Theorem cnegexlem2 8202

Description: Existence of a real number which produces a real number when multiplied by i. (Hint: zero is such a number, although we don't need to prove that yet). Lemma for cnegex 8204. (Contributed by Eric Schmidt, 22-May-2007.)

Assertion

Ref	Expression
cnegexlem2	⊢ ∃𝑦 ∈ ℝ (i · 𝑦) ∈ ℝ

Proof of Theorem cnegexlem2

Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0cn 8018	. 2 ⊢ 0 ∈ ℂ
2		cnre 8022	. 2 ⊢ (0 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 0 = (𝑥 + (i · 𝑦)))
3		ax-rnegex 7988	. . . . . 6 ⊢ (𝑥 ∈ ℝ → ∃𝑧 ∈ ℝ (𝑥 + 𝑧) = 0)
4	3	adantr 276	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ∃𝑧 ∈ ℝ (𝑥 + 𝑧) = 0)
5		recn 8012	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
6		ax-icn 7974	. . . . . . . . . . . 12 ⊢ i ∈ ℂ
7		recn 8012	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
8		mulcl 8006	. . . . . . . . . . . 12 ⊢ ((i ∈ ℂ ∧ 𝑦 ∈ ℂ) → (i · 𝑦) ∈ ℂ)
9	6, 7, 8	sylancr 414	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → (i · 𝑦) ∈ ℂ)
10		recn 8012	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℝ → 𝑧 ∈ ℂ)
11		addlid 8165	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℂ → (0 + 𝑧) = 𝑧)
12	11	3ad2ant3 1022	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ ∧ 𝑧 ∈ ℂ) → (0 + 𝑧) = 𝑧)
13	12	adantr 276	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ ∧ 𝑧 ∈ ℂ) ∧ ((𝑥 + 𝑧) = 0 ∧ 0 = (𝑥 + (i · 𝑦)))) → (0 + 𝑧) = 𝑧)
14		oveq1 5929	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 + 𝑧) = 0 → ((𝑥 + 𝑧) + (i · 𝑦)) = (0 + (i · 𝑦)))
15	14	ad2antrl 490	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ ∧ 𝑧 ∈ ℂ) ∧ ((𝑥 + 𝑧) = 0 ∧ 0 = (𝑥 + (i · 𝑦)))) → ((𝑥 + 𝑧) + (i · 𝑦)) = (0 + (i · 𝑦)))
16		add32 8185	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ) → ((𝑥 + 𝑧) + (i · 𝑦)) = ((𝑥 + (i · 𝑦)) + 𝑧))
17	16	3com23 1211	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑧) + (i · 𝑦)) = ((𝑥 + (i · 𝑦)) + 𝑧))
18		oveq1 5929	. . . . . . . . . . . . . . . . 17 ⊢ (0 = (𝑥 + (i · 𝑦)) → (0 + 𝑧) = ((𝑥 + (i · 𝑦)) + 𝑧))
19	18	eqcomd 2202	. . . . . . . . . . . . . . . 16 ⊢ (0 = (𝑥 + (i · 𝑦)) → ((𝑥 + (i · 𝑦)) + 𝑧) = (0 + 𝑧))
20	17, 19	sylan9eq 2249	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ ∧ 𝑧 ∈ ℂ) ∧ 0 = (𝑥 + (i · 𝑦))) → ((𝑥 + 𝑧) + (i · 𝑦)) = (0 + 𝑧))
21	20	adantrl 478	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ ∧ 𝑧 ∈ ℂ) ∧ ((𝑥 + 𝑧) = 0 ∧ 0 = (𝑥 + (i · 𝑦)))) → ((𝑥 + 𝑧) + (i · 𝑦)) = (0 + 𝑧))
22		addlid 8165	. . . . . . . . . . . . . . . 16 ⊢ ((i · 𝑦) ∈ ℂ → (0 + (i · 𝑦)) = (i · 𝑦))
23	22	3ad2ant2 1021	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ ∧ 𝑧 ∈ ℂ) → (0 + (i · 𝑦)) = (i · 𝑦))
24	23	adantr 276	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ ∧ 𝑧 ∈ ℂ) ∧ ((𝑥 + 𝑧) = 0 ∧ 0 = (𝑥 + (i · 𝑦)))) → (0 + (i · 𝑦)) = (i · 𝑦))
25	15, 21, 24	3eqtr3d 2237	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ ∧ 𝑧 ∈ ℂ) ∧ ((𝑥 + 𝑧) = 0 ∧ 0 = (𝑥 + (i · 𝑦)))) → (0 + 𝑧) = (i · 𝑦))
26	13, 25	eqtr3d 2231	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ ∧ 𝑧 ∈ ℂ) ∧ ((𝑥 + 𝑧) = 0 ∧ 0 = (𝑥 + (i · 𝑦)))) → 𝑧 = (i · 𝑦))
27	26	ex 115	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ ∧ 𝑧 ∈ ℂ) → (((𝑥 + 𝑧) = 0 ∧ 0 = (𝑥 + (i · 𝑦))) → 𝑧 = (i · 𝑦)))
28	5, 9, 10, 27	syl3an 1291	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (((𝑥 + 𝑧) = 0 ∧ 0 = (𝑥 + (i · 𝑦))) → 𝑧 = (i · 𝑦)))
29	28	3expa 1205	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → (((𝑥 + 𝑧) = 0 ∧ 0 = (𝑥 + (i · 𝑦))) → 𝑧 = (i · 𝑦)))
30	29	imp 124	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ((𝑥 + 𝑧) = 0 ∧ 0 = (𝑥 + (i · 𝑦)))) → 𝑧 = (i · 𝑦))
31		simplr 528	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ((𝑥 + 𝑧) = 0 ∧ 0 = (𝑥 + (i · 𝑦)))) → 𝑧 ∈ ℝ)
32	30, 31	eqeltrrd 2274	. . . . . . 7 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ((𝑥 + 𝑧) = 0 ∧ 0 = (𝑥 + (i · 𝑦)))) → (i · 𝑦) ∈ ℝ)
33	32	exp32 365	. . . . . 6 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → ((𝑥 + 𝑧) = 0 → (0 = (𝑥 + (i · 𝑦)) → (i · 𝑦) ∈ ℝ)))
34	33	rexlimdva 2614	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (∃𝑧 ∈ ℝ (𝑥 + 𝑧) = 0 → (0 = (𝑥 + (i · 𝑦)) → (i · 𝑦) ∈ ℝ)))
35	4, 34	mpd 13	. . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 = (𝑥 + (i · 𝑦)) → (i · 𝑦) ∈ ℝ))
36	35	reximdva 2599	. . 3 ⊢ (𝑥 ∈ ℝ → (∃𝑦 ∈ ℝ 0 = (𝑥 + (i · 𝑦)) → ∃𝑦 ∈ ℝ (i · 𝑦) ∈ ℝ))
37	36	rexlimiv 2608	. 2 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 0 = (𝑥 + (i · 𝑦)) → ∃𝑦 ∈ ℝ (i · 𝑦) ∈ ℝ)
38	1, 2, 37	mp2b 8	1 ⊢ ∃𝑦 ∈ ℝ (i · 𝑦) ∈ ℝ

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  ∃wrex 2476  (class class class)co 5922  ℂcc 7877  ℝcr 7878  0cc0 7879  ici 7881   + caddc 7882   · cmul 7884

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-resscn 7971  ax-1cn 7972  ax-icn 7974  ax-addcl 7975  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-iota 5219  df-fv 5266  df-ov 5925

This theorem is referenced by:  cnegex  8204