Theorem aprcl 8975

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 4131	. . . 4 ⊢ (𝐴 # 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ # )
2		eqeq1 2245	. . . . . . . . . 10 ⊢ (𝑥 = (1st ‘⟨𝐴, 𝐵⟩) → (𝑥 = (𝑟 + (i · 𝑠)) ↔ (1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠))))
3	2	anbi1d 469	. . . . . . . . 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 469	. . . . . . . 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 2575	. . . . . . 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 2575	. . . . . 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 2245	. . . . . . . . . 10 ⊢ (𝑦 = (2nd ‘⟨𝐴, 𝐵⟩) → (𝑦 = (𝑡 + (i · 𝑢)) ↔ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢))))
8	7	anbi2d 468	. . . . . . . . 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 469	. . . . . . . 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 2575	. . . . . . 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 2575	. . . . . 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 6431	. . . . 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 8911	. . . . 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 2333	. . . 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 2647	. . . . . 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 2647	. . . . 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 2647	. . . 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 2647	. . 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 4905	. . . . . . . . . 10 ⊢ Rel #
23	22	brrelex1i 4818	. . . . . . . . 9 ⊢ (𝐴 # 𝐵 → 𝐴 ∈ V)
24	23	ad3antrrr 496	. . . . . . . 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 4819	. . . . . . . . 9 ⊢ (𝐴 # 𝐵 → 𝐵 ∈ V)
26	25	ad3antrrr 496	. . . . . . . 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 6384	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
28	24, 26, 27	syl2anc 415	. . . . . . 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 535	. . . . . . . 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 535	. . . . . . . . . . 11 ⊢ ((𝐴 # 𝐵 ∧ (𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ)) → 𝑟 ∈ ℝ)
31	30	ad2antrr 492	. . . . . . . . . 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 8354	. . . . . . . . 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 8274	. . . . . . . . . . 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 537	. . . . . . . . . . . 12 ⊢ ((𝐴 # 𝐵 ∧ (𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ)) → 𝑠 ∈ ℝ)
36	35	ad2antrr 492	. . . . . . . . . . 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 8354	. . . . . . . . . 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 8346	. . . . . . . . 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 8345	. . . . . . . 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 2315	. . . . . . 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 2316	. . . . . 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 6385	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
43	24, 26, 42	syl2anc 415	. . . . . . 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 537	. . . . . . . 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 8312	. . . . . . . . . . . 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 8312	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ ℝ → 𝑢 ∈ ℂ)
49	48	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ) → 𝑢 ∈ ℂ)
50	47, 49	mulcld 8346	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ) → (i · 𝑢) ∈ ℂ)
51	46, 50	addcld 8345	. . . . . . . . . 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 2315	. . . . . . 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 2316	. . . . . 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 2676	. . 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 2676	. 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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∨ wo 720   = wceq 1402   ∈ wcel 2209  ∃wrex 2529  Vcvv 2821  ⟨cop 3712   class class class wbr 4130  {copab 4191  ‘cfv 5377  (class class class)co 6085  1^st c1st 6372  2^nd c2nd 6373  ℂcc 8177  ℝcr 8178  ici 8181   + caddc 8182   · cmul 8184   #_ℝ creap 8903   # cap 8910

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  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 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-resscn 8271  ax-icn 8274  ax-addcl 8275  ax-mulcl 8277

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  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-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fo 5383  df-fv 5385  df-1st 6374  df-2nd 6375  df-ap 8911

This theorem is used by:  apsscn  8976