Theorem curfpropd 18139

Description: If two categories have the same set of objects, morphisms, and compositions, then they curry the same functor to the same result. (Contributed by Mario Carneiro, 26-Jan-2017.)

Hypotheses

Ref	Expression
curfpropd.1	⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))
curfpropd.2	⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))
curfpropd.3	⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
curfpropd.4	⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
curfpropd.a	⊢ (𝜑 → 𝐴 ∈ Cat)
curfpropd.b	⊢ (𝜑 → 𝐵 ∈ Cat)
curfpropd.c	⊢ (𝜑 → 𝐶 ∈ Cat)
curfpropd.d	⊢ (𝜑 → 𝐷 ∈ Cat)
curfpropd.f	⊢ (𝜑 → 𝐹 ∈ ((𝐴 ×c 𝐶) Func 𝐸))

Assertion

Ref	Expression
curfpropd	⊢ (𝜑 → (⟨𝐴, 𝐶⟩ curryF 𝐹) = (⟨𝐵, 𝐷⟩ curryF 𝐹))

Proof of Theorem curfpropd

Dummy variables 𝑥 𝑔 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		curfpropd.1	. . . . 5 ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))
2	1	homfeqbas 17602	. . . 4 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵))
3		curfpropd.3	. . . . . . . 8 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
4	3	homfeqbas 17602	. . . . . . 7 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
5	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐴)) → (Base‘𝐶) = (Base‘𝐷))
6	5	mpteq1d 5181	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐴)) → (𝑦 ∈ (Base‘𝐶) ↦ (𝑥(1st ‘𝐹)𝑦)) = (𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘𝐹)𝑦)))
7	5	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐶)) → (Base‘𝐶) = (Base‘𝐷))
8		eqid 2731	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		eqid 2731	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
10		eqid 2731	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
11	3	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐴)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (Homf ‘𝐶) = (Homf ‘𝐷))
12		simprl 770	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐴)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
13		simprr 772	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐴)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑧 ∈ (Base‘𝐶))
14	8, 9, 10, 11, 12, 13	homfeqval 17603	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐴)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑦(Hom ‘𝐶)𝑧) = (𝑦(Hom ‘𝐷)𝑧))
15		curfpropd.2	. . . . . . . . . . 11 ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))
16		curfpropd.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ Cat)
17		curfpropd.b	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ Cat)
18	1, 15, 16, 17	cidpropd 17616	. . . . . . . . . 10 ⊢ (𝜑 → (Id‘𝐴) = (Id‘𝐵))
19	18	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐴)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (Id‘𝐴) = (Id‘𝐵))
20	19	fveq1d 6824	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐴)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → ((Id‘𝐴)‘𝑥) = ((Id‘𝐵)‘𝑥))
21	20	oveq1d 7361	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐴)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (((Id‘𝐴)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔) = (((Id‘𝐵)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔))
22	14, 21	mpteq12dv 5178	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐴)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((Id‘𝐴)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)) = (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐵)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))
23	5, 7, 22	mpoeq123dva 7420	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐴)) → (𝑦 ∈ (Base‘𝐶), 𝑧 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((Id‘𝐴)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔))) = (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐵)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔))))
24	6, 23	opeq12d 4833	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐴)) → ⟨(𝑦 ∈ (Base‘𝐶) ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ (Base‘𝐶), 𝑧 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((Id‘𝐴)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩ = ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐵)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩)
25	2, 24	mpteq12dva 5177	. . 3 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐴) ↦ ⟨(𝑦 ∈ (Base‘𝐶) ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ (Base‘𝐶), 𝑧 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((Id‘𝐴)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩) = (𝑥 ∈ (Base‘𝐵) ↦ ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐵)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩))
26	2	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐴)) → (Base‘𝐴) = (Base‘𝐵))
27		eqid 2731	. . . . . 6 ⊢ (Base‘𝐴) = (Base‘𝐴)
28		eqid 2731	. . . . . 6 ⊢ (Hom ‘𝐴) = (Hom ‘𝐴)
29		eqid 2731	. . . . . 6 ⊢ (Hom ‘𝐵) = (Hom ‘𝐵)
30	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (Homf ‘𝐴) = (Homf ‘𝐵))
31		simprl 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘𝐴))
32		simprr 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑦 ∈ (Base‘𝐴))
33	27, 28, 29, 30, 31, 32	homfeqval 17603	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥(Hom ‘𝐴)𝑦) = (𝑥(Hom ‘𝐵)𝑦))
34	4	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐴)𝑦)) → (Base‘𝐶) = (Base‘𝐷))
35		curfpropd.4	. . . . . . . . . 10 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
36		curfpropd.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ Cat)
37		curfpropd.d	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ Cat)
38	3, 35, 36, 37	cidpropd 17616	. . . . . . . . 9 ⊢ (𝜑 → (Id‘𝐶) = (Id‘𝐷))
39	38	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐴)𝑦)) ∧ 𝑧 ∈ (Base‘𝐶)) → (Id‘𝐶) = (Id‘𝐷))
40	39	fveq1d 6824	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐴)𝑦)) ∧ 𝑧 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑧) = ((Id‘𝐷)‘𝑧))
41	40	oveq2d 7362	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐴)𝑦)) ∧ 𝑧 ∈ (Base‘𝐶)) → (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐶)‘𝑧)) = (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))
42	34, 41	mpteq12dva 5177	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐴)𝑦)) → (𝑧 ∈ (Base‘𝐶) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐶)‘𝑧))) = (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))))
43	33, 42	mpteq12dva 5177	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑔 ∈ (𝑥(Hom ‘𝐴)𝑦) ↦ (𝑧 ∈ (Base‘𝐶) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐶)‘𝑧)))) = (𝑔 ∈ (𝑥(Hom ‘𝐵)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))
44	2, 26, 43	mpoeq123dva 7420	. . 3 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐴), 𝑦 ∈ (Base‘𝐴) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐴)𝑦) ↦ (𝑧 ∈ (Base‘𝐶) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐶)‘𝑧))))) = (𝑥 ∈ (Base‘𝐵), 𝑦 ∈ (Base‘𝐵) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐵)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧))))))
45	25, 44	opeq12d 4833	. 2 ⊢ (𝜑 → ⟨(𝑥 ∈ (Base‘𝐴) ↦ ⟨(𝑦 ∈ (Base‘𝐶) ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ (Base‘𝐶), 𝑧 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((Id‘𝐴)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝐴), 𝑦 ∈ (Base‘𝐴) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐴)𝑦) ↦ (𝑧 ∈ (Base‘𝐶) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐶)‘𝑧)))))⟩ = ⟨(𝑥 ∈ (Base‘𝐵) ↦ ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐵)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝐵), 𝑦 ∈ (Base‘𝐵) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐵)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))⟩)
46		eqid 2731	. . 3 ⊢ (⟨𝐴, 𝐶⟩ curryF 𝐹) = (⟨𝐴, 𝐶⟩ curryF 𝐹)
47		curfpropd.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝐴 ×c 𝐶) Func 𝐸))
48		eqid 2731	. . 3 ⊢ (Id‘𝐴) = (Id‘𝐴)
49		eqid 2731	. . 3 ⊢ (Id‘𝐶) = (Id‘𝐶)
50	46, 27, 16, 36, 47, 8, 9, 48, 28, 49	curfval 18129	. 2 ⊢ (𝜑 → (⟨𝐴, 𝐶⟩ curryF 𝐹) = ⟨(𝑥 ∈ (Base‘𝐴) ↦ ⟨(𝑦 ∈ (Base‘𝐶) ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ (Base‘𝐶), 𝑧 ∈ (Base‘𝐶) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((Id‘𝐴)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝐴), 𝑦 ∈ (Base‘𝐴) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐴)𝑦) ↦ (𝑧 ∈ (Base‘𝐶) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐶)‘𝑧)))))⟩)
51		eqid 2731	. . 3 ⊢ (⟨𝐵, 𝐷⟩ curryF 𝐹) = (⟨𝐵, 𝐷⟩ curryF 𝐹)
52		eqid 2731	. . 3 ⊢ (Base‘𝐵) = (Base‘𝐵)
53	1, 15, 3, 35, 16, 17, 36, 37	xpcpropd 18114	. . . . 5 ⊢ (𝜑 → (𝐴 ×c 𝐶) = (𝐵 ×c 𝐷))
54	53	oveq1d 7361	. . . 4 ⊢ (𝜑 → ((𝐴 ×c 𝐶) Func 𝐸) = ((𝐵 ×c 𝐷) Func 𝐸))
55	47, 54	eleqtrd 2833	. . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝐵 ×c 𝐷) Func 𝐸))
56		eqid 2731	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
57		eqid 2731	. . 3 ⊢ (Id‘𝐵) = (Id‘𝐵)
58		eqid 2731	. . 3 ⊢ (Id‘𝐷) = (Id‘𝐷)
59	51, 52, 17, 37, 55, 56, 10, 57, 29, 58	curfval 18129	. 2 ⊢ (𝜑 → (⟨𝐵, 𝐷⟩ curryF 𝐹) = ⟨(𝑥 ∈ (Base‘𝐵) ↦ ⟨(𝑦 ∈ (Base‘𝐷) ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ (Base‘𝐷), 𝑧 ∈ (Base‘𝐷) ↦ (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↦ (((Id‘𝐵)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝐵), 𝑦 ∈ (Base‘𝐵) ↦ (𝑔 ∈ (𝑥(Hom ‘𝐵)𝑦) ↦ (𝑧 ∈ (Base‘𝐷) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)((Id‘𝐷)‘𝑧)))))⟩)
60	45, 50, 59	3eqtr4d 2776	1 ⊢ (𝜑 → (⟨𝐴, 𝐶⟩ curryF 𝐹) = (⟨𝐵, 𝐷⟩ curryF 𝐹))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ⟨cop 4582   ↦ cmpt 5172  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  1^st c1st 7919  2^nd c2nd 7920  Basecbs 17120  Hom chom 17172  Catccat 17570  Idccid 17571  Hom_f chomf 17572  comp_fccomf 17573   Func cfunc 17761   ×_c cxpc 18074   curry_F ccurf 18116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-3 12189  df-4 12190  df-5 12191  df-6 12192  df-7 12193  df-8 12194  df-9 12195  df-n0 12382  df-z 12469  df-dec 12589  df-uz 12733  df-fz 13408  df-struct 17058  df-slot 17093  df-ndx 17105  df-base 17121  df-hom 17185  df-cco 17186  df-cat 17574  df-cid 17575  df-homf 17576  df-comf 17577  df-xpc 18078  df-curf 18120

This theorem is referenced by:  yonpropd  18174  oppcyon  18175  diagpropd  49330