Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  funcrngcsetc Structured version   Visualization version   GIF version

Theorem funcrngcsetc 41283
 Description: The "natural forgetful functor" from the category of non-unital rings into the category of sets which sends each non-unital ring to its underlying set (base set) and the morphisms (non-unital ring homomorphisms) to mappings of the corresponding base sets. An alternate proof is provided in funcrngcsetcALT 41284, using cofuval2 16468 to construct the "natural forgetful functor" from the category of non-unital rings into the category of sets by composing the "inclusion functor" from the category of non-unital rings into the category of extensible structures, see rngcifuestrc 41282, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 16710. (Contributed by AV, 26-Mar-2020.)
Hypotheses
Ref Expression
funcrngcsetc.r 𝑅 = (RngCat‘𝑈)
funcrngcsetc.s 𝑆 = (SetCat‘𝑈)
funcrngcsetc.b 𝐵 = (Base‘𝑅)
funcrngcsetc.u (𝜑𝑈 ∈ WUni)
funcrngcsetc.f (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
funcrngcsetc.g (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))))
Assertion
Ref Expression
funcrngcsetc (𝜑𝐹(𝑅 Func 𝑆)𝐺)
Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆   𝑥,𝑈,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝑆(𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem funcrngcsetc
Dummy variables 𝑎 𝑏 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2621 . . . . . 6 (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
2 funcrngcsetc.s . . . . . 6 𝑆 = (SetCat‘𝑈)
3 eqid 2621 . . . . . 6 (Base‘(ExtStrCat‘𝑈)) = (Base‘(ExtStrCat‘𝑈))
4 eqid 2621 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
5 funcrngcsetc.u . . . . . 6 (𝜑𝑈 ∈ WUni)
61, 5estrcbas 16686 . . . . . . 7 (𝜑𝑈 = (Base‘(ExtStrCat‘𝑈)))
76mpteq1d 4698 . . . . . 6 (𝜑 → (𝑥𝑈 ↦ (Base‘𝑥)) = (𝑥 ∈ (Base‘(ExtStrCat‘𝑈)) ↦ (Base‘𝑥)))
8 mpt2eq12 6668 . . . . . . 7 ((𝑈 = (Base‘(ExtStrCat‘𝑈)) ∧ 𝑈 = (Base‘(ExtStrCat‘𝑈))) → (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))) = (𝑥 ∈ (Base‘(ExtStrCat‘𝑈)), 𝑦 ∈ (Base‘(ExtStrCat‘𝑈)) ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))
96, 6, 8syl2anc 692 . . . . . 6 (𝜑 → (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))) = (𝑥 ∈ (Base‘(ExtStrCat‘𝑈)), 𝑦 ∈ (Base‘(ExtStrCat‘𝑈)) ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))
101, 2, 3, 4, 5, 7, 9funcestrcsetc 16710 . . . . 5 (𝜑 → (𝑥𝑈 ↦ (Base‘𝑥))((ExtStrCat‘𝑈) Func 𝑆)(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))
11 df-br 4614 . . . . 5 ((𝑥𝑈 ↦ (Base‘𝑥))((ExtStrCat‘𝑈) Func 𝑆)(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))) ↔ ⟨(𝑥𝑈 ↦ (Base‘𝑥)), (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))⟩ ∈ ((ExtStrCat‘𝑈) Func 𝑆))
1210, 11sylib 208 . . . 4 (𝜑 → ⟨(𝑥𝑈 ↦ (Base‘𝑥)), (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))⟩ ∈ ((ExtStrCat‘𝑈) Func 𝑆))
13 funcrngcsetc.r . . . . . . 7 𝑅 = (RngCat‘𝑈)
14 eqid 2621 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
1513, 14, 5rngcbas 41250 . . . . . 6 (𝜑 → (Base‘𝑅) = (𝑈 ∩ Rng))
16 incom 3783 . . . . . 6 (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
1715, 16syl6eq 2671 . . . . 5 (𝜑 → (Base‘𝑅) = (Rng ∩ 𝑈))
18 eqid 2621 . . . . . 6 (Hom ‘𝑅) = (Hom ‘𝑅)
1913, 14, 5, 18rngchomfval 41251 . . . . 5 (𝜑 → (Hom ‘𝑅) = ( RngHomo ↾ ((Base‘𝑅) × (Base‘𝑅))))
201, 5, 17, 19rnghmsubcsetc 41262 . . . 4 (𝜑 → (Hom ‘𝑅) ∈ (Subcat‘(ExtStrCat‘𝑈)))
2112, 20funcres 16477 . . 3 (𝜑 → (⟨(𝑥𝑈 ↦ (Base‘𝑥)), (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))⟩ ↾f (Hom ‘𝑅)) ∈ (((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func 𝑆))
22 mptexg 6438 . . . . . 6 (𝑈 ∈ WUni → (𝑥𝑈 ↦ (Base‘𝑥)) ∈ V)
235, 22syl 17 . . . . 5 (𝜑 → (𝑥𝑈 ↦ (Base‘𝑥)) ∈ V)
24 fvex 6158 . . . . . 6 (Hom ‘𝑅) ∈ V
2524a1i 11 . . . . 5 (𝜑 → (Hom ‘𝑅) ∈ V)
26 mpt2exga 7191 . . . . . 6 ((𝑈 ∈ WUni ∧ 𝑈 ∈ WUni) → (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))) ∈ V)
275, 5, 26syl2anc 692 . . . . 5 (𝜑 → (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))) ∈ V)
2815, 19rnghmresfn 41248 . . . . 5 (𝜑 → (Hom ‘𝑅) Fn ((Base‘𝑅) × (Base‘𝑅)))
2923, 25, 27, 28resfval2 16474 . . . 4 (𝜑 → (⟨(𝑥𝑈 ↦ (Base‘𝑥)), (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))⟩ ↾f (Hom ‘𝑅)) = ⟨((𝑥𝑈 ↦ (Base‘𝑥)) ↾ (Base‘𝑅)), (𝑎 ∈ (Base‘𝑅), 𝑏 ∈ (Base‘𝑅) ↦ ((𝑎(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))𝑏) ↾ (𝑎(Hom ‘𝑅)𝑏)))⟩)
30 inss1 3811 . . . . . . . 8 (𝑈 ∩ Rng) ⊆ 𝑈
3115, 30syl6eqss 3634 . . . . . . 7 (𝜑 → (Base‘𝑅) ⊆ 𝑈)
3231resmptd 5411 . . . . . 6 (𝜑 → ((𝑥𝑈 ↦ (Base‘𝑥)) ↾ (Base‘𝑅)) = (𝑥 ∈ (Base‘𝑅) ↦ (Base‘𝑥)))
33 funcrngcsetc.f . . . . . . 7 (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
34 funcrngcsetc.b . . . . . . . . 9 𝐵 = (Base‘𝑅)
3534a1i 11 . . . . . . . 8 (𝜑𝐵 = (Base‘𝑅))
3635mpteq1d 4698 . . . . . . 7 (𝜑 → (𝑥𝐵 ↦ (Base‘𝑥)) = (𝑥 ∈ (Base‘𝑅) ↦ (Base‘𝑥)))
3733, 36eqtr2d 2656 . . . . . 6 (𝜑 → (𝑥 ∈ (Base‘𝑅) ↦ (Base‘𝑥)) = 𝐹)
3832, 37eqtrd 2655 . . . . 5 (𝜑 → ((𝑥𝑈 ↦ (Base‘𝑥)) ↾ (Base‘𝑅)) = 𝐹)
39 funcrngcsetc.g . . . . . 6 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))))
40 oveq1 6611 . . . . . . . . 9 (𝑥 = 𝑎 → (𝑥 RngHomo 𝑦) = (𝑎 RngHomo 𝑦))
4140reseq2d 5356 . . . . . . . 8 (𝑥 = 𝑎 → ( I ↾ (𝑥 RngHomo 𝑦)) = ( I ↾ (𝑎 RngHomo 𝑦)))
42 oveq2 6612 . . . . . . . . 9 (𝑦 = 𝑏 → (𝑎 RngHomo 𝑦) = (𝑎 RngHomo 𝑏))
4342reseq2d 5356 . . . . . . . 8 (𝑦 = 𝑏 → ( I ↾ (𝑎 RngHomo 𝑦)) = ( I ↾ (𝑎 RngHomo 𝑏)))
4441, 43cbvmpt2v 6688 . . . . . . 7 (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))) = (𝑎𝐵, 𝑏𝐵 ↦ ( I ↾ (𝑎 RngHomo 𝑏)))
4544a1i 11 . . . . . 6 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))) = (𝑎𝐵, 𝑏𝐵 ↦ ( I ↾ (𝑎 RngHomo 𝑏))))
4634a1i 11 . . . . . . 7 ((𝜑𝑎𝐵) → 𝐵 = (Base‘𝑅))
47 eqidd 2622 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))) = (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))
48 fveq2 6148 . . . . . . . . . . . . 13 (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏))
49 fveq2 6148 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎))
5048, 49oveqan12rd 6624 . . . . . . . . . . . 12 ((𝑥 = 𝑎𝑦 = 𝑏) → ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
5150reseq2d 5356 . . . . . . . . . . 11 ((𝑥 = 𝑎𝑦 = 𝑏) → ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = ( I ↾ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
5251adantl 482 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ (𝑥 = 𝑎𝑦 = 𝑏)) → ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = ( I ↾ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
5334, 31syl5eqss 3628 . . . . . . . . . . . . . 14 (𝜑𝐵𝑈)
5453sseld 3582 . . . . . . . . . . . . 13 (𝜑 → (𝑎𝐵𝑎𝑈))
5554com12 32 . . . . . . . . . . . 12 (𝑎𝐵 → (𝜑𝑎𝑈))
5655adantr 481 . . . . . . . . . . 11 ((𝑎𝐵𝑏𝐵) → (𝜑𝑎𝑈))
5756impcom 446 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝑈)
5853sseld 3582 . . . . . . . . . . . 12 (𝜑 → (𝑏𝐵𝑏𝑈))
5958adantld 483 . . . . . . . . . . 11 (𝜑 → ((𝑎𝐵𝑏𝐵) → 𝑏𝑈))
6059imp 445 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝑈)
61 ovex 6632 . . . . . . . . . . . 12 ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ∈ V
6261a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ∈ V)
6362resiexd 6434 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( I ↾ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))) ∈ V)
6447, 52, 57, 60, 63ovmpt2d 6741 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))𝑏) = ( I ↾ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
6564reseq1d 5355 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((𝑎(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))𝑏) ↾ (𝑎(Hom ‘𝑅)𝑏)) = (( I ↾ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))) ↾ (𝑎(Hom ‘𝑅)𝑏)))
665adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑈 ∈ WUni)
67 simprl 793 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝐵)
68 simprr 795 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝐵)
6913, 34, 66, 18, 67, 68rngchom 41252 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(Hom ‘𝑅)𝑏) = (𝑎 RngHomo 𝑏))
7069reseq2d 5356 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (( I ↾ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))) ↾ (𝑎(Hom ‘𝑅)𝑏)) = (( I ↾ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))) ↾ (𝑎 RngHomo 𝑏)))
71 eqid 2621 . . . . . . . . . . . 12 (Base‘𝑎) = (Base‘𝑎)
72 eqid 2621 . . . . . . . . . . . 12 (Base‘𝑏) = (Base‘𝑏)
7371, 72rnghmf 41184 . . . . . . . . . . 11 (𝑓 ∈ (𝑎 RngHomo 𝑏) → 𝑓:(Base‘𝑎)⟶(Base‘𝑏))
74 fvex 6158 . . . . . . . . . . . . . 14 (Base‘𝑏) ∈ V
75 fvex 6158 . . . . . . . . . . . . . 14 (Base‘𝑎) ∈ V
7674, 75pm3.2i 471 . . . . . . . . . . . . 13 ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
7776a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V))
78 elmapg 7815 . . . . . . . . . . . 12 (((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V) → (𝑓 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ↔ 𝑓:(Base‘𝑎)⟶(Base‘𝑏)))
7977, 78syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑓 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ↔ 𝑓:(Base‘𝑎)⟶(Base‘𝑏)))
8073, 79syl5ibr 236 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑓 ∈ (𝑎 RngHomo 𝑏) → 𝑓 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
8180ssrdv 3589 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 RngHomo 𝑏) ⊆ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
8281resabs1d 5387 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (( I ↾ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))) ↾ (𝑎 RngHomo 𝑏)) = ( I ↾ (𝑎 RngHomo 𝑏)))
8365, 70, 823eqtrrd 2660 . . . . . . 7 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( I ↾ (𝑎 RngHomo 𝑏)) = ((𝑎(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))𝑏) ↾ (𝑎(Hom ‘𝑅)𝑏)))
8435, 46, 83mpt2eq123dva 6669 . . . . . 6 (𝜑 → (𝑎𝐵, 𝑏𝐵 ↦ ( I ↾ (𝑎 RngHomo 𝑏))) = (𝑎 ∈ (Base‘𝑅), 𝑏 ∈ (Base‘𝑅) ↦ ((𝑎(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))𝑏) ↾ (𝑎(Hom ‘𝑅)𝑏))))
8539, 45, 843eqtrrd 2660 . . . . 5 (𝜑 → (𝑎 ∈ (Base‘𝑅), 𝑏 ∈ (Base‘𝑅) ↦ ((𝑎(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))𝑏) ↾ (𝑎(Hom ‘𝑅)𝑏))) = 𝐺)
8638, 85opeq12d 4378 . . . 4 (𝜑 → ⟨((𝑥𝑈 ↦ (Base‘𝑥)) ↾ (Base‘𝑅)), (𝑎 ∈ (Base‘𝑅), 𝑏 ∈ (Base‘𝑅) ↦ ((𝑎(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))𝑏) ↾ (𝑎(Hom ‘𝑅)𝑏)))⟩ = ⟨𝐹, 𝐺⟩)
8729, 86eqtr2d 2656 . . 3 (𝜑 → ⟨𝐹, 𝐺⟩ = (⟨(𝑥𝑈 ↦ (Base‘𝑥)), (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))⟩ ↾f (Hom ‘𝑅)))
8813, 5, 15, 19rngcval 41247 . . . 4 (𝜑𝑅 = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))
8988oveq1d 6619 . . 3 (𝜑 → (𝑅 Func 𝑆) = (((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func 𝑆))
9021, 87, 893eltr4d 2713 . 2 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
91 df-br 4614 . 2 (𝐹(𝑅 Func 𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
9290, 91sylibr 224 1 (𝜑𝐹(𝑅 Func 𝑆)𝐺)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  Vcvv 3186   ∩ cin 3554  ⟨cop 4154   class class class wbr 4613   ↦ cmpt 4673   I cid 4984   ↾ cres 5076  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604   ↦ cmpt2 6606   ↑𝑚 cmap 7802  WUnicwun 9466  Basecbs 15781  Hom chom 15873   ↾cat cresc 16389   Func cfunc 16435   ↾f cresf 16438  SetCatcsetc 16646  ExtStrCatcestrc 16683  Rngcrng 41159   RngHomo crngh 41170  RngCatcrngc 41242 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 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  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 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-wun 9468  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-fz 12269  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-hom 15887  df-cco 15888  df-0g 16023  df-cat 16250  df-cid 16251  df-homf 16252  df-ssc 16391  df-resc 16392  df-subc 16393  df-func 16439  df-resf 16442  df-setc 16647  df-estrc 16684  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-mhm 17256  df-grp 17346  df-ghm 17579  df-abl 18117  df-mgp 18411  df-mgmhm 41064  df-rng0 41160  df-rnghomo 41172  df-rngc 41244 This theorem is referenced by: (None)
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