Theorem fuco23 49320

Description: The morphism part of the functor composition bifunctor. See also fuco23a 49331. (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 49318	. 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 6864	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐹‘𝑥) = (𝐹‘𝑋))
9	8	fveq2d 6864	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐾‘(𝐹‘𝑥)) = (𝐾‘(𝐹‘𝑋)))
10	7	fveq2d 6864	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑀‘𝑥) = (𝑀‘𝑋))
11	10	fveq2d 6864	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐾‘(𝑀‘𝑥)) = (𝐾‘(𝑀‘𝑋)))
12	9, 11	opeq12d 4847	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝑀‘𝑥))⟩ = ⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩)
13	10	fveq2d 6864	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑅‘(𝑀‘𝑥)) = (𝑅‘(𝑀‘𝑋)))
14	12, 13	oveq12d 7407	. . . 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 2768	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝑀‘𝑥))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑥))) = ∗ )
18	10	fveq2d 6864	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐵‘(𝑀‘𝑥)) = (𝐵‘(𝑀‘𝑋)))
19	8, 10	oveq12d 7407	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((𝐹‘𝑥)𝐿(𝑀‘𝑥)) = ((𝐹‘𝑋)𝐿(𝑀‘𝑋)))
20	7	fveq2d 6864	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐴‘𝑥) = (𝐴‘𝑋))
21	19, 20	fveq12d 6867	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (((𝐹‘𝑥)𝐿(𝑀‘𝑥))‘(𝐴‘𝑥)) = (((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋)))
22	17, 18, 21	oveq123d 7410	. 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 7424	. 2 ⊢ (𝜑 → ((𝐵‘(𝑀‘𝑋)) ∗ (((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋))) ∈ V)
25	6, 22, 23, 24	fvmptd 6977	1 ⊢ (𝜑 → ((𝐵(𝑈𝑃𝑉)𝐴)‘𝑋) = ((𝐵‘(𝑀‘𝑋)) ∗ (((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ⟨cop 4597  ‘cfv 6513  (class class class)co 7389  Basecbs 17185  compcco 17238   Nat cnat 17912   ∘_F cfuco 49295

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 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-ov 7392  df-oprab 7393  df-mpo 7394  df-1st 7970  df-2nd 7971  df-ixp 8873  df-func 17826  df-cofu 17828  df-nat 17914  df-fuco 49296

This theorem is referenced by:  fuco22natlem3  49323  fuco22natlem  49324  fuco23a  49331