Theorem fuco23 50102

Description: The morphism part of the functor composition bifunctor. See also fuco23a 50113. (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 50100	. 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 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
8	7	fveq2d 6887	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐹‘𝑥) = (𝐹‘𝑋))
9	8	fveq2d 6887	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐾‘(𝐹‘𝑥)) = (𝐾‘(𝐹‘𝑋)))
10	7	fveq2d 6887	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑀‘𝑥) = (𝑀‘𝑋))
11	10	fveq2d 6887	. . . . . 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 6887	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑅‘(𝑀‘𝑥)) = (𝑅‘(𝑀‘𝑋)))
14	12, 13	oveq12d 7430	. . . 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 485	. . . 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 6887	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐵‘(𝑀‘𝑥)) = (𝐵‘(𝑀‘𝑋)))
19	8, 10	oveq12d 7430	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((𝐹‘𝑥)𝐿(𝑀‘𝑥)) = ((𝐹‘𝑋)𝐿(𝑀‘𝑋)))
20	7	fveq2d 6887	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐴‘𝑥) = (𝐴‘𝑋))
21	19, 20	fveq12d 6890	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (((𝐹‘𝑥)𝐿(𝑀‘𝑥))‘(𝐴‘𝑥)) = (((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋)))
22	17, 18, 21	oveq123d 7433	. 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 7447	. 2 ⊢ (𝜑 → ((𝐵‘(𝑀‘𝑋)) ∗ (((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋))) ∈ V)
25	6, 22, 23, 24	fvmptd 6999	1 ⊢ (𝜑 → ((𝐵(𝑈𝑃𝑉)𝐴)‘𝑋) = ((𝐵‘(𝑀‘𝑋)) ∗ (((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋))))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ⟨cop 4596  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  compcco 17323   Nat cnat 18002   ∘_F cfuco 50077

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-ixp 8897  df-func 17916  df-cofu 17918  df-nat 18004  df-fuco 50078

This theorem is referenced by:  fuco22natlem3  50105  fuco22natlem  50106  fuco23a  50113