fucofval - Mathbox for Zhi Wang

	Mathbox for Zhi Wang		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fucofval		Structured version Visualization version GIF version

Theorem fucofval 50138

Description: Value of the function giving the functor composition bifunctor. Hypotheses fucofval.c and fucofval.d are not redundant (fucofvalne 50144). (Contributed by Zhi Wang, 29-Sep-2025.)

**Hypotheses**
Ref	Expression
fucofval.c	⊢ (𝜑 → 𝐶 ∈ 𝑇)
fucofval.d	⊢ (𝜑 → 𝐷 ∈ 𝑈)
fucofval.e	⊢ (𝜑 → 𝐸 ∈ 𝑉)
fucofval.o	⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⚬ )
fucofval.w	⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))

**Assertion**
Ref	Expression
fucofval	⊢ (𝜑 → ⚬ = ⟨( ∘_func ↾ 𝑊), (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ ⦋(1^st ‘(2^nd ‘𝑢)) / 𝑓⦌⦋(1^st ‘(1^st ‘𝑢)) / 𝑘⦌⦋(2^nd ‘(1^st ‘𝑢)) / 𝑙⦌⦋(1^st ‘(2^nd ‘𝑣)) / 𝑚⦌⦋(1^st ‘(1^st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1^st ‘𝑢)(𝐷 Nat 𝐸)(1^st ‘𝑣)), 𝑎 ∈ ((2^nd ‘𝑢)(𝐶 Nat 𝐷)(2^nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝐸)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))))))⟩)

Distinct variable groups: 𝐶,𝑎,𝑏,𝑓,𝑘,𝑙,𝑚,𝑟,𝑢,𝑣,𝑥 𝐷,𝑎,𝑏,𝑓,𝑘,𝑙,𝑚,𝑟,𝑢,𝑣,𝑥 𝐸,𝑎,𝑏,𝑓,𝑘,𝑙,𝑚,𝑟,𝑢,𝑣,𝑥 𝑊,𝑎,𝑏,𝑓,𝑘,𝑙,𝑚,𝑟,𝑢,𝑣,𝑥 𝜑,𝑎,𝑏,𝑓,𝑘,𝑙,𝑚,𝑟,𝑢,𝑣,𝑥 Allowed substitution hints: 𝑇(𝑥, 𝑣, 𝑢, 𝑓, 𝑘, 𝑚, 𝑟, 𝑎, 𝑏, 𝑙) 𝑈(𝑥, 𝑣, 𝑢, 𝑓, 𝑘, 𝑚, 𝑟, 𝑎, 𝑏, 𝑙) 𝑉(𝑥, 𝑣, 𝑢, 𝑓, 𝑘, 𝑚, 𝑟, 𝑎, 𝑏, 𝑙) ⚬ (𝑥, 𝑣, 𝑢, 𝑓, 𝑘, 𝑚, 𝑟, 𝑎, 𝑏, 𝑙)

**Proof of Theorem fucofval**
Step	Hyp	Ref	Expression
1		opex 5450	. . 3 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
2	1	a1i 11	. 2 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ V)
3		fucofval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑇)
4		fucofval.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑈)
5		op1stg 8007	. . 3 ⊢ ((𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑈) → (1^st ‘⟨𝐶, 𝐷⟩) = 𝐶)
6	3, 4, 5	syl2anc 596	. 2 ⊢ (𝜑 → (1^st ‘⟨𝐶, 𝐷⟩) = 𝐶)
7		op2ndg 8008	. . 3 ⊢ ((𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑈) → (2^nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
8	3, 4, 7	syl2anc 596	. 2 ⊢ (𝜑 → (2^nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
9		fucofval.e	. 2 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
10		fucofval.o	. 2 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⚬ )
11		fucofval.w	. 2 ⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
12	2, 6, 8, 9, 10, 11	fucofvalg 50137	1 ⊢ (𝜑 → ⚬ = ⟨( ∘_func ↾ 𝑊), (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ ⦋(1^st ‘(2^nd ‘𝑢)) / 𝑓⦌⦋(1^st ‘(1^st ‘𝑢)) / 𝑘⦌⦋(2^nd ‘(1^st ‘𝑢)) / 𝑙⦌⦋(1^st ‘(2^nd ‘𝑣)) / 𝑚⦌⦋(1^st ‘(1^st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1^st ‘𝑢)(𝐷 Nat 𝐸)(1^st ‘𝑣)), 𝑎 ∈ ((2^nd ‘𝑢)(𝐶 Nat 𝐷)(2^nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝐸)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))))))⟩)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2146 Vcvv 3458 ⦋csb 3856 ⟨cop 4600 ↦ cmpt 5197 × cxp 5664 ↾ cres 5668 ‘cfv 6543 (class class class)co 7423 ∈ cmpo 7425 1^st c1st 7993 2^nd c2nd 7994 Basecbs 17294 compcco 17347 Func cfunc 17936 ∘_func ccofu 17938 Nat cnat 18026 ∘_F cfuco 50135

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745

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 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-ral 3083 df-rex 3093 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-br 5115 df-opab 5179 df-mpt 5198 df-id 5561 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-iota 6499 df-fun 6545 df-fv 6551 df-ov 7426 df-oprab 7427 df-mpo 7428 df-1st 7995 df-2nd 7996 df-fuco 50136

This theorem is used by: fucoelvv 50139 fuco1 50140 fuco2 50142

W3C validator