Theorem fuco23a 49593

Description: The morphism part of the functor composition bifunctor. An alternate definition of ∘_F. See also fuco23 49582. (Contributed by Zhi Wang, 3-Oct-2025.)

Hypotheses

Ref	Expression
fuco23a.a	⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))
fuco23a.b	⊢ (𝜑 → 𝐵 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩))
fuco23a.x	⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
fuco23a.p	⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)
fuco23a.u	⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)
fuco23a.v	⊢ (𝜑 → 𝑉 = ⟨⟨𝑅, 𝑆⟩, ⟨𝑀, 𝑁⟩⟩)
fuco23a.o	⊢ (𝜑 → ∗ = (⟨(𝐾‘(𝐹‘𝑋)), (𝑅‘(𝐹‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑋))))

Assertion

Ref	Expression
fuco23a	⊢ (𝜑 → ((𝐵(𝑈𝑃𝑉)𝐴)‘𝑋) = ((((𝐹‘𝑋)𝑆(𝑀‘𝑋))‘(𝐴‘𝑋)) ∗ (𝐵‘(𝐹‘𝑋))))

Proof of Theorem fuco23a

Step	Hyp	Ref	Expression
1		fuco23a.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))
2		fuco23a.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩))
3		fuco23a.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
4		eqid 2736	. . 3 ⊢ (comp‘𝐸) = (comp‘𝐸)
5	1, 2, 3, 4	fuco23alem 49592	. 2 ⊢ (𝜑 → ((𝐵‘(𝑀‘𝑋))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑋)))(((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋))) = ((((𝐹‘𝑋)𝑆(𝑀‘𝑋))‘(𝐴‘𝑋))(⟨(𝐾‘(𝐹‘𝑋)), (𝑅‘(𝐹‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑋)))(𝐵‘(𝐹‘𝑋))))
6		fuco23a.p	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)
7		fuco23a.u	. . 3 ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)
8		fuco23a.v	. . 3 ⊢ (𝜑 → 𝑉 = ⟨⟨𝑅, 𝑆⟩, ⟨𝑀, 𝑁⟩⟩)
9		eqidd 2737	. . 3 ⊢ (𝜑 → (⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑋))) = (⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑋))))
10	6, 7, 8, 1, 2, 3, 9	fuco23 49582	. 2 ⊢ (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))))
11		fuco23a.o	. . 3 ⊢ (𝜑 → ∗ = (⟨(𝐾‘(𝐹‘𝑋)), (𝑅‘(𝐹‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑋))))
12	11	oveqd 7375	. 2 ⊢ (𝜑 → ((((𝐹‘𝑋)𝑆(𝑀‘𝑋))‘(𝐴‘𝑋)) ∗ (𝐵‘(𝐹‘𝑋))) = ((((𝐹‘𝑋)𝑆(𝑀‘𝑋))‘(𝐴‘𝑋))(⟨(𝐾‘(𝐹‘𝑋)), (𝑅‘(𝐹‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑋)))(𝐵‘(𝐹‘𝑋))))
13	5, 10, 12	3eqtr4d 2781	1 ⊢ (𝜑 → ((𝐵(𝑈𝑃𝑉)𝐴)‘𝑋) = ((((𝐹‘𝑋)𝑆(𝑀‘𝑋))‘(𝐴‘𝑋)) ∗ (𝐵‘(𝐹‘𝑋))))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ⟨cop 4586  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  compcco 17189   Nat cnat 17868   ∘_F cfuco 49557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-map 8765  df-ixp 8836  df-func 17782  df-cofu 17784  df-nat 17870  df-fuco 49558

This theorem is referenced by: (None)