Theorem readdcan 8038

Description: Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013.)

Assertion

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

Proof of Theorem readdcan

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-rnegex 7862	. . . 4 ⊢ (𝐶 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐶 + 𝑥) = 0)
2	1	3ad2ant3 1010	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ∃𝑥 ∈ ℝ (𝐶 + 𝑥) = 0)
3		oveq2 5850	. . . . . . 7 ⊢ ((𝐶 + 𝐴) = (𝐶 + 𝐵) → (𝑥 + (𝐶 + 𝐴)) = (𝑥 + (𝐶 + 𝐵)))
4	3	adantl 275	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → (𝑥 + (𝐶 + 𝐴)) = (𝑥 + (𝐶 + 𝐵)))
5		simprl 521	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝑥 ∈ ℝ)
6	5	recnd 7927	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝑥 ∈ ℂ)
7		simpl3 992	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝐶 ∈ ℝ)
8	7	recnd 7927	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝐶 ∈ ℂ)
9		simpl1 990	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝐴 ∈ ℝ)
10	9	recnd 7927	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝐴 ∈ ℂ)
11	6, 8, 10	addassd 7921	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → ((𝑥 + 𝐶) + 𝐴) = (𝑥 + (𝐶 + 𝐴)))
12		simpl2 991	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝐵 ∈ ℝ)
13	12	recnd 7927	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝐵 ∈ ℂ)
14	6, 8, 13	addassd 7921	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → ((𝑥 + 𝐶) + 𝐵) = (𝑥 + (𝐶 + 𝐵)))
15	11, 14	eqeq12d 2180	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → (((𝑥 + 𝐶) + 𝐴) = ((𝑥 + 𝐶) + 𝐵) ↔ (𝑥 + (𝐶 + 𝐴)) = (𝑥 + (𝐶 + 𝐵))))
16	15	adantr 274	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → (((𝑥 + 𝐶) + 𝐴) = ((𝑥 + 𝐶) + 𝐵) ↔ (𝑥 + (𝐶 + 𝐴)) = (𝑥 + (𝐶 + 𝐵))))
17	4, 16	mpbird 166	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → ((𝑥 + 𝐶) + 𝐴) = ((𝑥 + 𝐶) + 𝐵))
18	8	adantr 274	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → 𝐶 ∈ ℂ)
19	6	adantr 274	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → 𝑥 ∈ ℂ)
20		addcom 8035	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝐶 + 𝑥) = (𝑥 + 𝐶))
21	18, 19, 20	syl2anc 409	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → (𝐶 + 𝑥) = (𝑥 + 𝐶))
22		simplrr 526	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → (𝐶 + 𝑥) = 0)
23	21, 22	eqtr3d 2200	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → (𝑥 + 𝐶) = 0)
24	23	oveq1d 5857	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → ((𝑥 + 𝐶) + 𝐴) = (0 + 𝐴))
25	10	adantr 274	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → 𝐴 ∈ ℂ)
26		addid2 8037	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)
27	25, 26	syl 14	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → (0 + 𝐴) = 𝐴)
28	24, 27	eqtrd 2198	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → ((𝑥 + 𝐶) + 𝐴) = 𝐴)
29	23	oveq1d 5857	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → ((𝑥 + 𝐶) + 𝐵) = (0 + 𝐵))
30	13	adantr 274	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → 𝐵 ∈ ℂ)
31		addid2 8037	. . . . . . 7 ⊢ (𝐵 ∈ ℂ → (0 + 𝐵) = 𝐵)
32	30, 31	syl 14	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → (0 + 𝐵) = 𝐵)
33	29, 32	eqtrd 2198	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → ((𝑥 + 𝐶) + 𝐵) = 𝐵)
34	17, 28, 33	3eqtr3d 2206	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → 𝐴 = 𝐵)
35	34	ex 114	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → ((𝐶 + 𝐴) = (𝐶 + 𝐵) → 𝐴 = 𝐵))
36	2, 35	rexlimddv 2588	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 + 𝐴) = (𝐶 + 𝐵) → 𝐴 = 𝐵))
37		oveq2 5850	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 + 𝐴) = (𝐶 + 𝐵))
38	36, 37	impbid1 141	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 + 𝐴) = (𝐶 + 𝐵) ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 968   = wceq 1343   ∈ wcel 2136  ∃wrex 2445  (class class class)co 5842  ℂcc 7751  ℝcr 7752  0cc0 7753   + caddc 7756

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-resscn 7845  ax-1cn 7846  ax-icn 7848  ax-addcl 7849  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-i2m1 7858  ax-0id 7861  ax-rnegex 7862

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-iota 5153  df-fv 5196  df-ov 5845

This theorem is referenced by: (None)