Theorem cnegexlem1 8098

Description: Addition cancellation of a real number from two complex numbers. Lemma for cnegex 8101. (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 7887	. . . 4 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
2	1	3ad2ant1 1014	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
3		recn 7911	. . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
4		recn 7911	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
5		oveq2 5865	. . . . . . . . . . 11 ⊢ ((𝐴 + 𝐵) = (𝐴 + 𝐶) → (𝑥 + (𝐴 + 𝐵)) = (𝑥 + (𝐴 + 𝐶)))
6		simpr 109	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
7		simpll 525	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ)
8		simplrl 531	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ ℂ)
9	6, 7, 8	addassd 7946	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → ((𝑥 + 𝐴) + 𝐵) = (𝑥 + (𝐴 + 𝐵)))
10		simplrr 532	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → 𝐶 ∈ ℂ)
11	6, 7, 10	addassd 7946	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → ((𝑥 + 𝐴) + 𝐶) = (𝑥 + (𝐴 + 𝐶)))
12	9, 11	eqeq12d 2186	. . . . . . . . . . 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 155	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) → ((𝑥 + 𝐴) + 𝐵) = ((𝑥 + 𝐴) + 𝐶)))
14	13	adantr 274	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) ∧ (𝐴 + 𝑥) = 0) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) → ((𝑥 + 𝐴) + 𝐵) = ((𝑥 + 𝐴) + 𝐶)))
15		addcom 8060	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝐴 + 𝑥) = (𝑥 + 𝐴))
16	15	eqeq1d 2180	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑥) = 0 ↔ (𝑥 + 𝐴) = 0))
17	16	adantlr 475	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑥) = 0 ↔ (𝑥 + 𝐴) = 0))
18		oveq1 5864	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 + 𝐴) = 0 → ((𝑥 + 𝐴) + 𝐵) = (0 + 𝐵))
19		oveq1 5864	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 + 𝐴) = 0 → ((𝑥 + 𝐴) + 𝐶) = (0 + 𝐶))
20	18, 19	eqeq12d 2186	. . . . . . . . . . . . . 14 ⊢ ((𝑥 + 𝐴) = 0 → (((𝑥 + 𝐴) + 𝐵) = ((𝑥 + 𝐴) + 𝐶) ↔ (0 + 𝐵) = (0 + 𝐶)))
21	20	adantl 275	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) ∧ (𝑥 + 𝐴) = 0) → (((𝑥 + 𝐴) + 𝐵) = ((𝑥 + 𝐴) + 𝐶) ↔ (0 + 𝐵) = (0 + 𝐶)))
22		addid2 8062	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℂ → (0 + 𝐵) = 𝐵)
23		addid2 8062	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ ℂ → (0 + 𝐶) = 𝐶)
24	22, 23	eqeqan12d 2187	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((0 + 𝐵) = (0 + 𝐶) ↔ 𝐵 = 𝐶))
25	24	adantl 275	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) → ((0 + 𝐵) = (0 + 𝐶) ↔ 𝐵 = 𝐶))
26	25	ad2antrr 486	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) ∧ (𝑥 + 𝐴) = 0) → ((0 + 𝐵) = (0 + 𝐶) ↔ 𝐵 = 𝐶))
27	21, 26	bitrd 187	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) ∧ (𝑥 + 𝐴) = 0) → (((𝑥 + 𝐴) + 𝐵) = ((𝑥 + 𝐴) + 𝐶) ↔ 𝐵 = 𝐶))
28	27	ex 114	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → ((𝑥 + 𝐴) = 0 → (((𝑥 + 𝐴) + 𝐵) = ((𝑥 + 𝐴) + 𝐶) ↔ 𝐵 = 𝐶)))
29	17, 28	sylbid 149	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑥) = 0 → (((𝑥 + 𝐴) + 𝐵) = ((𝑥 + 𝐴) + 𝐶) ↔ 𝐵 = 𝐶)))
30	29	imp 123	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) ∧ (𝐴 + 𝑥) = 0) → (((𝑥 + 𝐴) + 𝐵) = ((𝑥 + 𝐴) + 𝐶) ↔ 𝐵 = 𝐶))
31	14, 30	sylibd 148	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) ∧ (𝐴 + 𝑥) = 0) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) → 𝐵 = 𝐶))
32	31	ex 114	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑥) = 0 → ((𝐴 + 𝐵) = (𝐴 + 𝐶) → 𝐵 = 𝐶)))
33	4, 32	sylan2 284	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℝ) → ((𝐴 + 𝑥) = 0 → ((𝐴 + 𝐵) = (𝐴 + 𝐶) → 𝐵 = 𝐶)))
34	33	rexlimdva 2588	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) → (∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0 → ((𝐴 + 𝐵) = (𝐴 + 𝐶) → 𝐵 = 𝐶)))
35	34	3impb 1195	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0 → ((𝐴 + 𝐵) = (𝐴 + 𝐶) → 𝐵 = 𝐶)))
36	3, 35	syl3an1 1267	. . 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 5865	. 2 ⊢ (𝐵 = 𝐶 → (𝐴 + 𝐵) = (𝐴 + 𝐶))
39	37, 38	impbid1 141	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 974   = wceq 1349   ∈ wcel 2142  ∃wrex 2450  (class class class)co 5857  ℂcc 7776  ℝcr 7777  0cc0 7778   + caddc 7781

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 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-ext 2153  ax-resscn 7870  ax-1cn 7871  ax-icn 7873  ax-addcl 7874  ax-mulcl 7876  ax-addcom 7878  ax-addass 7880  ax-i2m1 7883  ax-0id 7886  ax-rnegex 7887

This theorem depends on definitions:  df-bi 116  df-3an 976  df-tru 1352  df-nf 1455  df-sb 1757  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ral 2454  df-rex 2455  df-v 2733  df-un 3126  df-in 3128  df-ss 3135  df-sn 3590  df-pr 3591  df-op 3593  df-uni 3798  df-br 3991  df-iota 5162  df-fv 5208  df-ov 5860

This theorem is referenced by:  cnegexlem3  8100