Theorem 2arwcat 50221

Description: The condition for a structure with at most one object and at most two morphisms being a category. "2arwcat.2" to "2arwcat.5" are also necessary conditions if 𝑋, 0, and 1 are all sets, due to catlid 17715, catrid 17716, and catcocl 17717. (Contributed by Zhi Wang, 5-Nov-2025.)

Hypotheses

Ref	Expression
2arwcat.b	⊢ (𝜑 → {𝑋} = (Base‘𝐶))
2arwcat.h	⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))
2arwcat.x	⊢ (𝜑 → · = (comp‘𝐶))
2arwcat.1	⊢ (𝑋𝐻𝑋) = { 0 , 1 }
2arwcat.2	⊢ (𝜑 → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = 1 )
2arwcat.3	⊢ (𝜑 → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋) 0 ) = 0 )
2arwcat.4	⊢ (𝜑 → ( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = 0 )
2arwcat.5	⊢ (𝜑 → ( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 0 ) ∈ { 0 , 1 })

Assertion

Ref	Expression
2arwcat	⊢ (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ {𝑋} ↦ 1 )))

Distinct variable groups:   𝑦, ·   𝑦,𝐶   𝑦,𝐻   𝑦,𝑋   𝜑,𝑦

Allowed substitution hints:   1 (𝑦)   0 (𝑦)

Proof of Theorem 2arwcat

Dummy variables 𝑓 𝑔 𝑘 𝑤 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2arwcat.b	. 2 ⊢ (𝜑 → {𝑋} = (Base‘𝐶))
2		2arwcat.h	. 2 ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))
3		2arwcat.x	. 2 ⊢ (𝜑 → · = (comp‘𝐶))
4		2arwcat.2	. . . . . . 7 ⊢ (𝜑 → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = 1 )
5		ovex 7429	. . . . . . 7 ⊢ ( 1 (⟨𝑋, 𝑋⟩ · 𝑋) 1 ) ∈ V
6	4, 5	eqeltrrdi 2871	. . . . . 6 ⊢ (𝜑 → 1 ∈ V)
7		prid2g 4720	. . . . . 6 ⊢ ( 1 ∈ V → 1 ∈ { 0 , 1 })
8	6, 7	syl 17	. . . . 5 ⊢ (𝜑 → 1 ∈ { 0 , 1 })
9		2arwcat.1	. . . . 5 ⊢ (𝑋𝐻𝑋) = { 0 , 1 }
10	8, 9	eleqtrrdi 2873	. . . 4 ⊢ (𝜑 → 1 ∈ (𝑋𝐻𝑋))
11		df-ov 7399	. . . . 5 ⊢ (𝑋𝐻𝑋) = (𝐻‘⟨𝑋, 𝑋⟩)
12	2	fveq1d 6869	. . . . 5 ⊢ (𝜑 → (𝐻‘⟨𝑋, 𝑋⟩) = ((Hom ‘𝐶)‘⟨𝑋, 𝑋⟩))
13	11, 12	eqtrid 2809	. . . 4 ⊢ (𝜑 → (𝑋𝐻𝑋) = ((Hom ‘𝐶)‘⟨𝑋, 𝑋⟩))
14	10, 13	eleqtrd 2864	. . 3 ⊢ (𝜑 → 1 ∈ ((Hom ‘𝐶)‘⟨𝑋, 𝑋⟩))
15		elfv2ex 6910	. . 3 ⊢ ( 1 ∈ ((Hom ‘𝐶)‘⟨𝑋, 𝑋⟩) → 𝐶 ∈ V)
16	14, 15	syl 17	. 2 ⊢ (𝜑 → 𝐶 ∈ V)
17	9	2arwcatlem1 50216	. 2 ⊢ ((((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 ))) ↔ ((𝑥 ∈ {𝑋} ∧ 𝑦 ∈ {𝑋}) ∧ (𝑧 ∈ {𝑋} ∧ 𝑤 ∈ {𝑋}) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
18	8	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑋}) → 1 ∈ { 0 , 1 })
19		velsn 4598	. . . . 5 ⊢ (𝑦 ∈ {𝑋} ↔ 𝑦 = 𝑋)
20		id 22	. . . . . . 7 ⊢ (𝑦 = 𝑋 → 𝑦 = 𝑋)
21	20, 20	oveq12d 7414	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑦𝐻𝑦) = (𝑋𝐻𝑋))
22	21, 9	eqtrdi 2813	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦𝐻𝑦) = { 0 , 1 })
23	19, 22	sylbi 219	. . . 4 ⊢ (𝑦 ∈ {𝑋} → (𝑦𝐻𝑦) = { 0 , 1 })
24	23	adantl 485	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑋}) → (𝑦𝐻𝑦) = { 0 , 1 })
25	18, 24	eleqtrrd 2865	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑋}) → 1 ∈ (𝑦𝐻𝑦))
26		simprll 788	. . . 4 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑥 = 𝑋 ∧ 𝑦 = 𝑋))
27	26	simpld 498	. . 3 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → 𝑥 = 𝑋)
28	26	simprd 499	. . 3 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → 𝑦 = 𝑋)
29		simprr1 1235	. . 3 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑓 = 0 ∨ 𝑓 = 1 ))
30	4	adantr 484	. . 3 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = 1 )
31		2arwcat.3	. . . 4 ⊢ (𝜑 → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋) 0 ) = 0 )
32	31	adantr 484	. . 3 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋) 0 ) = 0 )
33	27, 28, 28, 29, 30, 32	2arwcatlem2 50217	. 2 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑓) = 𝑓)
34		simprlr 789	. . . 4 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑧 = 𝑋 ∧ 𝑤 = 𝑋))
35	34	simpld 498	. . 3 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → 𝑧 = 𝑋)
36		simprr2 1236	. . 3 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑔 = 0 ∨ 𝑔 = 1 ))
37		2arwcat.4	. . . 4 ⊢ (𝜑 → ( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = 0 )
38	37	adantr 484	. . 3 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = 0 )
39	28, 28, 35, 36, 30, 38	2arwcatlem3 50218	. 2 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑔(⟨𝑦, 𝑦⟩ · 𝑧) 1 ) = 𝑔)
40		2arwcat.5	. . . . 5 ⊢ (𝜑 → ( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 0 ) ∈ { 0 , 1 })
41	40	adantr 484	. . . 4 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 0 ) ∈ { 0 , 1 })
42	27, 28, 35, 29, 30, 38, 32, 41, 36	2arwcatlem4 50219	. . 3 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ { 0 , 1 })
43	27, 35	oveq12d 7414	. . . 4 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑥𝐻𝑧) = (𝑋𝐻𝑋))
44	43, 9	eqtrdi 2813	. . 3 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑥𝐻𝑧) = { 0 , 1 })
45	42, 44	eleqtrrd 2865	. 2 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
46	31, 37, 40	2arwcatlem5 50220	. . . . . . . 8 ⊢ (𝜑 → (( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 0 )(⟨𝑋, 𝑋⟩ · 𝑋) 0 ) = ( 0 (⟨𝑋, 𝑋⟩ · 𝑋)( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 0 )))
47	46	ad4antr 742	. . . . . . 7 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 0 ) → (( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 0 )(⟨𝑋, 𝑋⟩ · 𝑋) 0 ) = ( 0 (⟨𝑋, 𝑋⟩ · 𝑋)( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 0 )))
48		simpr 488	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 0 ) → 𝑘 = 0 )
49		simplr 778	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 0 ) → 𝑔 = 0 )
50	48, 49	oveq12d 7414	. . . . . . . 8 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 0 ) → (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = ( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 0 ))
51		simpr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) → 𝑓 = 0 )
52	51	ad2antrr 736	. . . . . . . 8 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 0 ) → 𝑓 = 0 )
53	50, 52	oveq12d 7414	. . . . . . 7 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 0 ) → ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = (( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 0 )(⟨𝑋, 𝑋⟩ · 𝑋) 0 ))
54	49, 52	oveq12d 7414	. . . . . . . 8 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 0 ) → (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = ( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 0 ))
55	48, 54	oveq12d 7414	. . . . . . 7 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 0 ) → (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)) = ( 0 (⟨𝑋, 𝑋⟩ · 𝑋)( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 0 )))
56	47, 53, 55	3eqtr4d 2807	. . . . . 6 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 0 ) → ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)))
57		eqidd 2763	. . . . . . . . 9 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → 𝑋 = 𝑋)
58	27, 28	opeq12d 4839	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑋⟩)
59	58, 35	oveq12d 7414	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (⟨𝑥, 𝑦⟩ · 𝑧) = (⟨𝑋, 𝑋⟩ · 𝑋))
60	59	oveqd 7413	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) = (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓))
61	60, 42	eqeltrrd 2863	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) ∈ { 0 , 1 })
62		ovex 7429	. . . . . . . . . . 11 ⊢ (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) ∈ V
63	62	elpr 4607	. . . . . . . . . 10 ⊢ ((𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) ∈ { 0 , 1 } ↔ ((𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 0 ∨ (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 1 ))
64	61, 63	sylib 220	. . . . . . . . 9 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ((𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 0 ∨ (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 1 ))
65	57, 57, 57, 64, 30, 32	2arwcatlem2 50217	. . . . . . . 8 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)) = (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓))
66	65	ad3antrrr 740	. . . . . . 7 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 1 ) → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)) = (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓))
67		simpr 488	. . . . . . . 8 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 1 ) → 𝑘 = 1 )
68	67	oveq1d 7411	. . . . . . 7 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 1 ) → (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)) = ( 1 (⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)))
69	67	oveq1d 7411	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 1 ) → (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = ( 1 (⟨𝑋, 𝑋⟩ · 𝑋)𝑔))
70	57, 57, 57, 36, 30, 32	2arwcatlem2 50217	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑔)
71	70	ad3antrrr 740	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 1 ) → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑔)
72	69, 71	eqtrd 2797	. . . . . . . 8 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 1 ) → (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑔)
73	72	oveq1d 7411	. . . . . . 7 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 1 ) → ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓))
74	66, 68, 73	3eqtr4rd 2808	. . . . . 6 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 1 ) → ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)))
75		simprr3 1237	. . . . . . 7 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑘 = 0 ∨ 𝑘 = 1 ))
76	75	ad2antrr 736	. . . . . 6 ⊢ ((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) → (𝑘 = 0 ∨ 𝑘 = 1 ))
77	56, 74, 76	mpjaodan 971	. . . . 5 ⊢ ((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) → ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)))
78		simpr 488	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 1 ) → 𝑔 = 1 )
79	78	oveq2d 7412	. . . . . . . 8 ⊢ ((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 1 ) → (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋) 1 ))
80	57, 57, 57, 75, 30, 38	2arwcatlem3 50218	. . . . . . . . 9 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑘(⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = 𝑘)
81	80	ad2antrr 736	. . . . . . . 8 ⊢ ((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 1 ) → (𝑘(⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = 𝑘)
82	79, 81	eqtrd 2797	. . . . . . 7 ⊢ ((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 1 ) → (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑘)
83	82	oveq1d 7411	. . . . . 6 ⊢ ((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 1 ) → ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑓))
84	78	oveq1d 7411	. . . . . . . 8 ⊢ ((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 1 ) → (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = ( 1 (⟨𝑋, 𝑋⟩ · 𝑋)𝑓))
85	57, 57, 57, 29, 30, 32	2arwcatlem2 50217	. . . . . . . . 9 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓)
86	85	ad2antrr 736	. . . . . . . 8 ⊢ ((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 1 ) → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓)
87	84, 86	eqtrd 2797	. . . . . . 7 ⊢ ((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 1 ) → (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓)
88	87	oveq2d 7412	. . . . . 6 ⊢ ((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 1 ) → (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑓))
89	83, 88	eqtr4d 2800	. . . . 5 ⊢ ((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 1 ) → ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)))
90	36	adantr 484	. . . . 5 ⊢ (((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) → (𝑔 = 0 ∨ 𝑔 = 1 ))
91	77, 89, 90	mpjaodan 971	. . . 4 ⊢ (((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) → ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)))
92	57, 57, 57, 36, 30, 38, 32, 41, 75	2arwcatlem4 50219	. . . . . . . 8 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) ∈ { 0 , 1 })
93		ovex 7429	. . . . . . . . 9 ⊢ (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) ∈ V
94	93	elpr 4607	. . . . . . . 8 ⊢ ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) ∈ { 0 , 1 } ↔ ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 0 ∨ (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 1 ))
95	92, 94	sylib 220	. . . . . . 7 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 0 ∨ (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 1 ))
96	57, 57, 57, 95, 30, 38	2arwcatlem3 50218	. . . . . 6 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔))
97	96	adantr 484	. . . . 5 ⊢ (((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 1 ) → ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔))
98		simpr 488	. . . . . 6 ⊢ (((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 1 ) → 𝑓 = 1 )
99	98	oveq2d 7412	. . . . 5 ⊢ (((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 1 ) → ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋) 1 ))
100	98	oveq2d 7412	. . . . . . 7 ⊢ (((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 1 ) → (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = (𝑔(⟨𝑋, 𝑋⟩ · 𝑋) 1 ))
101	57, 57, 57, 36, 30, 38	2arwcatlem3 50218	. . . . . . . 8 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑔(⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = 𝑔)
102	101	adantr 484	. . . . . . 7 ⊢ (((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 1 ) → (𝑔(⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = 𝑔)
103	100, 102	eqtrd 2797	. . . . . 6 ⊢ (((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 1 ) → (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑔)
104	103	oveq2d 7412	. . . . 5 ⊢ (((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 1 ) → (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔))
105	97, 99, 104	3eqtr4d 2807	. . . 4 ⊢ (((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 1 ) → ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)))
106	91, 105, 29	mpjaodan 971	. . 3 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)))
107	34	simprd 499	. . . . 5 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → 𝑤 = 𝑋)
108	58, 107	oveq12d 7414	. . . 4 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (⟨𝑥, 𝑦⟩ · 𝑤) = (⟨𝑋, 𝑋⟩ · 𝑋))
109	28, 35	opeq12d 4839	. . . . . 6 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ⟨𝑦, 𝑧⟩ = ⟨𝑋, 𝑋⟩)
110	109, 107	oveq12d 7414	. . . . 5 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (⟨𝑦, 𝑧⟩ · 𝑤) = (⟨𝑋, 𝑋⟩ · 𝑋))
111	110	oveqd 7413	. . . 4 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔))
112		eqidd 2763	. . . 4 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → 𝑓 = 𝑓)
113	108, 111, 112	oveq123d 7417	. . 3 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋)𝑓))
114	27, 35	opeq12d 4839	. . . . 5 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ⟨𝑥, 𝑧⟩ = ⟨𝑋, 𝑋⟩)
115	114, 107	oveq12d 7414	. . . 4 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (⟨𝑥, 𝑧⟩ · 𝑤) = (⟨𝑋, 𝑋⟩ · 𝑋))
116		eqidd 2763	. . . 4 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → 𝑘 = 𝑘)
117	115, 116, 60	oveq123d 7417	. . 3 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓)) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)))
118	106, 113, 117	3eqtr4d 2807	. 2 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓)))
119	1, 2, 3, 16, 17, 25, 33, 39, 45, 118	iscatd2 17713	1 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ {𝑋} ↦ 1 )))

Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 858   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  Vcvv 3454  {csn 4582  {cpr 4584  ⟨cop 4588   ↦ cmpt 5181  ‘cfv 6521  (class class class)co 7396  Basecbs 17245  Hom chom 17297  compcco 17298  Catccat 17696  Idccid 17697

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-cat 17700  df-cid 17701

This theorem is referenced by:  incat  50222