Theorem cnegexlem1 8122

Description: Addition cancellation of a real number from two complex numbers. Lemma for cnegex 8125. (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 7911	. . . 4 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
2	1	3ad2ant1 1018	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
3		recn 7935	. . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
4		recn 7935	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
5		oveq2 5877	. . . . . . . . . . 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 535	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ ℂ)
9	6, 7, 8	addassd 7970	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → ((𝑥 + 𝐴) + 𝐵) = (𝑥 + (𝐴 + 𝐵)))
10		simplrr 536	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → 𝐶 ∈ ℂ)
11	6, 7, 10	addassd 7970	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) ∧ 𝑥 ∈ ℂ) → ((𝑥 + 𝐴) + 𝐶) = (𝑥 + (𝐴 + 𝐶)))
12	9, 11	eqeq12d 2192	. . . . . . . . . . 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 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 8084	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝐴 + 𝑥) = (𝑥 + 𝐴))
16	15	eqeq1d 2186	. . . . . . . . . . . 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 5876	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 + 𝐴) = 0 → ((𝑥 + 𝐴) + 𝐵) = (0 + 𝐵))
19		oveq1 5876	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 + 𝐴) = 0 → ((𝑥 + 𝐴) + 𝐶) = (0 + 𝐶))
20	18, 19	eqeq12d 2192	. . . . . . . . . . . . . 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		addid2 8086	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℂ → (0 + 𝐵) = 𝐵)
23		addid2 8086	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ ℂ → (0 + 𝐶) = 𝐶)
24	22, 23	eqeqan12d 2193	. . . . . . . . . . . . . . 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 2594	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) → (∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0 → ((𝐴 + 𝐵) = (𝐴 + 𝐶) → 𝐵 = 𝐶)))
35	34	3impb 1199	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0 → ((𝐴 + 𝐵) = (𝐴 + 𝐶) → 𝐵 = 𝐶)))
36	3, 35	syl3an1 1271	. . 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 5877	. 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 978   = wceq 1353   ∈ wcel 2148  ∃wrex 2456  (class class class)co 5869  ℂcc 7800  ℝcr 7801  0cc0 7802   + caddc 7805

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-resscn 7894  ax-1cn 7895  ax-icn 7897  ax-addcl 7898  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-i2m1 7907  ax-0id 7910  ax-rnegex 7911

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-iota 5174  df-fv 5220  df-ov 5872

This theorem is referenced by:  cnegexlem3  8124