Theorem fuco22natlem3 50681

Description: Combine fuco22natlem2 50680 with fuco23 50678. (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))
fuco22natlem2.b	⊢ (𝜑 → 𝐵 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩))
fuco22natlem3.o	⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)
fuco22natlem3.u	⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)
fuco22natlem3.v	⊢ (𝜑 → 𝑉 = ⟨⟨𝑅, 𝑆⟩, ⟨𝑀, 𝑁⟩⟩)

Assertion

Ref	Expression
fuco22natlem3	⊢ (𝜑 → (((𝐵(𝑈𝑃𝑉)𝐴)‘𝑌)(⟨((𝐾 ∘ 𝐹)‘𝑋), ((𝐾 ∘ 𝐹)‘𝑌)⟩(comp‘𝐸)((𝑅 ∘ 𝑀)‘𝑌))((((𝐹‘𝑋)𝐿(𝐹‘𝑌)) ∘ (𝑋𝐺𝑌))‘𝐻)) = (((((𝑀‘𝑋)𝑆(𝑀‘𝑌)) ∘ (𝑋𝑁𝑌))‘𝐻)(⟨((𝐾 ∘ 𝐹)‘𝑋), ((𝑅 ∘ 𝑀)‘𝑋)⟩(comp‘𝐸)((𝑅 ∘ 𝑀)‘𝑌))((𝐵(𝑈𝑃𝑉)𝐴)‘𝑋)))

Proof of Theorem fuco22natlem3

Step	Hyp	Ref	Expression
1		fuco22natlem1.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
2		fuco22natlem1.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
3		fuco22natlem1.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))
4		fuco22natlem1.h	. . 3 ⊢ (𝜑 → 𝐻 ∈ (𝑋(Hom ‘𝐶)𝑌))
5		fuco22natlem2.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩))
6	1, 2, 3, 4, 5	fuco22natlem2 50680	. 2 ⊢ (𝜑 → (((𝐵‘(𝑀‘𝑌))(⟨(𝐾‘(𝐹‘𝑌)), (𝐾‘(𝑀‘𝑌))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))(((𝐹‘𝑌)𝐿(𝑀‘𝑌))‘(𝐴‘𝑌)))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝐹‘𝑌))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))(((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝐻))) = ((((𝑀‘𝑋)𝑆(𝑀‘𝑌))‘((𝑋𝑁𝑌)‘𝐻))(⟨(𝐾‘(𝐹‘𝑋)), (𝑅‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))((𝐵‘(𝑀‘𝑋))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑋)))(((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋)))))
7		eqid 2763	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
8		eqid 2763	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
9		eqid 2763	. . . . . . . 8 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
10	9, 3	natrcl2 50555	. . . . . . 7 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
11	7, 8, 10	funcf1 18073	. . . . . 6 ⊢ (𝜑 → 𝐹:(Base‘𝐶)⟶(Base‘𝐷))
12	11, 1	fvco3d 7012	. . . . 5 ⊢ (𝜑 → ((𝐾 ∘ 𝐹)‘𝑋) = (𝐾‘(𝐹‘𝑋)))
13	11, 2	fvco3d 7012	. . . . 5 ⊢ (𝜑 → ((𝐾 ∘ 𝐹)‘𝑌) = (𝐾‘(𝐹‘𝑌)))
14	12, 13	opeq12d 4851	. . . 4 ⊢ (𝜑 → ⟨((𝐾 ∘ 𝐹)‘𝑋), ((𝐾 ∘ 𝐹)‘𝑌)⟩ = ⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝐹‘𝑌))⟩)
15	9, 3	natrcl3 50556	. . . . . 6 ⊢ (𝜑 → 𝑀(𝐶 Func 𝐷)𝑁)
16	7, 8, 15	funcf1 18073	. . . . 5 ⊢ (𝜑 → 𝑀:(Base‘𝐶)⟶(Base‘𝐷))
17	16, 2	fvco3d 7012	. . . 4 ⊢ (𝜑 → ((𝑅 ∘ 𝑀)‘𝑌) = (𝑅‘(𝑀‘𝑌)))
18	14, 17	oveq12d 7464	. . 3 ⊢ (𝜑 → (⟨((𝐾 ∘ 𝐹)‘𝑋), ((𝐾 ∘ 𝐹)‘𝑌)⟩(comp‘𝐸)((𝑅 ∘ 𝑀)‘𝑌)) = (⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝐹‘𝑌))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌))))
19		fuco22natlem3.o	. . . 4 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)
20		fuco22natlem3.u	. . . 4 ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)
21		fuco22natlem3.v	. . . 4 ⊢ (𝜑 → 𝑉 = ⟨⟨𝑅, 𝑆⟩, ⟨𝑀, 𝑁⟩⟩)
22		eqidd 2764	. . . 4 ⊢ (𝜑 → (⟨(𝐾‘(𝐹‘𝑌)), (𝐾‘(𝑀‘𝑌))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌))) = (⟨(𝐾‘(𝐹‘𝑌)), (𝐾‘(𝑀‘𝑌))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌))))
23	19, 20, 21, 3, 5, 2, 22	fuco23 50678	. . 3 ⊢ (𝜑 → ((𝐵(𝑈𝑃𝑉)𝐴)‘𝑌) = ((𝐵‘(𝑀‘𝑌))(⟨(𝐾‘(𝐹‘𝑌)), (𝐾‘(𝑀‘𝑌))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))(((𝐹‘𝑌)𝐿(𝑀‘𝑌))‘(𝐴‘𝑌))))
24		eqid 2763	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
25		eqid 2763	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
26	7, 24, 25, 10, 1, 2	funcf2 18075	. . . 4 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐶)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝐷)(𝐹‘𝑌)))
27	26, 4	fvco3d 7012	. . 3 ⊢ (𝜑 → ((((𝐹‘𝑋)𝐿(𝐹‘𝑌)) ∘ (𝑋𝐺𝑌))‘𝐻) = (((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝐻)))
28	18, 23, 27	oveq123d 7467	. 2 ⊢ (𝜑 → (((𝐵(𝑈𝑃𝑉)𝐴)‘𝑌)(⟨((𝐾 ∘ 𝐹)‘𝑋), ((𝐾 ∘ 𝐹)‘𝑌)⟩(comp‘𝐸)((𝑅 ∘ 𝑀)‘𝑌))((((𝐹‘𝑋)𝐿(𝐹‘𝑌)) ∘ (𝑋𝐺𝑌))‘𝐻)) = (((𝐵‘(𝑀‘𝑌))(⟨(𝐾‘(𝐹‘𝑌)), (𝐾‘(𝑀‘𝑌))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))(((𝐹‘𝑌)𝐿(𝑀‘𝑌))‘(𝐴‘𝑌)))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝐹‘𝑌))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))(((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝐻))))
29	16, 1	fvco3d 7012	. . . . 5 ⊢ (𝜑 → ((𝑅 ∘ 𝑀)‘𝑋) = (𝑅‘(𝑀‘𝑋)))
30	12, 29	opeq12d 4851	. . . 4 ⊢ (𝜑 → ⟨((𝐾 ∘ 𝐹)‘𝑋), ((𝑅 ∘ 𝑀)‘𝑋)⟩ = ⟨(𝐾‘(𝐹‘𝑋)), (𝑅‘(𝑀‘𝑋))⟩)
31	30, 17	oveq12d 7464	. . 3 ⊢ (𝜑 → (⟨((𝐾 ∘ 𝐹)‘𝑋), ((𝑅 ∘ 𝑀)‘𝑋)⟩(comp‘𝐸)((𝑅 ∘ 𝑀)‘𝑌)) = (⟨(𝐾‘(𝐹‘𝑋)), (𝑅‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌))))
32	7, 24, 25, 15, 1, 2	funcf2 18075	. . . 4 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋(Hom ‘𝐶)𝑌)⟶((𝑀‘𝑋)(Hom ‘𝐷)(𝑀‘𝑌)))
33	32, 4	fvco3d 7012	. . 3 ⊢ (𝜑 → ((((𝑀‘𝑋)𝑆(𝑀‘𝑌)) ∘ (𝑋𝑁𝑌))‘𝐻) = (((𝑀‘𝑋)𝑆(𝑀‘𝑌))‘((𝑋𝑁𝑌)‘𝐻)))
34		eqidd 2764	. . . 4 ⊢ (𝜑 → (⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑋))) = (⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑋))))
35	19, 20, 21, 3, 5, 1, 34	fuco23 50678	. . 3 ⊢ (𝜑 → ((𝐵(𝑈𝑃𝑉)𝐴)‘𝑋) = ((𝐵‘(𝑀‘𝑋))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑋)))(((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋))))
36	31, 33, 35	oveq123d 7467	. 2 ⊢ (𝜑 → (((((𝑀‘𝑋)𝑆(𝑀‘𝑌)) ∘ (𝑋𝑁𝑌))‘𝐻)(⟨((𝐾 ∘ 𝐹)‘𝑋), ((𝑅 ∘ 𝑀)‘𝑋)⟩(comp‘𝐸)((𝑅 ∘ 𝑀)‘𝑌))((𝐵(𝑈𝑃𝑉)𝐴)‘𝑋)) = ((((𝑀‘𝑋)𝑆(𝑀‘𝑌))‘((𝑋𝑁𝑌)‘𝐻))(⟨(𝐾‘(𝐹‘𝑋)), (𝑅‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))((𝐵‘(𝑀‘𝑋))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑋)))(((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋)))))
37	6, 28, 36	3eqtr4d 2808	1 ⊢ (𝜑 → (((𝐵(𝑈𝑃𝑉)𝐴)‘𝑌)(⟨((𝐾 ∘ 𝐹)‘𝑋), ((𝐾 ∘ 𝐹)‘𝑌)⟩(comp‘𝐸)((𝑅 ∘ 𝑀)‘𝑌))((((𝐹‘𝑋)𝐿(𝐹‘𝑌)) ∘ (𝑋𝐺𝑌))‘𝐻)) = (((((𝑀‘𝑋)𝑆(𝑀‘𝑌)) ∘ (𝑋𝑁𝑌))‘𝐻)(⟨((𝐾 ∘ 𝐹)‘𝑋), ((𝑅 ∘ 𝑀)‘𝑋)⟩(comp‘𝐸)((𝑅 ∘ 𝑀)‘𝑌))((𝐵(𝑈𝑃𝑉)𝐴)‘𝑋)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ⟨cop 4838   ∘ ccom 5660  ‘cfv 6561  (class class class)co 7446  Basecbs 17419  Hom chom 17471  compcco 17472   Nat cnat 18151   ∘_F cfuco 50653

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 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

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 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1st 8028  df-2nd 8029  df-map 8880  df-ixp 8957  df-cat 17874  df-func 18065  df-cofu 18067  df-nat 18153  df-fuco 50654

This theorem is used by:  fuco22natlem  50682