Theorem cnegexlem2 8074

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 8076. (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 7891	. 2 ⊢ 0 ∈ ℂ
2		cnre 7895	. 2 ⊢ (0 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 0 = (𝑥 + (i · 𝑦)))
3		ax-rnegex 7862	. . . . . 6 ⊢ (𝑥 ∈ ℝ → ∃𝑧 ∈ ℝ (𝑥 + 𝑧) = 0)
4	3	adantr 274	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ∃𝑧 ∈ ℝ (𝑥 + 𝑧) = 0)
5		recn 7886	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
6		ax-icn 7848	. . . . . . . . . . . 12 ⊢ i ∈ ℂ
7		recn 7886	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
8		mulcl 7880	. . . . . . . . . . . 12 ⊢ ((i ∈ ℂ ∧ 𝑦 ∈ ℂ) → (i · 𝑦) ∈ ℂ)
9	6, 7, 8	sylancr 411	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → (i · 𝑦) ∈ ℂ)
10		recn 7886	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℝ → 𝑧 ∈ ℂ)
11		addid2 8037	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℂ → (0 + 𝑧) = 𝑧)
12	11	3ad2ant3 1010	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ ∧ 𝑧 ∈ ℂ) → (0 + 𝑧) = 𝑧)
13	12	adantr 274	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ ∧ 𝑧 ∈ ℂ) ∧ ((𝑥 + 𝑧) = 0 ∧ 0 = (𝑥 + (i · 𝑦)))) → (0 + 𝑧) = 𝑧)
14		oveq1 5849	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 + 𝑧) = 0 → ((𝑥 + 𝑧) + (i · 𝑦)) = (0 + (i · 𝑦)))
15	14	ad2antrl 482	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ ∧ 𝑧 ∈ ℂ) ∧ ((𝑥 + 𝑧) = 0 ∧ 0 = (𝑥 + (i · 𝑦)))) → ((𝑥 + 𝑧) + (i · 𝑦)) = (0 + (i · 𝑦)))
16		add32 8057	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ) → ((𝑥 + 𝑧) + (i · 𝑦)) = ((𝑥 + (i · 𝑦)) + 𝑧))
17	16	3com23 1199	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑧) + (i · 𝑦)) = ((𝑥 + (i · 𝑦)) + 𝑧))
18		oveq1 5849	. . . . . . . . . . . . . . . . 17 ⊢ (0 = (𝑥 + (i · 𝑦)) → (0 + 𝑧) = ((𝑥 + (i · 𝑦)) + 𝑧))
19	18	eqcomd 2171	. . . . . . . . . . . . . . . 16 ⊢ (0 = (𝑥 + (i · 𝑦)) → ((𝑥 + (i · 𝑦)) + 𝑧) = (0 + 𝑧))
20	17, 19	sylan9eq 2219	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ ∧ 𝑧 ∈ ℂ) ∧ 0 = (𝑥 + (i · 𝑦))) → ((𝑥 + 𝑧) + (i · 𝑦)) = (0 + 𝑧))
21	20	adantrl 470	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ ∧ 𝑧 ∈ ℂ) ∧ ((𝑥 + 𝑧) = 0 ∧ 0 = (𝑥 + (i · 𝑦)))) → ((𝑥 + 𝑧) + (i · 𝑦)) = (0 + 𝑧))
22		addid2 8037	. . . . . . . . . . . . . . . 16 ⊢ ((i · 𝑦) ∈ ℂ → (0 + (i · 𝑦)) = (i · 𝑦))
23	22	3ad2ant2 1009	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ ∧ 𝑧 ∈ ℂ) → (0 + (i · 𝑦)) = (i · 𝑦))
24	23	adantr 274	. . . . . . . . . . . . . 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 2206	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ ∧ 𝑧 ∈ ℂ) ∧ ((𝑥 + 𝑧) = 0 ∧ 0 = (𝑥 + (i · 𝑦)))) → (0 + 𝑧) = (i · 𝑦))
26	13, 25	eqtr3d 2200	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ ∧ 𝑧 ∈ ℂ) ∧ ((𝑥 + 𝑧) = 0 ∧ 0 = (𝑥 + (i · 𝑦)))) → 𝑧 = (i · 𝑦))
27	26	ex 114	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ ∧ 𝑧 ∈ ℂ) → (((𝑥 + 𝑧) = 0 ∧ 0 = (𝑥 + (i · 𝑦))) → 𝑧 = (i · 𝑦)))
28	5, 9, 10, 27	syl3an 1270	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (((𝑥 + 𝑧) = 0 ∧ 0 = (𝑥 + (i · 𝑦))) → 𝑧 = (i · 𝑦)))
29	28	3expa 1193	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → (((𝑥 + 𝑧) = 0 ∧ 0 = (𝑥 + (i · 𝑦))) → 𝑧 = (i · 𝑦)))
30	29	imp 123	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ((𝑥 + 𝑧) = 0 ∧ 0 = (𝑥 + (i · 𝑦)))) → 𝑧 = (i · 𝑦))
31		simplr 520	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ((𝑥 + 𝑧) = 0 ∧ 0 = (𝑥 + (i · 𝑦)))) → 𝑧 ∈ ℝ)
32	30, 31	eqeltrrd 2244	. . . . . . 7 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ((𝑥 + 𝑧) = 0 ∧ 0 = (𝑥 + (i · 𝑦)))) → (i · 𝑦) ∈ ℝ)
33	32	exp32 363	. . . . . 6 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ ℝ) → ((𝑥 + 𝑧) = 0 → (0 = (𝑥 + (i · 𝑦)) → (i · 𝑦) ∈ ℝ)))
34	33	rexlimdva 2583	. . . . 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 2568	. . 3 ⊢ (𝑥 ∈ ℝ → (∃𝑦 ∈ ℝ 0 = (𝑥 + (i · 𝑦)) → ∃𝑦 ∈ ℝ (i · 𝑦) ∈ ℝ))
37	36	rexlimiv 2577	. 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 103   ∧ w3a 968   = wceq 1343   ∈ wcel 2136  ∃wrex 2445  (class class class)co 5842  ℂcc 7751  ℝcr 7752  0cc0 7753  ici 7755   + caddc 7756   · cmul 7758

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-resscn 7845  ax-1cn 7846  ax-icn 7848  ax-addcl 7849  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-i2m1 7858  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-iota 5153  df-fv 5196  df-ov 5845

This theorem is referenced by:  cnegex  8076