Theorem parcoval2 50857

Description: Composition in a category with the parallel-arrow hom construction. Every composable pair contains an identity, so the category identity laws determine the value. (Contributed by Zhi Wang, 17-Sep-2026.)

Hypotheses

Ref	Expression
paridval.b	⊢ 𝐵 = (Base‘𝐶)
paridval.h	⊢ 𝐻 = (Hom ‘𝐶)
paridval.hv	⊢ 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))
paridval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
parcoval2.o	⊢ · = (comp‘𝐶)
parcoval2.f	⊢ (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))
parcoval2.g	⊢ (𝜑 → 𝐺 ∈ (𝑉𝐻𝑊))

Assertion

Ref	Expression
parcoval2	⊢ (𝜑 → (𝐺(⟨𝑈, 𝑉⟩ · 𝑊)𝐹) = if(𝑈 = 𝑉, 𝐺, 𝐹))

Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝐽,𝑦   𝑥,𝑆,𝑦   𝑥,𝑈,𝑦   𝑥,𝑉,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑥,𝑊,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝐶(𝑥, 𝑦)   · (𝑥, 𝑦)   𝐹(𝑥, 𝑦)   𝐺(𝑥, 𝑦)   𝐻(𝑥, 𝑦)

Proof of Theorem parcoval2

Step	Hyp	Ref	Expression
1		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → 𝑈 = 𝑉)
2	1	opeq1d 4838	. . . . . 6 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → ⟨𝑈, 𝑉⟩ = ⟨𝑉, 𝑉⟩)
3	2	oveq1d 7435	. . . . 5 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (⟨𝑈, 𝑉⟩ · 𝑊) = (⟨𝑉, 𝑉⟩ · 𝑊))
4		eqidd 2761	. . . . 5 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → 𝐺 = 𝐺)
5		paridval.hv	. . . . . . 7 ⊢ 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))
6		parcoval2.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))
7	6	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → 𝐹 ∈ (𝑈𝐻𝑉))
8	5, 7, 1	parhomideq 50841	. . . . . 6 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → 𝐹 = 𝐽)
9		paridval.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐶)
10		paridval.h	. . . . . . . 8 ⊢ 𝐻 = (Hom ‘𝐶)
11		paridval.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ Cat)
12	5	elmpocl2 7664	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑈𝐻𝑉) → 𝑉 ∈ 𝐵)
13	6, 12	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ 𝐵)
14	9, 10, 5, 11, 13	paridval 50856	. . . . . . 7 ⊢ (𝜑 → ((Id‘𝐶)‘𝑉) = 𝐽)
15	14	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → ((Id‘𝐶)‘𝑉) = 𝐽)
16	8, 15	eqtr4d 2798	. . . . 5 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → 𝐹 = ((Id‘𝐶)‘𝑉))
17	3, 4, 16	oveq123d 7441	. . . 4 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝐺(⟨𝑈, 𝑉⟩ · 𝑊)𝐹) = (𝐺(⟨𝑉, 𝑉⟩ · 𝑊)((Id‘𝐶)‘𝑉)))
18		eqid 2760	. . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶)
19		parcoval2.o	. . . . . 6 ⊢ · = (comp‘𝐶)
20		parcoval2.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ (𝑉𝐻𝑊))
21	5	elmpocl2 7664	. . . . . . 7 ⊢ (𝐺 ∈ (𝑉𝐻𝑊) → 𝑊 ∈ 𝐵)
22	20, 21	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
23	9, 10, 18, 11, 13, 19, 22, 20	catrid 17860	. . . . 5 ⊢ (𝜑 → (𝐺(⟨𝑉, 𝑉⟩ · 𝑊)((Id‘𝐶)‘𝑉)) = 𝐺)
24	23	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝐺(⟨𝑉, 𝑉⟩ · 𝑊)((Id‘𝐶)‘𝑉)) = 𝐺)
25	17, 24	eqtrd 2795	. . 3 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝐺(⟨𝑈, 𝑉⟩ · 𝑊)𝐹) = 𝐺)
26	1	iftrued 4489	. . 3 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → if(𝑈 = 𝑉, 𝐺, 𝐹) = 𝐺)
27	25, 26	eqtr4d 2798	. 2 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝐺(⟨𝑈, 𝑉⟩ · 𝑊)𝐹) = if(𝑈 = 𝑉, 𝐺, 𝐹))
28	5, 6, 20	parhomeqor 50850	. . . . . . . 8 ⊢ (𝜑 → (𝑈 = 𝑉 ∨ 𝑉 = 𝑊))
29	28	orcanai 1018	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → 𝑉 = 𝑊)
30	29	eqcomd 2766	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → 𝑊 = 𝑉)
31	30	oveq2d 7436	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → (⟨𝑈, 𝑉⟩ · 𝑊) = (⟨𝑈, 𝑉⟩ · 𝑉))
32	20	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → 𝐺 ∈ (𝑉𝐻𝑊))
33	5, 32, 29	parhomideq 50841	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → 𝐺 = 𝐽)
34	14	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → ((Id‘𝐶)‘𝑉) = 𝐽)
35	33, 34	eqtr4d 2798	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → 𝐺 = ((Id‘𝐶)‘𝑉))
36		eqidd 2761	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → 𝐹 = 𝐹)
37	31, 35, 36	oveq123d 7441	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → (𝐺(⟨𝑈, 𝑉⟩ · 𝑊)𝐹) = (((Id‘𝐶)‘𝑉)(⟨𝑈, 𝑉⟩ · 𝑉)𝐹))
38	5	elmpocl1 7663	. . . . . . 7 ⊢ (𝐹 ∈ (𝑈𝐻𝑉) → 𝑈 ∈ 𝐵)
39	6, 38	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝐵)
40	9, 10, 18, 11, 39, 19, 13, 6	catlid 17859	. . . . 5 ⊢ (𝜑 → (((Id‘𝐶)‘𝑉)(⟨𝑈, 𝑉⟩ · 𝑉)𝐹) = 𝐹)
41	40	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → (((Id‘𝐶)‘𝑉)(⟨𝑈, 𝑉⟩ · 𝑉)𝐹) = 𝐹)
42	37, 41	eqtrd 2795	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → (𝐺(⟨𝑈, 𝑉⟩ · 𝑊)𝐹) = 𝐹)
43		simpr 490	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → ¬ 𝑈 = 𝑉)
44	43	iffalsed 4492	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → if(𝑈 = 𝑉, 𝐺, 𝐹) = 𝐹)
45	42, 44	eqtr4d 2798	. 2 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → (𝐺(⟨𝑈, 𝑉⟩ · 𝑊)𝐹) = if(𝑈 = 𝑉, 𝐺, 𝐹))
46	27, 45	pm2.61dan 825	1 ⊢ (𝜑 → (𝐺(⟨𝑈, 𝑉⟩ · 𝑊)𝐹) = if(𝑈 = 𝑉, 𝐺, 𝐹))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∅c0 4278  ifcif 4481  {csn 4583  ⟨cop 4589  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  Basecbs 17389  Hom chom 17441  compcco 17442  Catccat 17840  Idccid 17841

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  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-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-cat 17844  df-cid 17845

This theorem is used by:  parmapfunc  50866