Theorem fuco22natlem1 49583

Description: Lemma for fuco22nat 49587. 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 2736	. . . 4 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
2		fuco22natlem1.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))
3		eqid 2736	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
4		eqid 2736	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
5		eqid 2736	. . . 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 17882	. . 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 2736	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
12		eqid 2736	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
13		eqid 2736	. . 3 ⊢ (comp‘𝐸) = (comp‘𝐸)
14		fuco22natlem1.k	. . 3 ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)
15	1, 2	natrcl2 49465	. . . . 5 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
16	3, 11, 15	funcf1 17790	. . . 4 ⊢ (𝜑 → 𝐹:(Base‘𝐶)⟶(Base‘𝐷))
17	16, 6	ffvelcdmd 7030	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐷))
18	16, 7	ffvelcdmd 7030	. . 3 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐷))
19	1, 2	natrcl3 49466	. . . . 5 ⊢ (𝜑 → 𝑀(𝐶 Func 𝐷)𝑁)
20	3, 11, 19	funcf1 17790	. . . 4 ⊢ (𝜑 → 𝑀:(Base‘𝐶)⟶(Base‘𝐷))
21	20, 7	ffvelcdmd 7030	. . 3 ⊢ (𝜑 → (𝑀‘𝑌) ∈ (Base‘𝐷))
22	3, 4, 12, 15, 6, 7	funcf2 17792	. . . 4 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐶)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝐷)(𝐹‘𝑌)))
23	22, 8	ffvelcdmd 7030	. . 3 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝐻) ∈ ((𝐹‘𝑋)(Hom ‘𝐷)(𝐹‘𝑌)))
24	1, 2, 3, 12, 7	natcl 17880	. . 3 ⊢ (𝜑 → (𝐴‘𝑌) ∈ ((𝐹‘𝑌)(Hom ‘𝐷)(𝑀‘𝑌)))
25	11, 12, 5, 13, 14, 17, 18, 21, 23, 24	funcco 17795	. 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 7030	. . 3 ⊢ (𝜑 → (𝑀‘𝑋) ∈ (Base‘𝐷))
27	1, 2, 3, 12, 6	natcl 17880	. . 3 ⊢ (𝜑 → (𝐴‘𝑋) ∈ ((𝐹‘𝑋)(Hom ‘𝐷)(𝑀‘𝑋)))
28	3, 4, 12, 19, 6, 7	funcf2 17792	. . . 4 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋(Hom ‘𝐶)𝑌)⟶((𝑀‘𝑋)(Hom ‘𝐷)(𝑀‘𝑌)))
29	28, 8	ffvelcdmd 7030	. . 3 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐻) ∈ ((𝑀‘𝑋)(Hom ‘𝐷)(𝑀‘𝑌)))
30	11, 12, 5, 13, 14, 17, 26, 21, 27, 29	funcco 17795	. 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 2779	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 4586   class class class wbr 5098  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  Hom chom 17188  compcco 17189   Func cfunc 17778   Nat cnat 17868

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 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  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 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-map 8765  df-ixp 8836  df-func 17782  df-nat 17870

This theorem is referenced by:  fuco22natlem2  49584