Theorem readdcan 8418

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 8241	. . . 4 ⊢ (𝐶 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐶 + 𝑥) = 0)
2	1	3ad2ant3 1047	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ∃𝑥 ∈ ℝ (𝐶 + 𝑥) = 0)
3		oveq2 6060	. . . . . . 7 ⊢ ((𝐶 + 𝐴) = (𝐶 + 𝐵) → (𝑥 + (𝐶 + 𝐴)) = (𝑥 + (𝐶 + 𝐵)))
4	3	adantl 277	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → (𝑥 + (𝐶 + 𝐴)) = (𝑥 + (𝐶 + 𝐵)))
5		simprl 531	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝑥 ∈ ℝ)
6	5	recnd 8307	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝑥 ∈ ℂ)
7		simpl3 1029	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝐶 ∈ ℝ)
8	7	recnd 8307	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝐶 ∈ ℂ)
9		simpl1 1027	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝐴 ∈ ℝ)
10	9	recnd 8307	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝐴 ∈ ℂ)
11	6, 8, 10	addassd 8301	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → ((𝑥 + 𝐶) + 𝐴) = (𝑥 + (𝐶 + 𝐴)))
12		simpl2 1028	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝐵 ∈ ℝ)
13	12	recnd 8307	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝐵 ∈ ℂ)
14	6, 8, 13	addassd 8301	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → ((𝑥 + 𝐶) + 𝐵) = (𝑥 + (𝐶 + 𝐵)))
15	11, 14	eqeq12d 2249	. . . . . . 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 276	. . . . . 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 167	. . . . 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 276	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → 𝐶 ∈ ℂ)
19	6	adantr 276	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → 𝑥 ∈ ℂ)
20		addcom 8415	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝐶 + 𝑥) = (𝑥 + 𝐶))
21	18, 19, 20	syl2anc 411	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → (𝐶 + 𝑥) = (𝑥 + 𝐶))
22		simplrr 538	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → (𝐶 + 𝑥) = 0)
23	21, 22	eqtr3d 2269	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → (𝑥 + 𝐶) = 0)
24	23	oveq1d 6067	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → ((𝑥 + 𝐶) + 𝐴) = (0 + 𝐴))
25	10	adantr 276	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → 𝐴 ∈ ℂ)
26		addlid 8417	. . . . . . 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 2267	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → ((𝑥 + 𝐶) + 𝐴) = 𝐴)
29	23	oveq1d 6067	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → ((𝑥 + 𝐶) + 𝐵) = (0 + 𝐵))
30	13	adantr 276	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → 𝐵 ∈ ℂ)
31		addlid 8417	. . . . . . 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 2267	. . . . 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 2275	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → 𝐴 = 𝐵)
35	34	ex 115	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → ((𝐶 + 𝐴) = (𝐶 + 𝐵) → 𝐴 = 𝐵))
36	2, 35	rexlimddv 2667	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 + 𝐴) = (𝐶 + 𝐵) → 𝐴 = 𝐵))
37		oveq2 6060	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 + 𝐴) = (𝐶 + 𝐵))
38	36, 37	impbid1 142	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 + 𝐴) = (𝐶 + 𝐵) ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205  ∃wrex 2523  (class class class)co 6052  ℂcc 8130  ℝcr 8131  0cc0 8132   + caddc 8135

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 2216  ax-resscn 8224  ax-1cn 8225  ax-icn 8227  ax-addcl 8228  ax-mulcl 8230  ax-addcom 8232  ax-addass 8234  ax-i2m1 8237  ax-0id 8240  ax-rnegex 8241

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-iota 5314  df-fv 5362  df-ov 6055

This theorem is referenced by: (None)