Theorem fuco22natlem3 48911

Description: Combine fuco22natlem2 48910 with fuco23 48908. (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 48910	. 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 2737	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
8		eqid 2737	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
9		eqid 2737	. . . . . . . 8 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
10	9, 3	natrcl2 48870	. . . . . . 7 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
11	7, 8, 10	funcf1 17926	. . . . . 6 ⊢ (𝜑 → 𝐹:(Base‘𝐶)⟶(Base‘𝐷))
12	11, 1	fvco3d 7016	. . . . 5 ⊢ (𝜑 → ((𝐾 ∘ 𝐹)‘𝑋) = (𝐾‘(𝐹‘𝑋)))
13	11, 2	fvco3d 7016	. . . . 5 ⊢ (𝜑 → ((𝐾 ∘ 𝐹)‘𝑌) = (𝐾‘(𝐹‘𝑌)))
14	12, 13	opeq12d 4889	. . . 4 ⊢ (𝜑 → ⟨((𝐾 ∘ 𝐹)‘𝑋), ((𝐾 ∘ 𝐹)‘𝑌)⟩ = ⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝐹‘𝑌))⟩)
15	9, 3	natrcl3 48871	. . . . . 6 ⊢ (𝜑 → 𝑀(𝐶 Func 𝐷)𝑁)
16	7, 8, 15	funcf1 17926	. . . . 5 ⊢ (𝜑 → 𝑀:(Base‘𝐶)⟶(Base‘𝐷))
17	16, 2	fvco3d 7016	. . . 4 ⊢ (𝜑 → ((𝑅 ∘ 𝑀)‘𝑌) = (𝑅‘(𝑀‘𝑌)))
18	14, 17	oveq12d 7456	. . 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 2738	. . . 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 48908	. . 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 2737	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
25		eqid 2737	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
26	7, 24, 25, 10, 1, 2	funcf2 17928	. . . 4 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐶)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝐷)(𝐹‘𝑌)))
27	26, 4	fvco3d 7016	. . 3 ⊢ (𝜑 → ((((𝐹‘𝑋)𝐿(𝐹‘𝑌)) ∘ (𝑋𝐺𝑌))‘𝐻) = (((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝐻)))
28	18, 23, 27	oveq123d 7459	. 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 7016	. . . . 5 ⊢ (𝜑 → ((𝑅 ∘ 𝑀)‘𝑋) = (𝑅‘(𝑀‘𝑋)))
30	12, 29	opeq12d 4889	. . . 4 ⊢ (𝜑 → ⟨((𝐾 ∘ 𝐹)‘𝑋), ((𝑅 ∘ 𝑀)‘𝑋)⟩ = ⟨(𝐾‘(𝐹‘𝑋)), (𝑅‘(𝑀‘𝑋))⟩)
31	30, 17	oveq12d 7456	. . 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 17928	. . . 4 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋(Hom ‘𝐶)𝑌)⟶((𝑀‘𝑋)(Hom ‘𝐷)(𝑀‘𝑌)))
33	32, 4	fvco3d 7016	. . 3 ⊢ (𝜑 → ((((𝑀‘𝑋)𝑆(𝑀‘𝑌)) ∘ (𝑋𝑁𝑌))‘𝐻) = (((𝑀‘𝑋)𝑆(𝑀‘𝑌))‘((𝑋𝑁𝑌)‘𝐻)))
34		eqidd 2738	. . . 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 48908	. . 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 7459	. 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 2787	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)))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  ⟨cop 4640   ∘ ccom 5697  ‘cfv 6569  (class class class)co 7438  Basecbs 17254  Hom chom 17318  compcco 17319   Nat cnat 18005   ∘_F cfuco 48885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5288  ax-sep 5305  ax-nul 5315  ax-pow 5374  ax-pr 5441  ax-un 7761

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3483  df-sbc 3795  df-csb 3912  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-iun 5001  df-br 5152  df-opab 5214  df-mpt 5235  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706  df-iota 6522  df-fun 6571  df-fn 6572  df-f 6573  df-f1 6574  df-fo 6575  df-f1o 6576  df-fv 6577  df-ov 7441  df-oprab 7442  df-mpo 7443  df-1st 8022  df-2nd 8023  df-map 8876  df-ixp 8946  df-cat 17722  df-func 17918  df-cofu 17920  df-nat 18007  df-fuco 48886

This theorem is referenced by:  fuco22natlem  48912