Theorem aprcl 8408

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 3930	. . . 4 ⊢ (𝐴 # 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ # )
2		eqeq1 2146	. . . . . . . . . 10 ⊢ (𝑥 = (1st ‘⟨𝐴, 𝐵⟩) → (𝑥 = (𝑟 + (i · 𝑠)) ↔ (1st ‘⟨𝐴, 𝐵⟩) = (𝑟 + (i · 𝑠))))
3	2	anbi1d 460	. . . . . . . . 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 460	. . . . . . . 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 2460	. . . . . . 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 2460	. . . . . 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 2146	. . . . . . . . . 10 ⊢ (𝑦 = (2nd ‘⟨𝐴, 𝐵⟩) → (𝑦 = (𝑡 + (i · 𝑢)) ↔ (2nd ‘⟨𝐴, 𝐵⟩) = (𝑡 + (i · 𝑢))))
8	7	anbi2d 459	. . . . . . . . 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 460	. . . . . . . 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 2460	. . . . . . 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 2460	. . . . . 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 6093	. . . . 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 8344	. . . . 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 2234	. . . 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 120	. . 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 108	. . . . . . 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 2529	. . . . . 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 2529	. . . . 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 2529	. . . 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 2529	. . 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 4665	. . . . . . . . . 10 ⊢ Rel #
23	22	brrelex1i 4582	. . . . . . . . 9 ⊢ (𝐴 # 𝐵 → 𝐴 ∈ V)
24	23	ad3antrrr 483	. . . . . . . 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 4583	. . . . . . . . 9 ⊢ (𝐴 # 𝐵 → 𝐵 ∈ V)
26	25	ad3antrrr 483	. . . . . . . 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 6048	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
28	24, 26, 27	syl2anc 408	. . . . . . 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 520	. . . . . . . 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 520	. . . . . . . . . . 11 ⊢ ((𝐴 # 𝐵 ∧ (𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ)) → 𝑟 ∈ ℝ)
31	30	ad2antrr 479	. . . . . . . . . 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 7794	. . . . . . . . 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 7715	. . . . . . . . . . 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 521	. . . . . . . . . . . 12 ⊢ ((𝐴 # 𝐵 ∧ (𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ)) → 𝑠 ∈ ℝ)
36	35	ad2antrr 479	. . . . . . . . . . 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 7794	. . . . . . . . . 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 7786	. . . . . . . . 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 7785	. . . . . . . 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 2216	. . . . . . 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 2217	. . . . . 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 6049	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
43	24, 26, 42	syl2anc 408	. . . . . . 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 521	. . . . . . . 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 7753	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ℝ → 𝑡 ∈ ℂ)
46	45	adantr 274	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ) → 𝑡 ∈ ℂ)
47	33	a1i 9	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ) → i ∈ ℂ)
48		recn 7753	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ ℝ → 𝑢 ∈ ℂ)
49	48	adantl 275	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ) → 𝑢 ∈ ℂ)
50	47, 49	mulcld 7786	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ) → (i · 𝑢) ∈ ℂ)
51	46, 50	addcld 7785	. . . . . . . . . 10 ⊢ ((𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑡 + (i · 𝑢)) ∈ ℂ)
52	51	adantl 275	. . . . . . . . 9 ⊢ (((𝐴 # 𝐵 ∧ (𝑟 ∈ ℝ ∧ 𝑠 ∈ ℝ)) ∧ (𝑡 ∈ ℝ ∧ 𝑢 ∈ ℝ)) → (𝑡 + (i · 𝑢)) ∈ ℂ)
53	52	adantr 274	. . . . . . . 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 2216	. . . . . . 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 2217	. . . . . 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 304	. . . . 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 114	. . . 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 2557	. . 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 2557	. 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 103   ∨ wo 697   = wceq 1331   ∈ wcel 1480  ∃wrex 2417  Vcvv 2686  ⟨cop 3530   class class class wbr 3929  {copab 3988  ‘cfv 5123  (class class class)co 5774  1^st c1st 6036  2^nd c2nd 6037  ℂcc 7618  ℝcr 7619  ici 7622   + caddc 7623   · cmul 7625   #_ℝ creap 8336   # cap 8343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-resscn 7712  ax-icn 7715  ax-addcl 7716  ax-mulcl 7718

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fo 5129  df-fv 5131  df-1st 6038  df-2nd 6039  df-ap 8344

This theorem is referenced by:  apsscn  8409