Theorem fuco22natlem1 49839

Description: Lemma for fuco22nat 49843. 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 2740	. . . 4 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
2		fuco22natlem1.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))
3		eqid 2740	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
4		eqid 2740	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
5		eqid 2740	. . . 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 17923	. . 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 6838	. 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 2740	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
12		eqid 2740	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
13		eqid 2740	. . 3 ⊢ (comp‘𝐸) = (comp‘𝐸)
14		fuco22natlem1.k	. . 3 ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)
15	1, 2	natrcl2 49721	. . . . 5 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
16	3, 11, 15	funcf1 17831	. . . 4 ⊢ (𝜑 → 𝐹:(Base‘𝐶)⟶(Base‘𝐷))
17	16, 6	ffvelcdmd 7033	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐷))
18	16, 7	ffvelcdmd 7033	. . 3 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐷))
19	1, 2	natrcl3 49722	. . . . 5 ⊢ (𝜑 → 𝑀(𝐶 Func 𝐷)𝑁)
20	3, 11, 19	funcf1 17831	. . . 4 ⊢ (𝜑 → 𝑀:(Base‘𝐶)⟶(Base‘𝐷))
21	20, 7	ffvelcdmd 7033	. . 3 ⊢ (𝜑 → (𝑀‘𝑌) ∈ (Base‘𝐷))
22	3, 4, 12, 15, 6, 7	funcf2 17833	. . . 4 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐶)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝐷)(𝐹‘𝑌)))
23	22, 8	ffvelcdmd 7033	. . 3 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝐻) ∈ ((𝐹‘𝑋)(Hom ‘𝐷)(𝐹‘𝑌)))
24	1, 2, 3, 12, 7	natcl 17921	. . 3 ⊢ (𝜑 → (𝐴‘𝑌) ∈ ((𝐹‘𝑌)(Hom ‘𝐷)(𝑀‘𝑌)))
25	11, 12, 5, 13, 14, 17, 18, 21, 23, 24	funcco 17836	. 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 7033	. . 3 ⊢ (𝜑 → (𝑀‘𝑋) ∈ (Base‘𝐷))
27	1, 2, 3, 12, 6	natcl 17921	. . 3 ⊢ (𝜑 → (𝐴‘𝑋) ∈ ((𝐹‘𝑋)(Hom ‘𝐷)(𝑀‘𝑋)))
28	3, 4, 12, 19, 6, 7	funcf2 17833	. . . 4 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋(Hom ‘𝐶)𝑌)⟶((𝑀‘𝑋)(Hom ‘𝐷)(𝑀‘𝑌)))
29	28, 8	ffvelcdmd 7033	. . 3 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐻) ∈ ((𝑀‘𝑋)(Hom ‘𝐷)(𝑀‘𝑌)))
30	11, 12, 5, 13, 14, 17, 26, 21, 27, 29	funcco 17836	. 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 2783	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 1547   ∈ wcel 2119  ⟨cop 4568   class class class wbr 5079  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  Hom chom 17229  compcco 17230   Func cfunc 17819   Nat cnat 17909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-map 8772  df-ixp 8843  df-func 17823  df-nat 17911

This theorem is referenced by:  fuco22natlem2  49840