Theorem cnegexlem1 8448

Description: Addition cancellation of a real number from two complex numbers. Lemma for cnegex 8451. (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 8236	. . . 4 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
2	1	3ad2ant1 1045	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
3		recn 8260	. . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
4		recn 8260	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
5		oveq2 6058	. . . . . . . . . . 11 ⊢ ((𝐴 + 𝐵) = (𝐴 + 𝐶) → (𝑥 + (𝐴 + 𝐵)) = (𝑥 + (𝐴 + 𝐶)))
6		simpr 110	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
7		simpll 527	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ)
8		simplrl 537	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ ℂ)
9	6, 7, 8	addassd 8296	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → ((𝑥 + 𝐴) + 𝐵) = (𝑥 + (𝐴 + 𝐵)))
10		simplrr 538	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → 𝐶 ∈ ℂ)
11	6, 7, 10	addassd 8296	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → ((𝑥 + 𝐴) + 𝐶) = (𝑥 + (𝐴 + 𝐶)))
12	9, 11	eqeq12d 2247	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → (((𝑥 + 𝐴) + 𝐵) = ((𝑥 + 𝐴) + 𝐶) ↔ (𝑥 + (𝐴 + 𝐵)) = (𝑥 + (𝐴 + 𝐶))))
13	5, 12	imbitrrid 156	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) → ((𝑥 + 𝐴) + 𝐵) = ((𝑥 + 𝐴) + 𝐶)))
14	13	adantr 276	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) ∧ (𝐴 + 𝑥) = 0) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) → ((𝑥 + 𝐴) + 𝐵) = ((𝑥 + 𝐴) + 𝐶)))
15		addcom 8410	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝐴 + 𝑥) = (𝑥 + 𝐴))
16	15	eqeq1d 2241	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑥) = 0 ↔ (𝑥 + 𝐴) = 0))
17	16	adantlr 477	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑥) = 0 ↔ (𝑥 + 𝐴) = 0))
18		oveq1 6057	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 + 𝐴) = 0 → ((𝑥 + 𝐴) + 𝐵) = (0 + 𝐵))
19		oveq1 6057	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 + 𝐴) = 0 → ((𝑥 + 𝐴) + 𝐶) = (0 + 𝐶))
20	18, 19	eqeq12d 2247	. . . . . . . . . . . . . 14 ⊢ ((𝑥 + 𝐴) = 0 → (((𝑥 + 𝐴) + 𝐵) = ((𝑥 + 𝐴) + 𝐶) ↔ (0 + 𝐵) = (0 + 𝐶)))
21	20	adantl 277	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) ∧ (𝑥 + 𝐴) = 0) → (((𝑥 + 𝐴) + 𝐵) = ((𝑥 + 𝐴) + 𝐶) ↔ (0 + 𝐵) = (0 + 𝐶)))
22		addlid 8412	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℂ → (0 + 𝐵) = 𝐵)
23		addlid 8412	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ ℂ → (0 + 𝐶) = 𝐶)
24	22, 23	eqeqan12d 2248	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((0 + 𝐵) = (0 + 𝐶) ↔ 𝐵 = 𝐶))
25	24	adantl 277	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) → ((0 + 𝐵) = (0 + 𝐶) ↔ 𝐵 = 𝐶))
26	25	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) ∧ (𝑥 + 𝐴) = 0) → ((0 + 𝐵) = (0 + 𝐶) ↔ 𝐵 = 𝐶))
27	21, 26	bitrd 188	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) ∧ (𝑥 + 𝐴) = 0) → (((𝑥 + 𝐴) + 𝐵) = ((𝑥 + 𝐴) + 𝐶) ↔ 𝐵 = 𝐶))
28	27	ex 115	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → ((𝑥 + 𝐴) = 0 → (((𝑥 + 𝐴) + 𝐵) = ((𝑥 + 𝐴) + 𝐶) ↔ 𝐵 = 𝐶)))
29	17, 28	sylbid 150	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑥) = 0 → (((𝑥 + 𝐴) + 𝐵) = ((𝑥 + 𝐴) + 𝐶) ↔ 𝐵 = 𝐶)))
30	29	imp 124	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) ∧ (𝐴 + 𝑥) = 0) → (((𝑥 + 𝐴) + 𝐵) = ((𝑥 + 𝐴) + 𝐶) ↔ 𝐵 = 𝐶))
31	14, 30	sylibd 149	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) ∧ (𝐴 + 𝑥) = 0) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) → 𝐵 = 𝐶))
32	31	ex 115	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑥) = 0 → ((𝐴 + 𝐵) = (𝐴 + 𝐶) → 𝐵 = 𝐶)))
33	4, 32	sylan2 286	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℝ) → ((𝐴 + 𝑥) = 0 → ((𝐴 + 𝐵) = (𝐴 + 𝐶) → 𝐵 = 𝐶)))
34	33	rexlimdva 2660	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) → (∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0 → ((𝐴 + 𝐵) = (𝐴 + 𝐶) → 𝐵 = 𝐶)))
35	34	3impb 1226	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0 → ((𝐴 + 𝐵) = (𝐴 + 𝐶) → 𝐵 = 𝐶)))
36	3, 35	syl3an1 1307	. . 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 6058	. 2 ⊢ (𝐵 = 𝐶 → (𝐴 + 𝐵) = (𝐴 + 𝐶))
39	37, 38	impbid1 142	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203  ∃wrex 2521  (class class class)co 6050  ℂcc 8125  ℝcr 8126  0cc0 8127   + caddc 8130

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-resscn 8219  ax-1cn 8220  ax-icn 8222  ax-addcl 8223  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0id 8235  ax-rnegex 8236

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-iota 5312  df-fv 5360  df-ov 6053

This theorem is referenced by:  cnegexlem3  8450