Theorem fuco22natlem1 49467

Description: Lemma for fuco22nat 49471. The commutative square of natural transformation 𝐴 in category 𝐷, mapped to category 𝐸 by the morphism part 𝐿 of the functor. (Contributed by Zhi Wang, 30-Sep-2025.)

Hypotheses

Ref	Expression
fuco22natlem1.x	⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
fuco22natlem1.y	⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
fuco22natlem1.a	⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))
fuco22natlem1.h	⊢ (𝜑 → 𝐻 ∈ (𝑋(Hom ‘𝐶)𝑌))
fuco22natlem1.k	⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)

Assertion

Ref	Expression
fuco22natlem1	⊢ (𝜑 → ((((𝐹‘𝑌)𝐿(𝑀‘𝑌))‘(𝐴‘𝑌))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝐹‘𝑌))⟩(comp‘𝐸)(𝐾‘(𝑀‘𝑌)))(((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝐻))) = ((((𝑀‘𝑋)𝐿(𝑀‘𝑌))‘((𝑋𝑁𝑌)‘𝐻))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝐾‘(𝑀‘𝑌)))(((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋))))

Proof of Theorem fuco22natlem1

Step	Hyp	Ref	Expression
1		eqid 2733	. . . 4 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
2		fuco22natlem1.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))
3		eqid 2733	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
4		eqid 2733	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
5		eqid 2733	. . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷)
6		fuco22natlem1.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
7		fuco22natlem1.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
8		fuco22natlem1.h	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (𝑋(Hom ‘𝐶)𝑌))
9	1, 2, 3, 4, 5, 6, 7, 8	nati 17867	. . 3 ⊢ (𝜑 → ((𝐴‘𝑌)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝐷)(𝑀‘𝑌))((𝑋𝐺𝑌)‘𝐻)) = (((𝑋𝑁𝑌)‘𝐻)(⟨(𝐹‘𝑋), (𝑀‘𝑋)⟩(comp‘𝐷)(𝑀‘𝑌))(𝐴‘𝑋)))
10	9	fveq2d 6832	. 2 ⊢ (𝜑 → (((𝐹‘𝑋)𝐿(𝑀‘𝑌))‘((𝐴‘𝑌)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝐷)(𝑀‘𝑌))((𝑋𝐺𝑌)‘𝐻))) = (((𝐹‘𝑋)𝐿(𝑀‘𝑌))‘(((𝑋𝑁𝑌)‘𝐻)(⟨(𝐹‘𝑋), (𝑀‘𝑋)⟩(comp‘𝐷)(𝑀‘𝑌))(𝐴‘𝑋))))
11		eqid 2733	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
12		eqid 2733	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
13		eqid 2733	. . 3 ⊢ (comp‘𝐸) = (comp‘𝐸)
14		fuco22natlem1.k	. . 3 ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)
15	1, 2	natrcl2 49349	. . . . 5 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
16	3, 11, 15	funcf1 17775	. . . 4 ⊢ (𝜑 → 𝐹:(Base‘𝐶)⟶(Base‘𝐷))
17	16, 6	ffvelcdmd 7024	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐷))
18	16, 7	ffvelcdmd 7024	. . 3 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐷))
19	1, 2	natrcl3 49350	. . . . 5 ⊢ (𝜑 → 𝑀(𝐶 Func 𝐷)𝑁)
20	3, 11, 19	funcf1 17775	. . . 4 ⊢ (𝜑 → 𝑀:(Base‘𝐶)⟶(Base‘𝐷))
21	20, 7	ffvelcdmd 7024	. . 3 ⊢ (𝜑 → (𝑀‘𝑌) ∈ (Base‘𝐷))
22	3, 4, 12, 15, 6, 7	funcf2 17777	. . . 4 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐶)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝐷)(𝐹‘𝑌)))
23	22, 8	ffvelcdmd 7024	. . 3 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝐻) ∈ ((𝐹‘𝑋)(Hom ‘𝐷)(𝐹‘𝑌)))
24	1, 2, 3, 12, 7	natcl 17865	. . 3 ⊢ (𝜑 → (𝐴‘𝑌) ∈ ((𝐹‘𝑌)(Hom ‘𝐷)(𝑀‘𝑌)))
25	11, 12, 5, 13, 14, 17, 18, 21, 23, 24	funcco 17780	. 2 ⊢ (𝜑 → (((𝐹‘𝑋)𝐿(𝑀‘𝑌))‘((𝐴‘𝑌)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝐷)(𝑀‘𝑌))((𝑋𝐺𝑌)‘𝐻))) = ((((𝐹‘𝑌)𝐿(𝑀‘𝑌))‘(𝐴‘𝑌))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝐹‘𝑌))⟩(comp‘𝐸)(𝐾‘(𝑀‘𝑌)))(((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝐻))))
26	20, 6	ffvelcdmd 7024	. . 3 ⊢ (𝜑 → (𝑀‘𝑋) ∈ (Base‘𝐷))
27	1, 2, 3, 12, 6	natcl 17865	. . 3 ⊢ (𝜑 → (𝐴‘𝑋) ∈ ((𝐹‘𝑋)(Hom ‘𝐷)(𝑀‘𝑋)))
28	3, 4, 12, 19, 6, 7	funcf2 17777	. . . 4 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋(Hom ‘𝐶)𝑌)⟶((𝑀‘𝑋)(Hom ‘𝐷)(𝑀‘𝑌)))
29	28, 8	ffvelcdmd 7024	. . 3 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐻) ∈ ((𝑀‘𝑋)(Hom ‘𝐷)(𝑀‘𝑌)))
30	11, 12, 5, 13, 14, 17, 26, 21, 27, 29	funcco 17780	. 2 ⊢ (𝜑 → (((𝐹‘𝑋)𝐿(𝑀‘𝑌))‘(((𝑋𝑁𝑌)‘𝐻)(⟨(𝐹‘𝑋), (𝑀‘𝑋)⟩(comp‘𝐷)(𝑀‘𝑌))(𝐴‘𝑋))) = ((((𝑀‘𝑋)𝐿(𝑀‘𝑌))‘((𝑋𝑁𝑌)‘𝐻))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝐾‘(𝑀‘𝑌)))(((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋))))
31	10, 25, 30	3eqtr3d 2776	1 ⊢ (𝜑 → ((((𝐹‘𝑌)𝐿(𝑀‘𝑌))‘(𝐴‘𝑌))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝐹‘𝑌))⟩(comp‘𝐸)(𝐾‘(𝑀‘𝑌)))(((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝐻))) = ((((𝑀‘𝑋)𝐿(𝑀‘𝑌))‘((𝑋𝑁𝑌)‘𝐻))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝐾‘(𝑀‘𝑌)))(((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋))))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ⟨cop 4581   class class class wbr 5093  ‘cfv 6486  (class class class)co 7352  Basecbs 17122  Hom chom 17174  compcco 17175   Func cfunc 17763   Nat cnat 17853

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 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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-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-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-map 8758  df-ixp 8828  df-func 17767  df-nat 17855

This theorem is referenced by:  fuco22natlem2  49468