Theorem fuco23a 50279

Description: The morphism part of the functor composition bifunctor. An alternate definition of ∘_F. See also fuco23 50268. (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 2760	. . 3 ⊢ (comp‘𝐸) = (comp‘𝐸)
5	1, 2, 3, 4	fuco23alem 50278	. 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 2761	. . 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 50268	. 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 7431	. 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 2805	1 ⊢ (𝜑 → ((𝐵(𝑈𝑃𝑉)𝐴)‘𝑋) = ((((𝐹‘𝑋)𝑆(𝑀‘𝑋))‘(𝐴‘𝑋)) ∗ (𝐵‘(𝐹‘𝑋))))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ⟨cop 4590  ‘cfv 6533  (class class class)co 7414  Basecbs 17302  compcco 17355   Nat cnat 18034   ∘_F cfuco 50243

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-map 8829  df-ixp 8906  df-func 17948  df-cofu 17950  df-nat 18036  df-fuco 50244

This theorem is used by: (None)