Theorem fuco23 50678

Description: The morphism part of the functor composition bifunctor. See also fuco23a 50689. (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 50676	. 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 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
8	7	fveq2d 6915	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐹‘𝑥) = (𝐹‘𝑋))
9	8	fveq2d 6915	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐾‘(𝐹‘𝑥)) = (𝐾‘(𝐹‘𝑋)))
10	7	fveq2d 6915	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑀‘𝑥) = (𝑀‘𝑋))
11	10	fveq2d 6915	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐾‘(𝑀‘𝑥)) = (𝐾‘(𝑀‘𝑋)))
12	9, 11	opeq12d 4851	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝑀‘𝑥))⟩ = ⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩)
13	10	fveq2d 6915	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑅‘(𝑀‘𝑥)) = (𝑅‘(𝑀‘𝑋)))
14	12, 13	oveq12d 7464	. . . 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 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ∗ = (⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑋))))
17	14, 16	eqtr4d 2801	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝑀‘𝑥))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑥))) = ∗ )
18	10	fveq2d 6915	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐵‘(𝑀‘𝑥)) = (𝐵‘(𝑀‘𝑋)))
19	8, 10	oveq12d 7464	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((𝐹‘𝑥)𝐿(𝑀‘𝑥)) = ((𝐹‘𝑋)𝐿(𝑀‘𝑋)))
20	7	fveq2d 6915	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐴‘𝑥) = (𝐴‘𝑋))
21	19, 20	fveq12d 6918	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (((𝐹‘𝑥)𝐿(𝑀‘𝑥))‘(𝐴‘𝑥)) = (((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋)))
22	17, 18, 21	oveq123d 7467	. 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 7481	. 2 ⊢ (𝜑 → ((𝐵‘(𝑀‘𝑋)) ∗ (((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋))) ∈ V)
25	6, 22, 23, 24	fvmptd 7027	1 ⊢ (𝜑 → ((𝐵(𝑈𝑃𝑉)𝐴)‘𝑋) = ((𝐵‘(𝑀‘𝑋)) ∗ (((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ⟨cop 4838  ‘cfv 6561  (class class class)co 7446  Basecbs 17419  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-ixp 8957  df-func 18065  df-cofu 18067  df-nat 18153  df-fuco 50654

This theorem is used by:  fuco22natlem3  50681  fuco22natlem  50682  fuco23a  50689