Theorem fuco22natlem3 49333

Description: Combine fuco22natlem2 49332 with fuco23 49330. (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 49332	. 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 2729	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
8		eqid 2729	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
9		eqid 2729	. . . . . . . 8 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
10	9, 3	natrcl2 49213	. . . . . . 7 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
11	7, 8, 10	funcf1 17828	. . . . . 6 ⊢ (𝜑 → 𝐹:(Base‘𝐶)⟶(Base‘𝐷))
12	11, 1	fvco3d 6961	. . . . 5 ⊢ (𝜑 → ((𝐾 ∘ 𝐹)‘𝑋) = (𝐾‘(𝐹‘𝑋)))
13	11, 2	fvco3d 6961	. . . . 5 ⊢ (𝜑 → ((𝐾 ∘ 𝐹)‘𝑌) = (𝐾‘(𝐹‘𝑌)))
14	12, 13	opeq12d 4845	. . . 4 ⊢ (𝜑 → ⟨((𝐾 ∘ 𝐹)‘𝑋), ((𝐾 ∘ 𝐹)‘𝑌)⟩ = ⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝐹‘𝑌))⟩)
15	9, 3	natrcl3 49214	. . . . . 6 ⊢ (𝜑 → 𝑀(𝐶 Func 𝐷)𝑁)
16	7, 8, 15	funcf1 17828	. . . . 5 ⊢ (𝜑 → 𝑀:(Base‘𝐶)⟶(Base‘𝐷))
17	16, 2	fvco3d 6961	. . . 4 ⊢ (𝜑 → ((𝑅 ∘ 𝑀)‘𝑌) = (𝑅‘(𝑀‘𝑌)))
18	14, 17	oveq12d 7405	. . 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 2730	. . . 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 49330	. . 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 2729	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
25		eqid 2729	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
26	7, 24, 25, 10, 1, 2	funcf2 17830	. . . 4 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐶)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝐷)(𝐹‘𝑌)))
27	26, 4	fvco3d 6961	. . 3 ⊢ (𝜑 → ((((𝐹‘𝑋)𝐿(𝐹‘𝑌)) ∘ (𝑋𝐺𝑌))‘𝐻) = (((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝐻)))
28	18, 23, 27	oveq123d 7408	. 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 6961	. . . . 5 ⊢ (𝜑 → ((𝑅 ∘ 𝑀)‘𝑋) = (𝑅‘(𝑀‘𝑋)))
30	12, 29	opeq12d 4845	. . . 4 ⊢ (𝜑 → ⟨((𝐾 ∘ 𝐹)‘𝑋), ((𝑅 ∘ 𝑀)‘𝑋)⟩ = ⟨(𝐾‘(𝐹‘𝑋)), (𝑅‘(𝑀‘𝑋))⟩)
31	30, 17	oveq12d 7405	. . 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 17830	. . . 4 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋(Hom ‘𝐶)𝑌)⟶((𝑀‘𝑋)(Hom ‘𝐷)(𝑀‘𝑌)))
33	32, 4	fvco3d 6961	. . 3 ⊢ (𝜑 → ((((𝑀‘𝑋)𝑆(𝑀‘𝑌)) ∘ (𝑋𝑁𝑌))‘𝐻) = (((𝑀‘𝑋)𝑆(𝑀‘𝑌))‘((𝑋𝑁𝑌)‘𝐻)))
34		eqidd 2730	. . . 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 49330	. . 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 7408	. 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 2774	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 1540   ∈ wcel 2109  ⟨cop 4595   ∘ ccom 5642  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  Hom chom 17231  compcco 17232   Nat cnat 17906   ∘_F cfuco 49305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-map 8801  df-ixp 8871  df-cat 17629  df-func 17820  df-cofu 17822  df-nat 17908  df-fuco 49306

This theorem is referenced by:  fuco22natlem  49334