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Theorem readdcan 7601
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 7433 . . . 4 (𝐶 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐶 + 𝑥) = 0)
213ad2ant3 966 . . 3 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ∃𝑥 ∈ ℝ (𝐶 + 𝑥) = 0)
3 oveq2 5642 . . . . . . 7 ((𝐶 + 𝐴) = (𝐶 + 𝐵) → (𝑥 + (𝐶 + 𝐴)) = (𝑥 + (𝐶 + 𝐵)))
43adantl 271 . . . . . 6 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → (𝑥 + (𝐶 + 𝐴)) = (𝑥 + (𝐶 + 𝐵)))
5 simprl 498 . . . . . . . . . 10 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝑥 ∈ ℝ)
65recnd 7495 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝑥 ∈ ℂ)
7 simpl3 948 . . . . . . . . . 10 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝐶 ∈ ℝ)
87recnd 7495 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝐶 ∈ ℂ)
9 simpl1 946 . . . . . . . . . 10 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝐴 ∈ ℝ)
109recnd 7495 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝐴 ∈ ℂ)
116, 8, 10addassd 7489 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → ((𝑥 + 𝐶) + 𝐴) = (𝑥 + (𝐶 + 𝐴)))
12 simpl2 947 . . . . . . . . . 10 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝐵 ∈ ℝ)
1312recnd 7495 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → 𝐵 ∈ ℂ)
146, 8, 13addassd 7489 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → ((𝑥 + 𝐶) + 𝐵) = (𝑥 + (𝐶 + 𝐵)))
1511, 14eqeq12d 2102 . . . . . . 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 270 . . . . . 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 165 . . . . 5 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → ((𝑥 + 𝐶) + 𝐴) = ((𝑥 + 𝐶) + 𝐵))
188adantr 270 . . . . . . . . 9 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → 𝐶 ∈ ℂ)
196adantr 270 . . . . . . . . 9 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → 𝑥 ∈ ℂ)
20 addcom 7598 . . . . . . . . 9 ((𝐶 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝐶 + 𝑥) = (𝑥 + 𝐶))
2118, 19, 20syl2anc 403 . . . . . . . 8 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → (𝐶 + 𝑥) = (𝑥 + 𝐶))
22 simplrr 503 . . . . . . . 8 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → (𝐶 + 𝑥) = 0)
2321, 22eqtr3d 2122 . . . . . . 7 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → (𝑥 + 𝐶) = 0)
2423oveq1d 5649 . . . . . 6 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → ((𝑥 + 𝐶) + 𝐴) = (0 + 𝐴))
2510adantr 270 . . . . . . 7 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → 𝐴 ∈ ℂ)
26 addid2 7600 . . . . . . 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 2120 . . . . 5 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → ((𝑥 + 𝐶) + 𝐴) = 𝐴)
2923oveq1d 5649 . . . . . 6 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → ((𝑥 + 𝐶) + 𝐵) = (0 + 𝐵))
3013adantr 270 . . . . . . 7 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → 𝐵 ∈ ℂ)
31 addid2 7600 . . . . . . 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 2120 . . . . 5 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → ((𝑥 + 𝐶) + 𝐵) = 𝐵)
3417, 28, 333eqtr3d 2128 . . . 4 ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) ∧ (𝐶 + 𝐴) = (𝐶 + 𝐵)) → 𝐴 = 𝐵)
3534ex 113 . . 3 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ (𝐶 + 𝑥) = 0)) → ((𝐶 + 𝐴) = (𝐶 + 𝐵) → 𝐴 = 𝐵))
362, 35rexlimddv 2493 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 + 𝐴) = (𝐶 + 𝐵) → 𝐴 = 𝐵))
37 oveq2 5642 . 2 (𝐴 = 𝐵 → (𝐶 + 𝐴) = (𝐶 + 𝐵))
3836, 37impbid1 140 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 + 𝐴) = (𝐶 + 𝐵) ↔ 𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 924   = wceq 1289  wcel 1438  wrex 2360  (class class class)co 5634  cc 7327  cr 7328  0cc0 7329   + caddc 7332
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-resscn 7416  ax-1cn 7417  ax-icn 7419  ax-addcl 7420  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-i2m1 7429  ax-0id 7432  ax-rnegex 7433
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-iota 4967  df-fv 5010  df-ov 5637
This theorem is referenced by: (None)
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