Theorem fuco23 48908

Description: The morphism part of the functor composition bifunctor. See also fuco23a 48919. (Contributed by Zhi Wang, 29-Sep-2025.)

Hypotheses

Ref	Expression
fuco22.o	⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)
fuco22.u	⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)
fuco22.v	⊢ (𝜑 → 𝑉 = ⟨⟨𝑅, 𝑆⟩, ⟨𝑀, 𝑁⟩⟩)
fuco22.a	⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))
fuco22.b	⊢ (𝜑 → 𝐵 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩))
fuco23.x	⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
fuco23.o	⊢ (𝜑 → ∗ = (⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑋))))

Assertion

Ref	Expression
fuco23	⊢ (𝜑 → ((𝐵(𝑈𝑃𝑉)𝐴)‘𝑋) = ((𝐵‘(𝑀‘𝑋)) ∗ (((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋))))

Proof of Theorem fuco23

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fuco22.o	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)
2		fuco22.u	. . 3 ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)
3		fuco22.v	. . 3 ⊢ (𝜑 → 𝑉 = ⟨⟨𝑅, 𝑆⟩, ⟨𝑀, 𝑁⟩⟩)
4		fuco22.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))
5		fuco22.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩))
6	1, 2, 3, 4, 5	fuco22 48906	. 2 ⊢ (𝜑 → (𝐵(𝑈𝑃𝑉)𝐴) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝐵‘(𝑀‘𝑥))(⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝑀‘𝑥))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑥)))(((𝐹‘𝑥)𝐿(𝑀‘𝑥))‘(𝐴‘𝑥)))))
7		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
8	7	fveq2d 6918	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐹‘𝑥) = (𝐹‘𝑋))
9	8	fveq2d 6918	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐾‘(𝐹‘𝑥)) = (𝐾‘(𝐹‘𝑋)))
10	7	fveq2d 6918	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑀‘𝑥) = (𝑀‘𝑋))
11	10	fveq2d 6918	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐾‘(𝑀‘𝑥)) = (𝐾‘(𝑀‘𝑋)))
12	9, 11	opeq12d 4889	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝑀‘𝑥))⟩ = ⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩)
13	10	fveq2d 6918	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑅‘(𝑀‘𝑥)) = (𝑅‘(𝑀‘𝑋)))
14	12, 13	oveq12d 7456	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝑀‘𝑥))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑥))) = (⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑋))))
15		fuco23.o	. . . . 5 ⊢ (𝜑 → ∗ = (⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑋))))
16	15	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ∗ = (⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑋))))
17	14, 16	eqtr4d 2780	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝑀‘𝑥))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑥))) = ∗ )
18	10	fveq2d 6918	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐵‘(𝑀‘𝑥)) = (𝐵‘(𝑀‘𝑋)))
19	8, 10	oveq12d 7456	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((𝐹‘𝑥)𝐿(𝑀‘𝑥)) = ((𝐹‘𝑋)𝐿(𝑀‘𝑋)))
20	7	fveq2d 6918	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐴‘𝑥) = (𝐴‘𝑋))
21	19, 20	fveq12d 6921	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (((𝐹‘𝑥)𝐿(𝑀‘𝑥))‘(𝐴‘𝑥)) = (((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋)))
22	17, 18, 21	oveq123d 7459	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((𝐵‘(𝑀‘𝑥))(⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝑀‘𝑥))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑥)))(((𝐹‘𝑥)𝐿(𝑀‘𝑥))‘(𝐴‘𝑥))) = ((𝐵‘(𝑀‘𝑋)) ∗ (((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋))))
23		fuco23.x	. 2 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
24		ovexd 7473	. 2 ⊢ (𝜑 → ((𝐵‘(𝑀‘𝑋)) ∗ (((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋))) ∈ V)
25	6, 22, 23, 24	fvmptd 7030	1 ⊢ (𝜑 → ((𝐵(𝑈𝑃𝑉)𝐴)‘𝑋) = ((𝐵‘(𝑀‘𝑋)) ∗ (((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3481  ⟨cop 4640  ‘cfv 6569  (class class class)co 7438  Basecbs 17254  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-ixp 8946  df-func 17918  df-cofu 17920  df-nat 18007  df-fuco 48886

This theorem is referenced by:  fuco22natlem3  48911  fuco22natlem  48912  fuco23a  48919