Theorem lanfval 50103

Description: Value of the function generating the set of left Kan extensions. (Contributed by Zhi Wang, 3-Nov-2025.)

Hypotheses

Ref	Expression
lanfval.r	⊢ 𝑅 = (𝐷 FuncCat 𝐸)
lanfval.s	⊢ 𝑆 = (𝐶 FuncCat 𝐸)
lanfval.c	⊢ (𝜑 → 𝐶 ∈ 𝑈)
lanfval.d	⊢ (𝜑 → 𝐷 ∈ 𝑉)
lanfval.e	⊢ (𝜑 → 𝐸 ∈ 𝑊)

Assertion

Ref	Expression
lanfval	⊢ (𝜑 → (⟨𝐶, 𝐷⟩ Lan 𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ ((⟨𝐷, 𝐸⟩ −∘F 𝑓)(𝑅 UP 𝑆)𝑥)))

Distinct variable groups:   𝐶,𝑓,𝑥   𝐷,𝑓,𝑥   𝑓,𝐸,𝑥   𝜑,𝑓,𝑥

Allowed substitution hints:   𝑅(𝑥,𝑓)   𝑆(𝑥,𝑓)   𝑈(𝑥,𝑓)   𝑉(𝑥,𝑓)   𝑊(𝑥,𝑓)

Proof of Theorem lanfval

Dummy variables 𝑐 𝑑 𝑒 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-lan 50097	. . 3 ⊢ Lan = (𝑝 ∈ (V × V), 𝑒 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑐⦌⦋(2nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ ((⟨𝑑, 𝑒⟩ −∘F 𝑓)((𝑑 FuncCat 𝑒) UP (𝑐 FuncCat 𝑒))𝑥)))
2	1	a1i 11	. 2 ⊢ (𝜑 → Lan = (𝑝 ∈ (V × V), 𝑒 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑐⦌⦋(2nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ ((⟨𝑑, 𝑒⟩ −∘F 𝑓)((𝑑 FuncCat 𝑒) UP (𝑐 FuncCat 𝑒))𝑥))))
3		fvexd 6842	. . 3 ⊢ ((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) → (1st ‘𝑝) ∈ V)
4		simprl 776	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) → 𝑝 = ⟨𝐶, 𝐷⟩)
5	4	fveq2d 6831	. . . 4 ⊢ ((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) → (1st ‘𝑝) = (1st ‘⟨𝐶, 𝐷⟩))
6		lanfval.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
7		lanfval.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
8		op1stg 7943	. . . . . 6 ⊢ ((𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
9	6, 7, 8	syl2anc 590	. . . . 5 ⊢ (𝜑 → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
10	9	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
11	5, 10	eqtrd 2774	. . 3 ⊢ ((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) → (1st ‘𝑝) = 𝐶)
12		fvexd 6842	. . . 4 ⊢ (((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) → (2nd ‘𝑝) ∈ V)
13		simplrl 782	. . . . . 6 ⊢ (((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) → 𝑝 = ⟨𝐶, 𝐷⟩)
14	13	fveq2d 6831	. . . . 5 ⊢ (((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) → (2nd ‘𝑝) = (2nd ‘⟨𝐶, 𝐷⟩))
15		op2ndg 7944	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
16	6, 7, 15	syl2anc 590	. . . . . 6 ⊢ (𝜑 → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
17	16	ad2antrr 732	. . . . 5 ⊢ (((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
18	14, 17	eqtrd 2774	. . . 4 ⊢ (((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) → (2nd ‘𝑝) = 𝐷)
19		simplr 774	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → 𝑐 = 𝐶)
20		simpr 485	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
21	19, 20	oveq12d 7374	. . . . 5 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑐 Func 𝑑) = (𝐶 Func 𝐷))
22		simpllr 781	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸))
23	22	simprd 496	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → 𝑒 = 𝐸)
24	19, 23	oveq12d 7374	. . . . 5 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑐 Func 𝑒) = (𝐶 Func 𝐸))
25	20, 23	oveq12d 7374	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑑 FuncCat 𝑒) = (𝐷 FuncCat 𝐸))
26		lanfval.r	. . . . . . . 8 ⊢ 𝑅 = (𝐷 FuncCat 𝐸)
27	25, 26	eqtr4di 2792	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑑 FuncCat 𝑒) = 𝑅)
28	19, 23	oveq12d 7374	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑐 FuncCat 𝑒) = (𝐶 FuncCat 𝐸))
29		lanfval.s	. . . . . . . 8 ⊢ 𝑆 = (𝐶 FuncCat 𝐸)
30	28, 29	eqtr4di 2792	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑐 FuncCat 𝑒) = 𝑆)
31	27, 30	oveq12d 7374	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → ((𝑑 FuncCat 𝑒) UP (𝑐 FuncCat 𝑒)) = (𝑅 UP 𝑆))
32	20, 23	opeq12d 4812	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → ⟨𝑑, 𝑒⟩ = ⟨𝐷, 𝐸⟩)
33	32	oveq1d 7371	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (⟨𝑑, 𝑒⟩ −∘F 𝑓) = (⟨𝐷, 𝐸⟩ −∘F 𝑓))
34		eqidd 2740	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → 𝑥 = 𝑥)
35	31, 33, 34	oveq123d 7377	. . . . 5 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → ((⟨𝑑, 𝑒⟩ −∘F 𝑓)((𝑑 FuncCat 𝑒) UP (𝑐 FuncCat 𝑒))𝑥) = ((⟨𝐷, 𝐸⟩ −∘F 𝑓)(𝑅 UP 𝑆)𝑥))
36	21, 24, 35	mpoeq123dv 7431	. . . 4 ⊢ ((((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) ∧ 𝑑 = 𝐷) → (𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ ((⟨𝑑, 𝑒⟩ −∘F 𝑓)((𝑑 FuncCat 𝑒) UP (𝑐 FuncCat 𝑒))𝑥)) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ ((⟨𝐷, 𝐸⟩ −∘F 𝑓)(𝑅 UP 𝑆)𝑥)))
37	12, 18, 36	csbied2 3868	. . 3 ⊢ (((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) ∧ 𝑐 = 𝐶) → ⦋(2nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ ((⟨𝑑, 𝑒⟩ −∘F 𝑓)((𝑑 FuncCat 𝑒) UP (𝑐 FuncCat 𝑒))𝑥)) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ ((⟨𝐷, 𝐸⟩ −∘F 𝑓)(𝑅 UP 𝑆)𝑥)))
38	3, 11, 37	csbied2 3868	. 2 ⊢ ((𝜑 ∧ (𝑝 = ⟨𝐶, 𝐷⟩ ∧ 𝑒 = 𝐸)) → ⦋(1st ‘𝑝) / 𝑐⦌⦋(2nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ ((⟨𝑑, 𝑒⟩ −∘F 𝑓)((𝑑 FuncCat 𝑒) UP (𝑐 FuncCat 𝑒))𝑥)) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ ((⟨𝐷, 𝐸⟩ −∘F 𝑓)(𝑅 UP 𝑆)𝑥)))
39	6	elexd 3454	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
40	7	elexd 3454	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
41	39, 40	opelxpd 5657	. 2 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (V × V))
42		lanfval.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑊)
43	42	elexd 3454	. 2 ⊢ (𝜑 → 𝐸 ∈ V)
44		ovex 7389	. . . 4 ⊢ (𝐶 Func 𝐷) ∈ V
45		ovex 7389	. . . 4 ⊢ (𝐶 Func 𝐸) ∈ V
46	44, 45	mpoex 8021	. . 3 ⊢ (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ ((⟨𝐷, 𝐸⟩ −∘F 𝑓)(𝑅 UP 𝑆)𝑥)) ∈ V
47	46	a1i 11	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ ((⟨𝐷, 𝐸⟩ −∘F 𝑓)(𝑅 UP 𝑆)𝑥)) ∈ V)
48	2, 38, 41, 43, 47	ovmpod 7508	1 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ Lan 𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ ((⟨𝐷, 𝐸⟩ −∘F 𝑓)(𝑅 UP 𝑆)𝑥)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3431  ⦋csb 3831  ⟨cop 4561   × cxp 5616  ‘cfv 6485  (class class class)co 7356   ∈ cmpo 7358  1^st c1st 7929  2^nd c2nd 7930   Func cfunc 17812   FuncCat cfuc 17903   UP cup 49663   −∘_F cprcof 49863   Lan clan 50095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-lan 50097

This theorem is referenced by:  lanpropd  50105  reldmlan2  50107  lanval  50109  lanrcl  50111