isnat2 - Metamath Proof Explorer

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Theorem isnat2 18176

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 18087	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
2		isnat2.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
3		1st2nd 8084	. . . . 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 8084	. . . . 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 7469	. . 3 ⊢ (𝜑 → (𝐹𝑁𝐺) = (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
9	8	eleq2d 2850	. 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 8087	. . . 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 8087	. . . 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 18175	. 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 2147 ∀wral 3080 ⟨cop 4840 class class class wbr 5108 Rel wrel 5663 ‘cfv 6566 (class class class)co 7451 1^st c1st 8033 2^nd c2nd 8034 Xcixp 8963 Basecbs 17431 Hom chom 17483 compcco 17484 Func cfunc 18079 Nat cnat 18169

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-id 5553 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-ov 7454 df-oprab 7455 df-mpo 7456 df-1st 8035 df-2nd 8036 df-ixp 8964 df-func 18083 df-nat 18171

This theorem is used by: fuccocl 18192 fucidcl 18193 invfuc 18202 curf2cl 18455 yonedalem4c 18501 yonedalem3 18504 isnat3 50673

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