isnat2 - Metamath Proof Explorer

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Theorem isnat2 18040

Description: Property of being a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)

**Hypotheses**
Ref	Expression
natfval.1	⊢ 𝑁 = (𝐶 Nat 𝐷)
natfval.b	⊢ 𝐵 = (Base‘𝐶)
natfval.h	⊢ 𝐻 = (Hom ‘𝐶)
natfval.j	⊢ 𝐽 = (Hom ‘𝐷)
natfval.o	⊢ · = (comp‘𝐷)
isnat2.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
isnat2.g	⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))

**Assertion**
Ref	Expression
isnat2	⊢ (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ↔ (𝐴 ∈ X𝑥 ∈ 𝐵 (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩ · ((1^st ‘𝐺)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘ℎ)) = (((𝑥(2^nd ‘𝐺)𝑦)‘ℎ)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐺)‘𝑦))(𝐴‘𝑥)))))

Distinct variable groups: 𝑥,ℎ,𝑦,𝐴 𝑥,𝐵,𝑦 𝐶,ℎ,𝑥,𝑦 ℎ,𝐹,𝑥,𝑦 ℎ,𝐺,𝑥,𝑦 ℎ,𝐻 𝜑,ℎ,𝑥,𝑦 𝐷,ℎ,𝑥,𝑦 Allowed substitution hints: 𝐵(ℎ) · (𝑥, 𝑦, ℎ) 𝐻(𝑥, 𝑦) 𝐽(𝑥, 𝑦, ℎ) 𝑁(𝑥, 𝑦, ℎ)

**Proof of Theorem isnat2**
Step	Hyp	Ref	Expression
1		relfunc 17951	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
2		isnat2.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
3		1st2nd 8036	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
4	1, 2, 3	sylancr 599	. . . 4 ⊢ (𝜑 → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
5		isnat2.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
6		1st2nd 8036	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → 𝐺 = ⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩)
7	1, 5, 6	sylancr 599	. . . 4 ⊢ (𝜑 → 𝐺 = ⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩)
8	4, 7	oveq12d 7431	. . 3 ⊢ (𝜑 → (𝐹𝑁𝐺) = (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
9	8	eleq2d 2846	. 2 ⊢ (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ↔ 𝐴 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩)))
10		natfval.1	. . 3 ⊢ 𝑁 = (𝐶 Nat 𝐷)
11		natfval.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
12		natfval.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
13		natfval.j	. . 3 ⊢ 𝐽 = (Hom ‘𝐷)
14		natfval.o	. . 3 ⊢ · = (comp‘𝐷)
15		1st2ndbr 8039	. . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
16	1, 2, 15	sylancr 599	. . 3 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
17		1st2ndbr 8039	. . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
18	1, 5, 17	sylancr 599	. . 3 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
19	10, 11, 12, 13, 14, 16, 18	isnat 18039	. 2 ⊢ (𝜑 → (𝐴 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩) ↔ (𝐴 ∈ X𝑥 ∈ 𝐵 (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩ · ((1^st ‘𝐺)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘ℎ)) = (((𝑥(2^nd ‘𝐺)𝑦)‘ℎ)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐺)‘𝑦))(𝐴‘𝑥)))))
20	9, 19	bitrd 282	1 ⊢ (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ↔ (𝐴 ∈ X𝑥 ∈ 𝐵 (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩ · ((1^st ‘𝐺)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘ℎ)) = (((𝑥(2^nd ‘𝐺)𝑦)‘ℎ)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐺)‘𝑦))(𝐴‘𝑥)))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∀wral 3076 ⟨cop 4590 class class class wbr 5103 Rel wrel 5660 ‘cfv 6533 (class class class)co 7413 1^st c1st 7984 2^nd c2nd 7985 Xcixp 8904 Basecbs 17301 Hom chom 17353 compcco 17354 Func cfunc 17943 Nat cnat 18033

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7416 df-oprab 7417 df-mpo 7418 df-1st 7986 df-2nd 7987 df-ixp 8905 df-func 17947 df-nat 18035

This theorem is used by: fuccocl 18056 fucidcl 18057 invfuc 18066 curf2cl 18319 yonedalem4c 18365 yonedalem3 18368

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