Theorem fuco22natlem1 49829

Description: Lemma for fuco22nat 49833. 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 2737	. . . 4 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
2		fuco22natlem1.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))
3		eqid 2737	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
4		eqid 2737	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
5		eqid 2737	. . . 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 17916	. . 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 2737	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
12		eqid 2737	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
13		eqid 2737	. . 3 ⊢ (comp‘𝐸) = (comp‘𝐸)
14		fuco22natlem1.k	. . 3 ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)
15	1, 2	natrcl2 49711	. . . . 5 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
16	3, 11, 15	funcf1 17824	. . . 4 ⊢ (𝜑 → 𝐹:(Base‘𝐶)⟶(Base‘𝐷))
17	16, 6	ffvelcdmd 7031	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐷))
18	16, 7	ffvelcdmd 7031	. . 3 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐷))
19	1, 2	natrcl3 49712	. . . . 5 ⊢ (𝜑 → 𝑀(𝐶 Func 𝐷)𝑁)
20	3, 11, 19	funcf1 17824	. . . 4 ⊢ (𝜑 → 𝑀:(Base‘𝐶)⟶(Base‘𝐷))
21	20, 7	ffvelcdmd 7031	. . 3 ⊢ (𝜑 → (𝑀‘𝑌) ∈ (Base‘𝐷))
22	3, 4, 12, 15, 6, 7	funcf2 17826	. . . 4 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐶)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝐷)(𝐹‘𝑌)))
23	22, 8	ffvelcdmd 7031	. . 3 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝐻) ∈ ((𝐹‘𝑋)(Hom ‘𝐷)(𝐹‘𝑌)))
24	1, 2, 3, 12, 7	natcl 17914	. . 3 ⊢ (𝜑 → (𝐴‘𝑌) ∈ ((𝐹‘𝑌)(Hom ‘𝐷)(𝑀‘𝑌)))
25	11, 12, 5, 13, 14, 17, 18, 21, 23, 24	funcco 17829	. 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 7031	. . 3 ⊢ (𝜑 → (𝑀‘𝑋) ∈ (Base‘𝐷))
27	1, 2, 3, 12, 6	natcl 17914	. . 3 ⊢ (𝜑 → (𝐴‘𝑋) ∈ ((𝐹‘𝑋)(Hom ‘𝐷)(𝑀‘𝑋)))
28	3, 4, 12, 19, 6, 7	funcf2 17826	. . . 4 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋(Hom ‘𝐶)𝑌)⟶((𝑀‘𝑋)(Hom ‘𝐷)(𝑀‘𝑌)))
29	28, 8	ffvelcdmd 7031	. . 3 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐻) ∈ ((𝑀‘𝑋)(Hom ‘𝐷)(𝑀‘𝑌)))
30	11, 12, 5, 13, 14, 17, 26, 21, 27, 29	funcco 17829	. 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 2780	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 1542   ∈ wcel 2114  ⟨cop 4574   class class class wbr 5086  ‘cfv 6492  (class class class)co 7360  Basecbs 17170  Hom chom 17222  compcco 17223   Func cfunc 17812   Nat cnat 17902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  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 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-map 8768  df-ixp 8839  df-func 17816  df-nat 17904

This theorem is referenced by:  fuco22natlem2  49830