Theorem cnegexlem2 8095

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 8097. (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 7912	. 2 ⊢ 0 ∈ ℂ
2		cnre 7916	. 2 ⊢ (0 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 0 = (𝑥 + (i · 𝑦)))
3		ax-rnegex 7883	. . . . . 6 ⊢ (𝑥 ∈ ℝ → ∃𝑧 ∈ ℝ (𝑥 + 𝑧) = 0)
4	3	adantr 274	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ∃𝑧 ∈ ℝ (𝑥 + 𝑧) = 0)
5		recn 7907	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
6		ax-icn 7869	. . . . . . . . . . . 12 ⊢ i ∈ ℂ
7		recn 7907	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
8		mulcl 7901	. . . . . . . . . . . 12 ⊢ ((i ∈ ℂ ∧ 𝑦 ∈ ℂ) → (i · 𝑦) ∈ ℂ)
9	6, 7, 8	sylancr 412	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → (i · 𝑦) ∈ ℂ)
10		recn 7907	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℝ → 𝑧 ∈ ℂ)
11		addid2 8058	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℂ → (0 + 𝑧) = 𝑧)
12	11	3ad2ant3 1015	. . . . . . . . . . . . . 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 5860	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 + 𝑧) = 0 → ((𝑥 + 𝑧) + (i · 𝑦)) = (0 + (i · 𝑦)))
15	14	ad2antrl 487	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ ∧ 𝑧 ∈ ℂ) ∧ ((𝑥 + 𝑧) = 0 ∧ 0 = (𝑥 + (i · 𝑦)))) → ((𝑥 + 𝑧) + (i · 𝑦)) = (0 + (i · 𝑦)))
16		add32 8078	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ) → ((𝑥 + 𝑧) + (i · 𝑦)) = ((𝑥 + (i · 𝑦)) + 𝑧))
17	16	3com23 1204	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑧) + (i · 𝑦)) = ((𝑥 + (i · 𝑦)) + 𝑧))
18		oveq1 5860	. . . . . . . . . . . . . . . . 17 ⊢ (0 = (𝑥 + (i · 𝑦)) → (0 + 𝑧) = ((𝑥 + (i · 𝑦)) + 𝑧))
19	18	eqcomd 2176	. . . . . . . . . . . . . . . 16 ⊢ (0 = (𝑥 + (i · 𝑦)) → ((𝑥 + (i · 𝑦)) + 𝑧) = (0 + 𝑧))
20	17, 19	sylan9eq 2223	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ ∧ 𝑧 ∈ ℂ) ∧ 0 = (𝑥 + (i · 𝑦))) → ((𝑥 + 𝑧) + (i · 𝑦)) = (0 + 𝑧))
21	20	adantrl 475	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ ∧ 𝑧 ∈ ℂ) ∧ ((𝑥 + 𝑧) = 0 ∧ 0 = (𝑥 + (i · 𝑦)))) → ((𝑥 + 𝑧) + (i · 𝑦)) = (0 + 𝑧))
22		addid2 8058	. . . . . . . . . . . . . . . 16 ⊢ ((i · 𝑦) ∈ ℂ → (0 + (i · 𝑦)) = (i · 𝑦))
23	22	3ad2ant2 1014	. . . . . . . . . . . . . . 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 2211	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ (i · 𝑦) ∈ ℂ ∧ 𝑧 ∈ ℂ) ∧ ((𝑥 + 𝑧) = 0 ∧ 0 = (𝑥 + (i · 𝑦)))) → (0 + 𝑧) = (i · 𝑦))
26	13, 25	eqtr3d 2205	. . . . . . . . . . . 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 1275	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (((𝑥 + 𝑧) = 0 ∧ 0 = (𝑥 + (i · 𝑦))) → 𝑧 = (i · 𝑦)))
29	28	3expa 1198	. . . . . . . . 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 525	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ((𝑥 + 𝑧) = 0 ∧ 0 = (𝑥 + (i · 𝑦)))) → 𝑧 ∈ ℝ)
32	30, 31	eqeltrrd 2248	. . . . . . 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 2587	. . . . 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 2572	. . 3 ⊢ (𝑥 ∈ ℝ → (∃𝑦 ∈ ℝ 0 = (𝑥 + (i · 𝑦)) → ∃𝑦 ∈ ℝ (i · 𝑦) ∈ ℝ))
37	36	rexlimiv 2581	. 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 973   = wceq 1348   ∈ wcel 2141  ∃wrex 2449  (class class class)co 5853  ℂcc 7772  ℝcr 7773  0cc0 7774  ici 7776   + caddc 7777   · cmul 7779

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-resscn 7866  ax-1cn 7867  ax-icn 7869  ax-addcl 7870  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-i2m1 7879  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856

This theorem is referenced by:  cnegex  8097