Theorem aptap 8968

Description: Complex apartness (as defined at df-ap 8900) is a tight apartness (as defined at df-tap 7605). (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 2245	. . . . . . . . . 10 ⊢ (𝑢 = (1st ‘𝑡) → (𝑢 = (𝑝 + (i · 𝑞)) ↔ (1st ‘𝑡) = (𝑝 + (i · 𝑞))))
2	1	anbi1d 469	. . . . . . . . 9 ⊢ (𝑢 = (1st ‘𝑡) → ((𝑢 = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠))) ↔ ((1st ‘𝑡) = (𝑝 + (i · 𝑞)) ∧ 𝑣 = (𝑟 + (i · 𝑠)))))
3	2	anbi1d 469	. . . . . . . 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 2575	. . . . . . 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 2575	. . . . . 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 2245	. . . . . . . . . 10 ⊢ (𝑣 = (2nd ‘𝑡) → (𝑣 = (𝑟 + (i · 𝑠)) ↔ (2nd ‘𝑡) = (𝑟 + (i · 𝑠))))
7	6	anbi2d 468	. . . . . . . . 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 469	. . . . . . . 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 2575	. . . . . . 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 2575	. . . . . 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 6421	. . . . 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 8900	. . . . 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 2333	. . . 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 4900	. . . . . . . . . 10 ⊢ Rel #
15		simp-5l 549	. . . . . . . . . 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 6405	. . . . . . . . . 10 ⊢ ((Rel # ∧ 𝑡 ∈ # ) → 𝑡 = ⟨(1st ‘𝑡), (2nd ‘𝑡)⟩)
17	14, 15, 16	sylancr 418	. . . . . . . . 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 543	. . . . . . . . . . 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 550	. . . . . . . . . . . . 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 8344	. . . . . . . . . . . 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 8264	. . . . . . . . . . . . . 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 548	. . . . . . . . . . . . . 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 8344	. . . . . . . . . . . . 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 8336	. . . . . . . . . . . 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 8335	. . . . . . . . . . 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 2315	. . . . . . . . . 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 544	. . . . . . . . . . 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 540	. . . . . . . . . . . . 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 8344	. . . . . . . . . . . 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 533	. . . . . . . . . . . . . 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 8344	. . . . . . . . . . . . 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 8336	. . . . . . . . . . . 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 8335	. . . . . . . . . . 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 2315	. . . . . . . . . 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 6393	. . . . . . . . 9 ⊢ (𝑡 ∈ (ℂ × ℂ) ↔ (𝑡 = ⟨(1st ‘𝑡), (2nd ‘𝑡)⟩ ∧ ((1st ‘𝑡) ∈ ℂ ∧ (2nd ‘𝑡) ∈ ℂ)))
38	17, 36, 37	sylanbrc 421	. . . . . . . 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 2671	. . . . . . 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 2668	. . . . . 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 2668	. . . . 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 2668	. . . 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 3252	. 2 ⊢ # ⊆ (ℂ × ℂ)
45		apirr 8923	. . . 4 ⊢ (𝑥 ∈ ℂ → ¬ 𝑥 # 𝑥)
46	45	rgen 2603	. . 3 ⊢ ∀𝑥 ∈ ℂ ¬ 𝑥 # 𝑥
47		apsym 8924	. . . . 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 2636	. . 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 8925	. . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 # 𝑦 → (𝑥 # 𝑧 ∨ 𝑦 # 𝑧)))
52	51	rgen3 2637	. . 3 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ ∀𝑧 ∈ ℂ (𝑥 # 𝑦 → (𝑥 # 𝑧 ∨ 𝑦 # 𝑧))
53		apti 8940	. . . . 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 2636	. . 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 7607	. 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 1210	1 ⊢ # TAp ℂ

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 720   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529   ⊆ wss 3220  ⟨cop 3708   class class class wbr 4125  {copab 4186   × cxp 4767  Rel wrel 4774  ‘cfv 5372  (class class class)co 6075  1^st c1st 6362  2^nd c2nd 6363   TAp wtap 7604  ℂcc 8167  ℝcr 8168  ici 8171   + caddc 8172   · cmul 8174   #_ℝ creap 8892   # cap 8899

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fo 5378  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-pap 7598  df-tap 7605  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900

This theorem is referenced by: (None)