Theorem aprcl 8620

Description: Reverse closure for apartness. (Contributed by Jim Kingdon, 19-Dec-2023.)

Assertion

Ref	Expression
aprcl	⊢ (𝐴 # 𝐵 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ))

Proof of Theorem aprcl

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

Step	Hyp	Ref	Expression
1		df-br 4018	. . . 4 ⊢ (𝐴 # 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ # )
2		eqeq1 2195	. . . . . . . . . 10 ⊢ (𝑥 = (1st ‘⟨𝐴, 𝐵⟩) → (𝑥 = (𝑟 + (i · 𝑠)) ↔ (1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠))))
3	2	anbi1d 465	. . . . . . . . 9 ⊢ (𝑥 = (1st ‘⟨𝐴, 𝐵⟩) → ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ↔ ((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢)))))
4	3	anbi1d 465	. . . . . . . 8 ⊢ (𝑥 = (1st ‘⟨𝐴, 𝐵⟩) → (((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢)) ↔ (((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢))))
5	4	2rexbidv 2514	. . . . . . 7 ⊢ (𝑥 = (1st ‘⟨𝐴, 𝐵⟩) → (∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢)) ↔ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ (((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢))))
6	5	2rexbidv 2514	. . . . . 6 ⊢ (𝑥 = (1st ‘⟨𝐴, 𝐵⟩) → (∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢)) ↔ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ (((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢))))
7		eqeq1 2195	. . . . . . . . . 10 ⊢ (𝑦 = (2nd ‘⟨𝐴, 𝐵⟩) → (𝑦 = (𝑡 + (i · 𝑢)) ↔ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢))))
8	7	anbi2d 464	. . . . . . . . 9 ⊢ (𝑦 = (2nd ‘⟨𝐴, 𝐵⟩) → (((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ↔ ((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢)))))
9	8	anbi1d 465	. . . . . . . 8 ⊢ (𝑦 = (2nd ‘⟨𝐴, 𝐵⟩) → ((((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢)) ↔ (((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢))))
10	9	2rexbidv 2514	. . . . . . 7 ⊢ (𝑦 = (2nd ‘⟨𝐴, 𝐵⟩) → (∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ (((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢)) ↔ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ (((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢))))
11	10	2rexbidv 2514	. . . . . 6 ⊢ (𝑦 = (2nd ‘⟨𝐴, 𝐵⟩) → (∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ (((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢)) ↔ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ (((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢))))
12	6, 11	elopabi 6213	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢))} → ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ (((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢)))
13		df-ap 8556	. . . . 5 ⊢ # = {⟨𝑥, 𝑦⟩ ∣ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢))}
14	12, 13	eleq2s 2283	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ # → ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ (((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢)))
15	1, 14	sylbi 121	. . 3 ⊢ (𝐴 # 𝐵 → ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ (((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢)))
16		simpl 109	. . . . . . 7 ⊢ ((((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢)) → ((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢))))
17	16	reximi 2586	. . . . . 6 ⊢ (∃𝑢 ∈ ℝ (((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢)) → ∃𝑢 ∈ ℝ ((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢))))
18	17	reximi 2586	. . . . 5 ⊢ (∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ (((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢)) → ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢))))
19	18	reximi 2586	. . . 4 ⊢ (∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ (((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢)) → ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢))))
20	19	reximi 2586	. . 3 ⊢ (∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ (((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢)) → ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢))))
21	15, 20	syl 14	. 2 ⊢ (𝐴 # 𝐵 → ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢))))
22	13	relopabi 4766	. . . . . . . . . 10 ⊢ Rel #
23	22	brrelex1i 4683	. . . . . . . . 9 ⊢ (𝐴 # 𝐵 → 𝐴 ∈ V)
24	23	ad3antrrr 492	. . . . . . . 8 ⊢ ((((𝐴 # 𝐵 ∧ (𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ)) ∧ (𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ)) ∧ ((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢)))) → 𝐴 ∈ V)
25	22	brrelex2i 4684	. . . . . . . . 9 ⊢ (𝐴 # 𝐵 → 𝐵 ∈ V)
26	25	ad3antrrr 492	. . . . . . . 8 ⊢ ((((𝐴 # 𝐵 ∧ (𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ)) ∧ (𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ)) ∧ ((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢)))) → 𝐵 ∈ V)
27		op1stg 6168	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
28	24, 26, 27	syl2anc 411	. . . . . . 7 ⊢ ((((𝐴 # 𝐵 ∧ (𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ)) ∧ (𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ)) ∧ ((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢)))) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
29		simprl 529	. . . . . . . 8 ⊢ ((((𝐴 # 𝐵 ∧ (𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ)) ∧ (𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ)) ∧ ((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢)))) → (1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)))
30		simprl 529	. . . . . . . . . . 11 ⊢ ((𝐴 # 𝐵 ∧ (𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ)) → 𝑟 ∈ ℝ)
31	30	ad2antrr 488	. . . . . . . . . 10 ⊢ ((((𝐴 # 𝐵 ∧ (𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ)) ∧ (𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ)) ∧ ((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢)))) → 𝑟 ∈ ℝ)
32	31	recnd 8003	. . . . . . . . 9 ⊢ ((((𝐴 # 𝐵 ∧ (𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ)) ∧ (𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ)) ∧ ((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢)))) → 𝑟 ∈ ℂ)
33		ax-icn 7923	. . . . . . . . . . 11 ⊢ i ∈ ℂ
34	33	a1i 9	. . . . . . . . . 10 ⊢ ((((𝐴 # 𝐵 ∧ (𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ)) ∧ (𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ)) ∧ ((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢)))) → i ∈ ℂ)
35		simprr 531	. . . . . . . . . . . 12 ⊢ ((𝐴 # 𝐵 ∧ (𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ)) → 𝑠 ∈ ℝ)
36	35	ad2antrr 488	. . . . . . . . . . 11 ⊢ ((((𝐴 # 𝐵 ∧ (𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ)) ∧ (𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ)) ∧ ((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢)))) → 𝑠 ∈ ℝ)
37	36	recnd 8003	. . . . . . . . . 10 ⊢ ((((𝐴 # 𝐵 ∧ (𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ)) ∧ (𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ)) ∧ ((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢)))) → 𝑠 ∈ ℂ)
38	34, 37	mulcld 7995	. . . . . . . . 9 ⊢ ((((𝐴 # 𝐵 ∧ (𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ)) ∧ (𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ)) ∧ ((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢)))) → (i · 𝑠) ∈ ℂ)
39	32, 38	addcld 7994	. . . . . . . 8 ⊢ ((((𝐴 # 𝐵 ∧ (𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ)) ∧ (𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ)) ∧ ((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢)))) → (𝑟 + (i · 𝑠)) ∈ ℂ)
40	29, 39	eqeltrd 2265	. . . . . . 7 ⊢ ((((𝐴 # 𝐵 ∧ (𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ)) ∧ (𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ)) ∧ ((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢)))) → (1st ‘⟨𝐴, 𝐵⟩) ∈ ℂ)
41	28, 40	eqeltrrd 2266	. . . . . 6 ⊢ ((((𝐴 # 𝐵 ∧ (𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ)) ∧ (𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ)) ∧ ((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢)))) → 𝐴 ∈ ℂ)
42		op2ndg 6169	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
43	24, 26, 42	syl2anc 411	. . . . . . 7 ⊢ ((((𝐴 # 𝐵 ∧ (𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ)) ∧ (𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ)) ∧ ((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢)))) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
44		simprr 531	. . . . . . . 8 ⊢ ((((𝐴 # 𝐵 ∧ (𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ)) ∧ (𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ)) ∧ ((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢)))) → (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢)))
45		recn 7961	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ℝ → 𝑡 ∈ ℂ)
46	45	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ) → 𝑡 ∈ ℂ)
47	33	a1i 9	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ) → i ∈ ℂ)
48		recn 7961	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ ℝ → 𝑢 ∈ ℂ)
49	48	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ) → 𝑢 ∈ ℂ)
50	47, 49	mulcld 7995	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ) → (i · 𝑢) ∈ ℂ)
51	46, 50	addcld 7994	. . . . . . . . . 10 ⊢ ((𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑡 + (i · 𝑢)) ∈ ℂ)
52	51	adantl 277	. . . . . . . . 9 ⊢ (((𝐴 # 𝐵 ∧ (𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ)) ∧ (𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ)) → (𝑡 + (i · 𝑢)) ∈ ℂ)
53	52	adantr 276	. . . . . . . 8 ⊢ ((((𝐴 # 𝐵 ∧ (𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ)) ∧ (𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ)) ∧ ((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢)))) → (𝑡 + (i · 𝑢)) ∈ ℂ)
54	44, 53	eqeltrd 2265	. . . . . . 7 ⊢ ((((𝐴 # 𝐵 ∧ (𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ)) ∧ (𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ)) ∧ ((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢)))) → (2nd ‘⟨𝐴, 𝐵⟩) ∈ ℂ)
55	43, 54	eqeltrrd 2266	. . . . . 6 ⊢ ((((𝐴 # 𝐵 ∧ (𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ)) ∧ (𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ)) ∧ ((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢)))) → 𝐵 ∈ ℂ)
56	41, 55	jca 306	. . . . 5 ⊢ ((((𝐴 # 𝐵 ∧ (𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ)) ∧ (𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ)) ∧ ((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢)))) → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ))
57	56	ex 115	. . . 4 ⊢ (((𝐴 # 𝐵 ∧ (𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ)) ∧ (𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ)) → (((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢))) → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)))
58	57	rexlimdvva 2614	. . 3 ⊢ ((𝐴 # 𝐵 ∧ (𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ)) → (∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢))) → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)))
59	58	rexlimdvva 2614	. 2 ⊢ (𝐴 # 𝐵 → (∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠)) ∧ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢))) → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)))
60	21, 59	mpd 13	1 ⊢ (𝐴 # 𝐵 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 709   = wceq 1363   ∈ wcel 2159  ∃wrex 2468  Vcvv 2751  ⟨cop 3609   class class class wbr 4017  {copab 4077  ‘cfv 5230  (class class class)co 5890  1^st c1st 6156  2^nd c2nd 6157  ℂcc 7826  ℝcr 7827  ici 7830   + caddc 7831   · cmul 7833   #_ℝ creap 8548   # cap 8555

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-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-resscn 7920  ax-icn 7923  ax-addcl 7924  ax-mulcl 7926

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rex 2473  df-v 2753  df-sbc 2977  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-br 4018  df-opab 4079  df-mpt 4080  df-id 4307  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-fo 5236  df-fv 5238  df-1st 6158  df-2nd 6159  df-ap 8556

This theorem is referenced by:  apsscn  8621