺)) = (0g𝐻))
378adantr 481 . . . . . . . . 9 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → 𝑌 ∈ V)
384symgid 19183 . . . . . . . . 9 (𝑌 ∈ V → ( I ↾ 𝑌) = (0g𝐻))
3937, 38syl 17 . . . . . . . 8 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → ( I ↾ 𝑌) = (0g𝐻))
4036, 39eqtr4d 2779 . . . . . . 7 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → (𝐹‘(0g𝐺)) = ( I ↾ 𝑌))
4140fveq1d 6844 . . . . . 6 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → ((𝐹‘(0g𝐺))‘𝑧) = (( I ↾ 𝑌)‘𝑧))
42 fvresi 7119 . . . . . . 7 (𝑧𝑌 → (( I ↾ 𝑌)‘𝑧) = 𝑧)
4342adantl 482 . . . . . 6 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → (( I ↾ 𝑌)‘𝑧) = 𝑧)
4433, 41, 433eqtrd 2780 . . . . 5 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → ((0g𝐺) 𝑧) = 𝑧)
4511ad2antrr 724 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → 𝐹:𝑋⟶(Base‘𝐻))
46 simprr 771 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → 𝑣𝑋)
4745, 46ffvelcdmd 7036 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝐹𝑣) ∈ (Base‘𝐻))
488ad2antrr 724 . . . . . . . . . . . 12 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → 𝑌 ∈ V)
494, 10elsymgbas 19155 . . . . . . . . . . . 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 6784 . . . . . . . . . 10 ((𝐹𝑣):𝑌1-1-onto𝑌 → (𝐹𝑣):𝑌𝑌)
5351, 52syl 17 . . . . . . . . 9 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝐹𝑣):𝑌𝑌)
54 simplr 767 . . . . . . . . 9 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → 𝑧𝑌)
55 fvco3 6940 . . . . . . . . 9 (((𝐹𝑣):𝑌𝑌𝑧𝑌) → (((𝐹𝑢) ∘ (𝐹𝑣))‘𝑧) = ((𝐹𝑢)‘((𝐹𝑣)‘𝑧)))
5653, 54, 55syl2anc 584 . . . . . . . 8 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (((𝐹𝑢) ∘ (𝐹𝑣))‘𝑧) = ((𝐹𝑢)‘((𝐹𝑣)‘𝑧)))
57 simpll 765 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
58 simprl 769 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → 𝑢𝑋)
59 eqid 2736 . . . . . . . . . . . 12 (+g𝐺) = (+g𝐺)
60 eqid 2736 . . . . . . . . . . . 12 (+g𝐻) = (+g𝐻)
619, 59, 60ghmlin 19013 . . . . . . . . . . 11 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑢𝑋𝑣𝑋) → (𝐹‘(𝑢(+g𝐺)𝑣)) = ((𝐹𝑢)(+g𝐻)(𝐹𝑣)))
6257, 58, 46, 61syl3anc 1371 . . . . . . . . . 10 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝐹‘(𝑢(+g𝐺)𝑣)) = ((𝐹𝑢)(+g𝐻)(𝐹𝑣)))
6345, 58ffvelcdmd 7036 . . . . . . . . . . 11 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝐹𝑢) ∈ (Base‘𝐻))
644, 10, 60symgov 19165 . . . . . . . . . . 11 (((𝐹𝑢) ∈ (Base‘𝐻) ∧ (𝐹𝑣) ∈ (Base‘𝐻)) → ((𝐹𝑢)(+g𝐻)(𝐹𝑣)) = ((𝐹𝑢) ∘ (𝐹𝑣)))
6563, 47, 64syl2anc 584 . . . . . . . . . 10 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → ((𝐹𝑢)(+g𝐻)(𝐹𝑣)) = ((𝐹𝑢) ∘ (𝐹𝑣)))
6662, 65eqtrd 2776 . . . . . . . . 9 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝐹‘(𝑢(+g𝐺)𝑣)) = ((𝐹𝑢) ∘ (𝐹𝑣)))
6766fveq1d 6844 . . . . . . . 8 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑧) = (((𝐹𝑢) ∘ (𝐹𝑣))‘𝑧))
6853, 54ffvelcdmd 7036 . . . . . . . . 9 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → ((𝐹𝑣)‘𝑧) ∈ 𝑌)
69 fveq2 6842 . . . . . . . . . . 11 (𝑥 = 𝑢 → (𝐹𝑥) = (𝐹𝑢))
7069fveq1d 6844 . . . . . . . . . 10 (𝑥 = 𝑢 → ((𝐹𝑥)‘𝑦) = ((𝐹𝑢)‘𝑦))
71 fveq2 6842 . . . . . . . . . 10 (𝑦 = ((𝐹𝑣)‘𝑧) → ((𝐹𝑢)‘𝑦) = ((𝐹𝑢)‘((𝐹𝑣)‘𝑧)))
72 fvex 6855 . . . . . . . . . 10 ((𝐹𝑢)‘((𝐹𝑣)‘𝑧)) ∈ V
7370, 71, 22, 72ovmpo 7515 . . . . . . . . 9 ((𝑢𝑋 ∧ ((𝐹𝑣)‘𝑧) ∈ 𝑌) → (𝑢 ((𝐹𝑣)‘𝑧)) = ((𝐹𝑢)‘((𝐹𝑣)‘𝑧)))
7458, 68, 73syl2anc 584 . . . . . . . 8 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝑢 ((𝐹𝑣)‘𝑧)) = ((𝐹𝑢)‘((𝐹𝑣)‘𝑧)))
7556, 67, 743eqtr4d 2786 . . . . . . 7 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑧) = (𝑢 ((𝐹𝑣)‘𝑧)))
761ad2antrr 724 . . . . . . . . 9 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → 𝐺 ∈ Grp)
779, 59grpcl 18756 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑢𝑋𝑣𝑋) → (𝑢(+g𝐺)𝑣) ∈ 𝑋)
7876, 58, 46, 77syl3anc 1371 . . . . . . . 8 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝑢(+g𝐺)𝑣) ∈ 𝑋)
79 fveq2 6842 . . . . . . . . . 10 (𝑥 = (𝑢(+g𝐺)𝑣) → (𝐹𝑥) = (𝐹‘(𝑢(+g𝐺)𝑣)))
8079fveq1d 6844 . . . . . . . . 9 (𝑥 = (𝑢(+g𝐺)𝑣) → ((𝐹𝑥)‘𝑦) = ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑦))
81 fveq2 6842 . . . . . . . . 9 (𝑦 = 𝑧 → ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑦) = ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑧))
82 fvex 6855 . . . . . . . . 9 ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑧) ∈ V
8380, 81, 22, 82ovmpo 7515 . . . . . . . 8 (((𝑢(+g𝐺)𝑣) ∈ 𝑋𝑧𝑌) → ((𝑢(+g𝐺)𝑣) 𝑧) = ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑧))
8478, 54, 83syl2anc 584 . . . . . . 7 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → ((𝑢(+g𝐺)𝑣) 𝑧) = ((𝐹‘(𝑢(+g𝐺)𝑣))‘𝑧))
85 fveq2 6842 . . . . . . . . . . 11 (𝑥 = 𝑣 → (𝐹𝑥) = (𝐹𝑣))
8685fveq1d 6844 . . . . . . . . . 10 (𝑥 = 𝑣 → ((𝐹𝑥)‘𝑦) = ((𝐹𝑣)‘𝑦))
87 fveq2 6842 . . . . . . . . . 10 (𝑦 = 𝑧 → ((𝐹𝑣)‘𝑦) = ((𝐹𝑣)‘𝑧))
88 fvex 6855 . . . . . . . . . 10 ((𝐹𝑣)‘𝑧) ∈ V
8986, 87, 22, 88ovmpo 7515 . . . . . . . . 9 ((𝑣𝑋𝑧𝑌) → (𝑣 𝑧) = ((𝐹𝑣)‘𝑧))
9046, 54, 89syl2anc 584 . . . . . . . 8 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝑣 𝑧) = ((𝐹𝑣)‘𝑧))
9190oveq2d 7373 . . . . . . 7 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → (𝑢 (𝑣 𝑧)) = (𝑢 ((𝐹𝑣)‘𝑧)))
9275, 84, 913eqtr4d 2786 . . . . . 6 (((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) ∧ (𝑢𝑋𝑣𝑋)) → ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧)))
9392ralrimivva 3197 . . . . 5 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → ∀𝑢𝑋𝑣𝑋 ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧)))
9444, 93jca 512 . . . 4 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑧𝑌) → (((0g𝐺) 𝑧) = 𝑧 ∧ ∀𝑢𝑋𝑣𝑋 ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧))))
9594ralrimiva 3143 . . 3 (𝐹 ∈ (𝐺 GrpHom 𝐻) → ∀𝑧𝑌 (((0g𝐺) 𝑧) = 𝑧 ∧ ∀𝑢𝑋𝑣𝑋 ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧))))
9624, 95jca 512 . 2 (𝐹 ∈ (𝐺 GrpHom 𝐻) → ( :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑧𝑌 (((0g𝐺) 𝑧) = 𝑧 ∧ ∀𝑢𝑋𝑣𝑋 ((𝑢(+g𝐺)𝑣) 𝑧) = (𝑢 (𝑣 𝑧)))))
979, 59, 25isga 19071 . 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 1344 1 (𝐹 ∈ (𝐺 GrpHom 𝐻) → ∈ (𝐺 GrpAct 𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2943  wral 3064  Vcvv 3445  c0 4282   I cid 5530   × cxp 5631  cres 5635  ccom 5637  wf 6492  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  cmpo 7359  Basecbs 17083  +gcplusg 17133  0gc0g 17321  Grpcgrp 18748   GrpHom cghm 19005   GrpAct cga 19069  SymGrpcsymg 19148
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-rep 5242  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-tp 4591  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-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  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-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-tset 17152  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-efmnd 18679  df-grp 18751  df-ghm 19006  df-ga 19070  df-symg 19149
This theorem is referenced by:  symgga  19189
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