Theorem aptap 8941

Description: Complex apartness (as defined at df-ap 8873) is a tight apartness (as defined at df-tap 7579). (Contributed by Jim Kingdon, 16-Feb-2025.)

Assertion

Ref	Expression
aptap	⊢ # TAp ℂ

Proof of Theorem aptap

Dummy variables 𝑞 𝑝 𝑟 𝑠 𝑡 𝑢 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2241	. . . . . . . . . 10 ⊢ (𝑢 = (1st ‘𝑡) → (𝑢 = (𝑝 + (i · 𝑞)) ↔ (1st ‘𝑡) = (𝑝 + (i · 𝑞))))
2	1	anbi1d 465	. . . . . . . . 9 ⊢ (𝑢 = (1st ‘𝑡) → ((𝑢 = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠))) ↔ ((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠)))))
3	2	anbi1d 465	. . . . . . . 8 ⊢ (𝑢 = (1st ‘𝑡) → (((𝑢 = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠)) ↔ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))))
4	3	2rexbidv 2569	. . . . . . 7 ⊢ (𝑢 = (1st ‘𝑡) → (∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ((𝑢 = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠)) ↔ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))))
5	4	2rexbidv 2569	. . . . . 6 ⊢ (𝑢 = (1st ‘𝑡) → (∃𝑝 ∈ ℝ ∃𝑞 ∈ ℝ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ((𝑢 = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠)) ↔ ∃𝑝 ∈ ℝ ∃𝑞 ∈ ℝ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))))
6		eqeq1 2241	. . . . . . . . . 10 ⊢ (𝑣 = (2nd ‘𝑡) → (𝑣 = (𝑟 + (i · 𝑠)) ↔ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))))
7	6	anbi2d 464	. . . . . . . . 9 ⊢ (𝑣 = (2nd ‘𝑡) → (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠))) ↔ ((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠)))))
8	7	anbi1d 465	. . . . . . . 8 ⊢ (𝑣 = (2nd ‘𝑡) → ((((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠)) ↔ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))))
9	8	2rexbidv 2569	. . . . . . 7 ⊢ (𝑣 = (2nd ‘𝑡) → (∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠)) ↔ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))))
10	9	2rexbidv 2569	. . . . . 6 ⊢ (𝑣 = (2nd ‘𝑡) → (∃𝑝 ∈ ℝ ∃𝑞 ∈ ℝ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠)) ↔ ∃𝑝 ∈ ℝ ∃𝑞 ∈ ℝ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))))
11	5, 10	elopabi 6404	. . . . 5 ⊢ (𝑡 ∈ {⟨𝑢, 𝑣⟩ ∣ ∃𝑝 ∈ ℝ ∃𝑞 ∈ ℝ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ((𝑢 = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))} → ∃𝑝 ∈ ℝ ∃𝑞 ∈ ℝ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠)))
12		df-ap 8873	. . . . 5 ⊢ # = {⟨𝑢, 𝑣⟩ ∣ ∃𝑝 ∈ ℝ ∃𝑞 ∈ ℝ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ((𝑢 = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))}
13	11, 12	eleq2s 2329	. . . 4 ⊢ (𝑡 ∈ # → ∃𝑝 ∈ ℝ ∃𝑞 ∈ ℝ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠)))
14	12	relopabi 4885	. . . . . . . . . 10 ⊢ Rel #
15		simp-5l 545	. . . . . . . . . 10 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → 𝑡 ∈ # )
16		1st2nd 6388	. . . . . . . . . 10 ⊢ ((Rel # ∧ 𝑡 ∈ # ) → 𝑡 = ⟨(1st ‘𝑡), (2nd ‘𝑡)⟩)
17	14, 15, 16	sylancr 414	. . . . . . . . 9 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → 𝑡 = ⟨(1st ‘𝑡), (2nd ‘𝑡)⟩)
18		simprll 539	. . . . . . . . . . 11 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → (1st ‘𝑡) = (𝑝 + (i · 𝑞)))
19		simp-5r 546	. . . . . . . . . . . . 13 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → 𝑝 ∈ ℝ)
20	19	recnd 8318	. . . . . . . . . . . 12 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → 𝑝 ∈ ℂ)
21		ax-icn 8238	. . . . . . . . . . . . . 14 ⊢ i ∈ ℂ
22	21	a1i 9	. . . . . . . . . . . . 13 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → i ∈ ℂ)
23		simp-4r 544	. . . . . . . . . . . . . 14 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → 𝑞 ∈ ℝ)
24	23	recnd 8318	. . . . . . . . . . . . 13 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → 𝑞 ∈ ℂ)
25	22, 24	mulcld 8310	. . . . . . . . . . . 12 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → (i · 𝑞) ∈ ℂ)
26	20, 25	addcld 8309	. . . . . . . . . . 11 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → (𝑝 + (i · 𝑞)) ∈ ℂ)
27	18, 26	eqeltrd 2311	. . . . . . . . . 10 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → (1st ‘𝑡) ∈ ℂ)
28		simprlr 540	. . . . . . . . . . 11 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → (2nd ‘𝑡) = (𝑟 + (i · 𝑠)))
29		simpllr 536	. . . . . . . . . . . . 13 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → 𝑟 ∈ ℝ)
30	29	recnd 8318	. . . . . . . . . . . 12 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → 𝑟 ∈ ℂ)
31		simplr 529	. . . . . . . . . . . . . 14 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → 𝑠 ∈ ℝ)
32	31	recnd 8318	. . . . . . . . . . . . 13 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → 𝑠 ∈ ℂ)
33	22, 32	mulcld 8310	. . . . . . . . . . . 12 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → (i · 𝑠) ∈ ℂ)
34	30, 33	addcld 8309	. . . . . . . . . . 11 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → (𝑟 + (i · 𝑠)) ∈ ℂ)
35	28, 34	eqeltrd 2311	. . . . . . . . . 10 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → (2nd ‘𝑡) ∈ ℂ)
36	27, 35	jca 306	. . . . . . . . 9 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → ((1st ‘𝑡) ∈ ℂ ∧ (2nd ‘𝑡) ∈ ℂ))
37		elxp6 6376	. . . . . . . . 9 ⊢ (𝑡 ∈ (ℂ × ℂ) ↔ (𝑡 = ⟨(1st ‘𝑡), (2nd ‘𝑡)⟩ ∧ ((1st ‘𝑡) ∈ ℂ ∧ (2nd ‘𝑡) ∈ ℂ)))
38	17, 36, 37	sylanbrc 417	. . . . . . . 8 ⊢ ((((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) ∧ 𝑠 ∈ ℝ) ∧ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠))) → 𝑡 ∈ (ℂ × ℂ))
39	38	rexlimdva2 2665	. . . . . . 7 ⊢ ((((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) ∧ 𝑟 ∈ ℝ) → (∃𝑠 ∈ ℝ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠)) → 𝑡 ∈ (ℂ × ℂ)))
40	39	rexlimdva 2662	. . . . . 6 ⊢ (((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) ∧ 𝑞 ∈ ℝ) → (∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠)) → 𝑡 ∈ (ℂ × ℂ)))
41	40	rexlimdva 2662	. . . . 5 ⊢ ((𝑡 ∈ # ∧ 𝑝 ∈ ℝ) → (∃𝑞 ∈ ℝ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠)) → 𝑡 ∈ (ℂ × ℂ)))
42	41	rexlimdva 2662	. . . 4 ⊢ (𝑡 ∈ # → (∃𝑝 ∈ ℝ ∃𝑞 ∈ ℝ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ (((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))) ∧ (𝑝 #ℝ 𝑟 ∨ 𝑞 #ℝ 𝑠)) → 𝑡 ∈ (ℂ × ℂ)))
43	13, 42	mpd 13	. . 3 ⊢ (𝑡 ∈ # → 𝑡 ∈ (ℂ × ℂ))
44	43	ssriv 3246	. 2 ⊢ # ⊆ (ℂ × ℂ)
45		apirr 8896	. . . 4 ⊢ (𝑥 ∈ ℂ → ¬ 𝑥 # 𝑥)
46	45	rgen 2597	. . 3 ⊢ ∀𝑥 ∈ ℂ ¬ 𝑥 # 𝑥
47		apsym 8897	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 # 𝑦 ↔ 𝑦 # 𝑥))
48	47	biimpd 144	. . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 # 𝑦 → 𝑦 # 𝑥))
49	48	rgen2 2630	. . 3 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ (𝑥 # 𝑦 → 𝑦 # 𝑥)
50	46, 49	pm3.2i 272	. 2 ⊢ (∀𝑥 ∈ ℂ ¬ 𝑥 # 𝑥 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ (𝑥 # 𝑦 → 𝑦 # 𝑥))
51		apcotr 8898	. . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 # 𝑦 → (𝑥 # 𝑧 ∨ 𝑦 # 𝑧)))
52	51	rgen3 2631	. . 3 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ ∀𝑧 ∈ ℂ (𝑥 # 𝑦 → (𝑥 # 𝑧 ∨ 𝑦 # 𝑧))
53		apti 8913	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 = 𝑦 ↔ ¬ 𝑥 # 𝑦))
54	53	biimprd 158	. . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (¬ 𝑥 # 𝑦 → 𝑥 = 𝑦))
55	54	rgen2 2630	. . 3 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ (¬ 𝑥 # 𝑦 → 𝑥 = 𝑦)
56	52, 55	pm3.2i 272	. 2 ⊢ (∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ ∀𝑧 ∈ ℂ (𝑥 # 𝑦 → (𝑥 # 𝑧 ∨ 𝑦 # 𝑧)) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ (¬ 𝑥 # 𝑦 → 𝑥 = 𝑦))
57		dftap2 7581	. 2 ⊢ ( # TAp ℂ ↔ ( # ⊆ (ℂ × ℂ) ∧ (∀𝑥 ∈ ℂ ¬ 𝑥 # 𝑥 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ (𝑥 # 𝑦 → 𝑦 # 𝑥)) ∧ (∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ ∀𝑧 ∈ ℂ (𝑥 # 𝑦 → (𝑥 # 𝑧 ∨ 𝑦 # 𝑧)) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ (¬ 𝑥 # 𝑦 → 𝑥 = 𝑦))))
58	44, 50, 56, 57	mpbir3an 1206	1 ⊢ # TAp ℂ

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 716   = wceq 1398   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523   ⊆ wss 3214  ⟨cop 3697   class class class wbr 4114  {copab 4175   × cxp 4752  Rel wrel 4759  ‘cfv 5357  (class class class)co 6058  1^st c1st 6345  2^nd c2nd 6346   TAp wtap 7578  ℂcc 8141  ℝcr 8142  ici 8145   + caddc 8146   · cmul 8148   #_ℝ creap 8865   # cap 8872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fo 5363  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-pap 7572  df-tap 7579  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873

This theorem is referenced by: (None)