isnat2 - Metamath Proof Explorer

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Theorem isnat2 16431

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 16345	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
2		isnat2.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
3		1st2nd 7105	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
4	1, 2, 3	sylancr 694	. . . 4 ⊢ (𝜑 → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
5		isnat2.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
6		1st2nd 7105	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → 𝐺 = ⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩)
7	1, 5, 6	sylancr 694	. . . 4 ⊢ (𝜑 → 𝐺 = ⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩)
8	4, 7	oveq12d 6567	. . 3 ⊢ (𝜑 → (𝐹𝑁𝐺) = (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
9	8	eleq2d 2673	. 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 7108	. . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
16	1, 2, 15	sylancr 694	. . 3 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
17		1st2ndbr 7108	. . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
18	1, 5, 17	sylancr 694	. . 3 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
19	10, 11, 12, 13, 14, 16, 18	isnat 16430	. 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 267	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

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⟨cop 4131 class class class wbr 4583 Rel wrel 5043 ‘cfv 5804 (class class class)co 6549 1^st c1st 7057 2^nd c2nd 7058 Xcixp 7794 Basecbs 15695 Hom chom 15779 compcco 15780 Func cfunc 16337 Nat cnat 16424

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-ixp 7795 df-func 16341 df-nat 16426

This theorem is referenced by: fuccocl 16447 fucidcl 16448 invfuc 16457 curf2cl 16694 yonedalem4c 16740 yonedalem3 16743

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