Theorem cnegexlem1 7283

Description: Addition cancellation of a real number from two complex numbers. Lemma for cnegex 7286. (Contributed by Eric Schmidt, 22-May-2007.)

Assertion

Ref	Expression
cnegexlem1	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶))

Proof of Theorem cnegexlem1

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-rnegex 7085	. . . 4 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
2	1	3ad2ant1 959	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
3		recn 7106	. . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
4		recn 7106	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
5		oveq2 5540	. . . . . . . . . . 11 ⊢ ((𝐴 + 𝐵) = (𝐴 + 𝐶) → (𝑥 + (𝐴 + 𝐵)) = (𝑥 + (𝐴 + 𝐶)))
6		simpr 108	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
7		simpll 495	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ)
8		simplrl 501	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ ℂ)
9	6, 7, 8	addassd 7141	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → ((𝑥 + 𝐴) + 𝐵) = (𝑥 + (𝐴 + 𝐵)))
10		simplrr 502	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → 𝐶 ∈ ℂ)
11	6, 7, 10	addassd 7141	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → ((𝑥 + 𝐴) + 𝐶) = (𝑥 + (𝐴 + 𝐶)))
12	9, 11	eqeq12d 2095	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → (((𝑥 + 𝐴) + 𝐵) = ((𝑥 + 𝐴) + 𝐶) ↔ (𝑥 + (𝐴 + 𝐵)) = (𝑥 + (𝐴 + 𝐶))))
13	5, 12	syl5ibr 154	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) → ((𝑥 + 𝐴) + 𝐵) = ((𝑥 + 𝐴) + 𝐶)))
14	13	adantr 270	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) ∧ (𝐴 + 𝑥) = 0) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) → ((𝑥 + 𝐴) + 𝐵) = ((𝑥 + 𝐴) + 𝐶)))
15		addcom 7245	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝐴 + 𝑥) = (𝑥 + 𝐴))
16	15	eqeq1d 2089	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑥) = 0 ↔ (𝑥 + 𝐴) = 0))
17	16	adantlr 460	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑥) = 0 ↔ (𝑥 + 𝐴) = 0))
18		oveq1 5539	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 + 𝐴) = 0 → ((𝑥 + 𝐴) + 𝐵) = (0 + 𝐵))
19		oveq1 5539	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 + 𝐴) = 0 → ((𝑥 + 𝐴) + 𝐶) = (0 + 𝐶))
20	18, 19	eqeq12d 2095	. . . . . . . . . . . . . 14 ⊢ ((𝑥 + 𝐴) = 0 → (((𝑥 + 𝐴) + 𝐵) = ((𝑥 + 𝐴) + 𝐶) ↔ (0 + 𝐵) = (0 + 𝐶)))
21	20	adantl 271	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) ∧ (𝑥 + 𝐴) = 0) → (((𝑥 + 𝐴) + 𝐵) = ((𝑥 + 𝐴) + 𝐶) ↔ (0 + 𝐵) = (0 + 𝐶)))
22		addid2 7247	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℂ → (0 + 𝐵) = 𝐵)
23		addid2 7247	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ ℂ → (0 + 𝐶) = 𝐶)
24	22, 23	eqeqan12d 2096	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((0 + 𝐵) = (0 + 𝐶) ↔ 𝐵 = 𝐶))
25	24	adantl 271	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) → ((0 + 𝐵) = (0 + 𝐶) ↔ 𝐵 = 𝐶))
26	25	ad2antrr 471	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) ∧ (𝑥 + 𝐴) = 0) → ((0 + 𝐵) = (0 + 𝐶) ↔ 𝐵 = 𝐶))
27	21, 26	bitrd 186	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) ∧ (𝑥 + 𝐴) = 0) → (((𝑥 + 𝐴) + 𝐵) = ((𝑥 + 𝐴) + 𝐶) ↔ 𝐵 = 𝐶))
28	27	ex 113	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → ((𝑥 + 𝐴) = 0 → (((𝑥 + 𝐴) + 𝐵) = ((𝑥 + 𝐴) + 𝐶) ↔ 𝐵 = 𝐶)))
29	17, 28	sylbid 148	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑥) = 0 → (((𝑥 + 𝐴) + 𝐵) = ((𝑥 + 𝐴) + 𝐶) ↔ 𝐵 = 𝐶)))
30	29	imp 122	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) ∧ (𝐴 + 𝑥) = 0) → (((𝑥 + 𝐴) + 𝐵) = ((𝑥 + 𝐴) + 𝐶) ↔ 𝐵 = 𝐶))
31	14, 30	sylibd 147	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) ∧ (𝐴 + 𝑥) = 0) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) → 𝐵 = 𝐶))
32	31	ex 113	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑥) = 0 → ((𝐴 + 𝐵) = (𝐴 + 𝐶) → 𝐵 = 𝐶)))
33	4, 32	sylan2 280	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℝ) → ((𝐴 + 𝑥) = 0 → ((𝐴 + 𝐵) = (𝐴 + 𝐶) → 𝐵 = 𝐶)))
34	33	rexlimdva 2477	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) → (∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0 → ((𝐴 + 𝐵) = (𝐴 + 𝐶) → 𝐵 = 𝐶)))
35	34	3impb 1134	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0 → ((𝐴 + 𝐵) = (𝐴 + 𝐶) → 𝐵 = 𝐶)))
36	3, 35	syl3an1 1202	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0 → ((𝐴 + 𝐵) = (𝐴 + 𝐶) → 𝐵 = 𝐶)))
37	2, 36	mpd 13	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) → 𝐵 = 𝐶))
38		oveq2 5540	. 2 ⊢ (𝐵 = 𝐶 → (𝐴 + 𝐵) = (𝐴 + 𝐶))
39	37, 38	impbid1 140	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∧ w3a 919   = wceq 1284   ∈ wcel 1433  ∃wrex 2349  (class class class)co 5532  ℂcc 6979  ℝcr 6980  0cc0 6981   + caddc 6984

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-resscn 7068  ax-1cn 7069  ax-icn 7071  ax-addcl 7072  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-i2m1 7081  ax-0id 7084  ax-rnegex 7085

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-iota 4887  df-fv 4930  df-ov 5535

This theorem is referenced by:  cnegexlem3  7285