Theorem curfpropd 17932

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 17386	. . . 4 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵))
3		curfpropd.3	. . . . . . . 8 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
4	3	homfeqbas 17386	. . . . . . 7 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
5	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐴)) → (Base‘𝐶) = (Base‘𝐷))
6	5	mpteq1d 5173	. . . . 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 2739	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		eqid 2739	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
10		eqid 2739	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
11	3	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐴)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (Homf ‘𝐶) = (Homf ‘𝐷))
12		simprl 767	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐴)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
13		simprr 769	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐴)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑧 ∈ (Base‘𝐶))
14	8, 9, 10, 11, 12, 13	homfeqval 17387	. . . . . . 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 17400	. . . . . . . . . 10 ⊢ (𝜑 → (Id‘𝐴) = (Id‘𝐵))
19	18	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐴)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (Id‘𝐴) = (Id‘𝐵))
20	19	fveq1d 6770	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐴)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → ((Id‘𝐴)‘𝑥) = ((Id‘𝐵)‘𝑥))
21	20	oveq1d 7283	. . . . . . 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 5169	. . . . . 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 7340	. . . . 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 4817	. . . 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 5167	. . 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 2739	. . . . . 6 ⊢ (Base‘𝐴) = (Base‘𝐴)
28		eqid 2739	. . . . . 6 ⊢ (Hom ‘𝐴) = (Hom ‘𝐴)
29		eqid 2739	. . . . . 6 ⊢ (Hom ‘𝐵) = (Hom ‘𝐵)
30	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (Homf ‘𝐴) = (Homf ‘𝐵))
31		simprl 767	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘𝐴))
32		simprr 769	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑦 ∈ (Base‘𝐴))
33	27, 28, 29, 30, 31, 32	homfeqval 17387	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥(Hom ‘𝐴)𝑦) = (𝑥(Hom ‘𝐵)𝑦))
34	4	ad2antrr 722	. . . . . 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 17400	. . . . . . . . 9 ⊢ (𝜑 → (Id‘𝐶) = (Id‘𝐷))
39	38	ad3antrrr 726	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐴)𝑦)) ∧ 𝑧 ∈ (Base‘𝐶)) → (Id‘𝐶) = (Id‘𝐷))
40	39	fveq1d 6770	. . . . . . 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 7284	. . . . . 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 5167	. . . . 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 5167	. . . 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 7340	. . 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 4817	. 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 2739	. . 3 ⊢ (⟨𝐴, 𝐶⟩ curryF 𝐹) = (⟨𝐴, 𝐶⟩ curryF 𝐹)
47		curfpropd.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝐴 ×c 𝐶) Func 𝐸))
48		eqid 2739	. . 3 ⊢ (Id‘𝐴) = (Id‘𝐴)
49		eqid 2739	. . 3 ⊢ (Id‘𝐶) = (Id‘𝐶)
50	46, 27, 16, 36, 47, 8, 9, 48, 28, 49	curfval 17922	. 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 2739	. . 3 ⊢ (⟨𝐵, 𝐷⟩ curryF 𝐹) = (⟨𝐵, 𝐷⟩ curryF 𝐹)
52		eqid 2739	. . 3 ⊢ (Base‘𝐵) = (Base‘𝐵)
53	1, 15, 3, 35, 16, 17, 36, 37	xpcpropd 17907	. . . . 5 ⊢ (𝜑 → (𝐴 ×c 𝐶) = (𝐵 ×c 𝐷))
54	53	oveq1d 7283	. . . 4 ⊢ (𝜑 → ((𝐴 ×c 𝐶) Func 𝐸) = ((𝐵 ×c 𝐷) Func 𝐸))
55	47, 54	eleqtrd 2842	. . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝐵 ×c 𝐷) Func 𝐸))
56		eqid 2739	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
57		eqid 2739	. . 3 ⊢ (Id‘𝐵) = (Id‘𝐵)
58		eqid 2739	. . 3 ⊢ (Id‘𝐷) = (Id‘𝐷)
59	51, 52, 17, 37, 55, 56, 10, 57, 29, 58	curfval 17922	. 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 2789	1 ⊢ (𝜑 → (⟨𝐴, 𝐶⟩ curryF 𝐹) = (⟨𝐵, 𝐷⟩ curryF 𝐹))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2109  ⟨cop 4572   ↦ cmpt 5161  ‘cfv 6430  (class class class)co 7268   ∈ cmpo 7270  1^st c1st 7815  2^nd c2nd 7816  Basecbs 16893  Hom chom 16954  Catccat 17354  Idccid 17355  Hom_f chomf 17356  comp_fccomf 17357   Func cfunc 17550   ×_c cxpc 17866   curry_F ccurf 17909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579  ax-cnex 10911  ax-resscn 10912  ax-1cn 10913  ax-icn 10914  ax-addcl 10915  ax-addrcl 10916  ax-mulcl 10917  ax-mulrcl 10918  ax-mulcom 10919  ax-addass 10920  ax-mulass 10921  ax-distr 10922  ax-i2m1 10923  ax-1ne0 10924  ax-1rid 10925  ax-rnegex 10926  ax-rrecex 10927  ax-cnre 10928  ax-pre-lttri 10929  ax-pre-lttrn 10930  ax-pre-ltadd 10931  ax-pre-mulgt0 10932

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-pred 6199  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-om 7701  df-1st 7817  df-2nd 7818  df-frecs 8081  df-wrecs 8112  df-recs 8186  df-rdg 8225  df-1o 8281  df-er 8472  df-en 8708  df-dom 8709  df-sdom 8710  df-fin 8711  df-pnf 10995  df-mnf 10996  df-xr 10997  df-ltxr 10998  df-le 10999  df-sub 11190  df-neg 11191  df-nn 11957  df-2 12019  df-3 12020  df-4 12021  df-5 12022  df-6 12023  df-7 12024  df-8 12025  df-9 12026  df-n0 12217  df-z 12303  df-dec 12420  df-uz 12565  df-fz 13222  df-struct 16829  df-slot 16864  df-ndx 16876  df-base 16894  df-hom 16967  df-cco 16968  df-cat 17358  df-cid 17359  df-homf 17360  df-comf 17361  df-xpc 17870  df-curf 17913

This theorem is referenced by:  yonpropd  17967  oppcyon  17968