Theorem curfpropd 18143

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 17606	. . . 4 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵))
3		curfpropd.3	. . . . . . . 8 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
4	3	homfeqbas 17606	. . . . . . 7 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
5	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐴)) → (Base‘𝐶) = (Base‘𝐷))
6	5	mpteq1d 5185	. . . . 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 2733	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		eqid 2733	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
10		eqid 2733	. . . . . . . 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 17607	. . . . . . 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 17620	. . . . . . . . . 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 6832	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐴)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → ((Id‘𝐴)‘𝑥) = ((Id‘𝐵)‘𝑥))
21	20	oveq1d 7369	. . . . . . 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 5182	. . . . . 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 7428	. . . . 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 4834	. . . 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 5181	. . 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 2733	. . . . . 6 ⊢ (Base‘𝐴) = (Base‘𝐴)
28		eqid 2733	. . . . . 6 ⊢ (Hom ‘𝐴) = (Hom ‘𝐴)
29		eqid 2733	. . . . . 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 17607	. . . . 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 17620	. . . . . . . . 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 6832	. . . . . . 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 7370	. . . . . 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 5181	. . . . 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 5181	. . . 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 7428	. . 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 4834	. 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 2733	. . 3 ⊢ (⟨𝐴, 𝐶⟩ curryF 𝐹) = (⟨𝐴, 𝐶⟩ curryF 𝐹)
47		curfpropd.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝐴 ×c 𝐶) Func 𝐸))
48		eqid 2733	. . 3 ⊢ (Id‘𝐴) = (Id‘𝐴)
49		eqid 2733	. . 3 ⊢ (Id‘𝐶) = (Id‘𝐶)
50	46, 27, 16, 36, 47, 8, 9, 48, 28, 49	curfval 18133	. 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 2733	. . 3 ⊢ (⟨𝐵, 𝐷⟩ curryF 𝐹) = (⟨𝐵, 𝐷⟩ curryF 𝐹)
52		eqid 2733	. . 3 ⊢ (Base‘𝐵) = (Base‘𝐵)
53	1, 15, 3, 35, 16, 17, 36, 37	xpcpropd 18118	. . . . 5 ⊢ (𝜑 → (𝐴 ×c 𝐶) = (𝐵 ×c 𝐷))
54	53	oveq1d 7369	. . . 4 ⊢ (𝜑 → ((𝐴 ×c 𝐶) Func 𝐸) = ((𝐵 ×c 𝐷) Func 𝐸))
55	47, 54	eleqtrd 2835	. . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝐵 ×c 𝐷) Func 𝐸))
56		eqid 2733	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
57		eqid 2733	. . 3 ⊢ (Id‘𝐵) = (Id‘𝐵)
58		eqid 2733	. . 3 ⊢ (Id‘𝐷) = (Id‘𝐷)
59	51, 52, 17, 37, 55, 56, 10, 57, 29, 58	curfval 18133	. 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 2778	1 ⊢ (𝜑 → (⟨𝐴, 𝐶⟩ curryF 𝐹) = (⟨𝐵, 𝐷⟩ curryF 𝐹))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ⟨cop 4583   ↦ cmpt 5176  ‘cfv 6488  (class class class)co 7354   ∈ cmpo 7356  1^st c1st 7927  2^nd c2nd 7928  Basecbs 17124  Hom chom 17176  Catccat 17574  Idccid 17575  Hom_f chomf 17576  comp_fccomf 17577   Func cfunc 17765   ×_c cxpc 18078   curry_F ccurf 18120

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7676  ax-cnex 11071  ax-resscn 11072  ax-1cn 11073  ax-icn 11074  ax-addcl 11075  ax-addrcl 11076  ax-mulcl 11077  ax-mulrcl 11078  ax-mulcom 11079  ax-addass 11080  ax-mulass 11081  ax-distr 11082  ax-i2m1 11083  ax-1ne0 11084  ax-1rid 11085  ax-rnegex 11086  ax-rrecex 11087  ax-cnre 11088  ax-pre-lttri 11089  ax-pre-lttrn 11090  ax-pre-ltadd 11091  ax-pre-mulgt0 11092

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-riota 7311  df-ov 7357  df-oprab 7358  df-mpo 7359  df-om 7805  df-1st 7929  df-2nd 7930  df-frecs 8219  df-wrecs 8250  df-recs 8299  df-rdg 8337  df-1o 8393  df-er 8630  df-en 8878  df-dom 8879  df-sdom 8880  df-fin 8881  df-pnf 11157  df-mnf 11158  df-xr 11159  df-ltxr 11160  df-le 11161  df-sub 11355  df-neg 11356  df-nn 12135  df-2 12197  df-3 12198  df-4 12199  df-5 12200  df-6 12201  df-7 12202  df-8 12203  df-9 12204  df-n0 12391  df-z 12478  df-dec 12597  df-uz 12741  df-fz 13412  df-struct 17062  df-slot 17097  df-ndx 17109  df-base 17125  df-hom 17189  df-cco 17190  df-cat 17578  df-cid 17579  df-homf 17580  df-comf 17581  df-xpc 18082  df-curf 18124

This theorem is referenced by:  yonpropd  18178  oppcyon  18179  diagpropd  49420