Theorem addcan 8199

Description: Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Assertion

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

Proof of Theorem addcan

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnegex2 8198	. . 3 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝑥 + 𝐴) = 0)
2	1	3ad2ant1 1020	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ∃𝑥 ∈ ℂ (𝑥 + 𝐴) = 0)
3		oveq2 5926	. . . 4 ⊢ ((𝐴 + 𝐵) = (𝐴 + 𝐶) → (𝑥 + (𝐴 + 𝐵)) = (𝑥 + (𝐴 + 𝐶)))
4		simprr 531	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝐴) = 0)) → (𝑥 + 𝐴) = 0)
5	4	oveq1d 5933	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝐴) = 0)) → ((𝑥 + 𝐴) + 𝐵) = (0 + 𝐵))
6		simprl 529	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝐴) = 0)) → 𝑥 ∈ ℂ)
7		simpl1 1002	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝐴) = 0)) → 𝐴 ∈ ℂ)
8		simpl2 1003	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝐴) = 0)) → 𝐵 ∈ ℂ)
9	6, 7, 8	addassd 8042	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝐴) = 0)) → ((𝑥 + 𝐴) + 𝐵) = (𝑥 + (𝐴 + 𝐵)))
10		addlid 8158	. . . . . . 7 ⊢ (𝐵 ∈ ℂ → (0 + 𝐵) = 𝐵)
11	8, 10	syl 14	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝐴) = 0)) → (0 + 𝐵) = 𝐵)
12	5, 9, 11	3eqtr3d 2234	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝐴) = 0)) → (𝑥 + (𝐴 + 𝐵)) = 𝐵)
13	4	oveq1d 5933	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝐴) = 0)) → ((𝑥 + 𝐴) + 𝐶) = (0 + 𝐶))
14		simpl3 1004	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝐴) = 0)) → 𝐶 ∈ ℂ)
15	6, 7, 14	addassd 8042	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝐴) = 0)) → ((𝑥 + 𝐴) + 𝐶) = (𝑥 + (𝐴 + 𝐶)))
16		addlid 8158	. . . . . . 7 ⊢ (𝐶 ∈ ℂ → (0 + 𝐶) = 𝐶)
17	14, 16	syl 14	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝐴) = 0)) → (0 + 𝐶) = 𝐶)
18	13, 15, 17	3eqtr3d 2234	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝐴) = 0)) → (𝑥 + (𝐴 + 𝐶)) = 𝐶)
19	12, 18	eqeq12d 2208	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝐴) = 0)) → ((𝑥 + (𝐴 + 𝐵)) = (𝑥 + (𝐴 + 𝐶)) ↔ 𝐵 = 𝐶))
20	3, 19	imbitrid 154	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝐴) = 0)) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) → 𝐵 = 𝐶))
21		oveq2 5926	. . 3 ⊢ (𝐵 = 𝐶 → (𝐴 + 𝐵) = (𝐴 + 𝐶))
22	20, 21	impbid1 142	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝑥 ∈ ℂ ∧ (𝑥 + 𝐴) = 0)) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶))
23	2, 22	rexlimddv 2616	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  ∃wrex 2473  (class class class)co 5918  ℂcc 7870  0cc0 7872   + caddc 7875

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-resscn 7964  ax-1cn 7965  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5215  df-fv 5262  df-ov 5921

This theorem is referenced by:  addcani  8201  addcand  8203  subcan  8274