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Theorem readdcan 7895
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.
StepHypRef Expression
1 ax-rnegex 7722 . . . 4 (𝐶 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐶 + 𝑥) = 0)
213ad2ant3 1004 . . 3 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ∃𝑥 ∈ ℝ (𝐶 + 𝑥) = 0)
3 oveq2 5775 . . . . . . 7 ((𝐶 + 𝐴) = (𝐶 + 𝐵) → (𝑥 + (𝐶 + 𝐴)) = (𝑥 + (𝐶 + 𝐵)))
43adantl 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 520 . . . . . . . . . 10 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝑥 ∈ ℝ)
65recnd 7787 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝑥 ∈ ℂ)
7 simpl3 986 . . . . . . . . . 10 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝐶 ∈ ℝ)
87recnd 7787 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝐶 ∈ ℂ)
9 simpl1 984 . . . . . . . . . 10 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝐴 ∈ ℝ)
109recnd 7787 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝐴 ∈ ℂ)
116, 8, 10addassd 7781 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → ((𝑥 + 𝐶) + 𝐴) = (𝑥 + (𝐶 + 𝐴)))
12 simpl2 985 . . . . . . . . . 10 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝐵 ∈ ℝ)
1312recnd 7787 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝐵 ∈ ℂ)
146, 8, 13addassd 7781 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → ((𝑥 + 𝐶) + 𝐵) = (𝑥 + (𝐶 + 𝐵)))
1511, 14eqeq12d 2152 . . . . . . 7 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → (((𝑥 + 𝐶) + 𝐴) = ((𝑥 + 𝐶) + 𝐵) ↔ (𝑥 + (𝐶 + 𝐴)) = (𝑥 + (𝐶 + 𝐵))))
1615adantr 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))))
174, 16mpbird 166 . . . . 5 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → ((𝑥 + 𝐶) + 𝐴) = ((𝑥 + 𝐶) + 𝐵))
188adantr 274 . . . . . . . . 9 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → 𝐶 ∈ ℂ)
196adantr 274 . . . . . . . . 9 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → 𝑥 ∈ ℂ)
20 addcom 7892 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝐶 + 𝑥) = (𝑥 + 𝐶))
2118, 19, 20syl2anc 408 . . . . . . . 8 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → (𝐶 + 𝑥) = (𝑥 + 𝐶))
22 simplrr 525 . . . . . . . 8 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → (𝐶 + 𝑥) = 0)
2321, 22eqtr3d 2172 . . . . . . 7 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → (𝑥 + 𝐶) = 0)
2423oveq1d 5782 . . . . . 6 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → ((𝑥 + 𝐶) + 𝐴) = (0 + 𝐴))
2510adantr 274 . . . . . . 7 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → 𝐴 ∈ ℂ)
26 addid2 7894 . . . . . . 7 (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)
2725, 26syl 14 . . . . . 6 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → (0 + 𝐴) = 𝐴)
2824, 27eqtrd 2170 . . . . 5 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → ((𝑥 + 𝐶) + 𝐴) = 𝐴)
2923oveq1d 5782 . . . . . 6 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → ((𝑥 + 𝐶) + 𝐵) = (0 + 𝐵))
3013adantr 274 . . . . . . 7 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → 𝐵 ∈ ℂ)
31 addid2 7894 . . . . . . 7 (𝐵 ∈ ℂ → (0 + 𝐵) = 𝐵)
3230, 31syl 14 . . . . . 6 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → (0 + 𝐵) = 𝐵)
3329, 32eqtrd 2170 . . . . 5 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → ((𝑥 + 𝐶) + 𝐵) = 𝐵)
3417, 28, 333eqtr3d 2178 . . . 4 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → 𝐴 = 𝐵)
3534ex 114 . . 3 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → ((𝐶 + 𝐴) = (𝐶 + 𝐵) → 𝐴 = 𝐵))
362, 35rexlimddv 2552 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 + 𝐴) = (𝐶 + 𝐵) → 𝐴 = 𝐵))
37 oveq2 5775 . 2 (𝐴 = 𝐵 → (𝐶 + 𝐴) = (𝐶 + 𝐵))
3836, 37impbid1 141 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 + 𝐴) = (𝐶 + 𝐵) ↔ 𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 962   = wceq 1331  wcel 1480  wrex 2415  (class class class)co 5767  cc 7611  cr 7612  0cc0 7613   + caddc 7616
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-resscn 7705  ax-1cn 7706  ax-icn 7708  ax-addcl 7709  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-i2m1 7718  ax-0id 7721  ax-rnegex 7722
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-iota 5083  df-fv 5126  df-ov 5770
This theorem is referenced by: (None)
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