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Theorem readdcan 6910
Description: Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013.)
Assertion
Ref Expression
readdcan ((A B 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 6752 . . . 4 (𝐶 ℝ → x ℝ (𝐶 + x) = 0)
213ad2ant3 926 . . 3 ((A B 𝐶 ℝ) → x ℝ (𝐶 + x) = 0)
3 oveq2 5463 . . . . . . 7 ((𝐶 + A) = (𝐶 + B) → (x + (𝐶 + A)) = (x + (𝐶 + B)))
43adantl 262 . . . . . 6 ((((A B 𝐶 ℝ) (x (𝐶 + x) = 0)) (𝐶 + A) = (𝐶 + B)) → (x + (𝐶 + A)) = (x + (𝐶 + B)))
5 simprl 483 . . . . . . . . . 10 (((A B 𝐶 ℝ) (x (𝐶 + x) = 0)) → x ℝ)
65recnd 6811 . . . . . . . . 9 (((A B 𝐶 ℝ) (x (𝐶 + x) = 0)) → x ℂ)
7 simpl3 908 . . . . . . . . . 10 (((A B 𝐶 ℝ) (x (𝐶 + x) = 0)) → 𝐶 ℝ)
87recnd 6811 . . . . . . . . 9 (((A B 𝐶 ℝ) (x (𝐶 + x) = 0)) → 𝐶 ℂ)
9 simpl1 906 . . . . . . . . . 10 (((A B 𝐶 ℝ) (x (𝐶 + x) = 0)) → A ℝ)
109recnd 6811 . . . . . . . . 9 (((A B 𝐶 ℝ) (x (𝐶 + x) = 0)) → A ℂ)
116, 8, 10addassd 6807 . . . . . . . 8 (((A B 𝐶 ℝ) (x (𝐶 + x) = 0)) → ((x + 𝐶) + A) = (x + (𝐶 + A)))
12 simpl2 907 . . . . . . . . . 10 (((A B 𝐶 ℝ) (x (𝐶 + x) = 0)) → B ℝ)
1312recnd 6811 . . . . . . . . 9 (((A B 𝐶 ℝ) (x (𝐶 + x) = 0)) → B ℂ)
146, 8, 13addassd 6807 . . . . . . . 8 (((A B 𝐶 ℝ) (x (𝐶 + x) = 0)) → ((x + 𝐶) + B) = (x + (𝐶 + B)))
1511, 14eqeq12d 2051 . . . . . . 7 (((A B 𝐶 ℝ) (x (𝐶 + x) = 0)) → (((x + 𝐶) + A) = ((x + 𝐶) + B) ↔ (x + (𝐶 + A)) = (x + (𝐶 + B))))
1615adantr 261 . . . . . 6 ((((A B 𝐶 ℝ) (x (𝐶 + x) = 0)) (𝐶 + A) = (𝐶 + B)) → (((x + 𝐶) + A) = ((x + 𝐶) + B) ↔ (x + (𝐶 + A)) = (x + (𝐶 + B))))
174, 16mpbird 156 . . . . 5 ((((A B 𝐶 ℝ) (x (𝐶 + x) = 0)) (𝐶 + A) = (𝐶 + B)) → ((x + 𝐶) + A) = ((x + 𝐶) + B))
188adantr 261 . . . . . . . . 9 ((((A B 𝐶 ℝ) (x (𝐶 + x) = 0)) (𝐶 + A) = (𝐶 + B)) → 𝐶 ℂ)
196adantr 261 . . . . . . . . 9 ((((A B 𝐶 ℝ) (x (𝐶 + x) = 0)) (𝐶 + A) = (𝐶 + B)) → x ℂ)
20 addcom 6907 . . . . . . . . 9 ((𝐶 x ℂ) → (𝐶 + x) = (x + 𝐶))
2118, 19, 20syl2anc 391 . . . . . . . 8 ((((A B 𝐶 ℝ) (x (𝐶 + x) = 0)) (𝐶 + A) = (𝐶 + B)) → (𝐶 + x) = (x + 𝐶))
22 simplrr 488 . . . . . . . 8 ((((A B 𝐶 ℝ) (x (𝐶 + x) = 0)) (𝐶 + A) = (𝐶 + B)) → (𝐶 + x) = 0)
2321, 22eqtr3d 2071 . . . . . . 7 ((((A B 𝐶 ℝ) (x (𝐶 + x) = 0)) (𝐶 + A) = (𝐶 + B)) → (x + 𝐶) = 0)
2423oveq1d 5470 . . . . . 6 ((((A B 𝐶 ℝ) (x (𝐶 + x) = 0)) (𝐶 + A) = (𝐶 + B)) → ((x + 𝐶) + A) = (0 + A))
2510adantr 261 . . . . . . 7 ((((A B 𝐶 ℝ) (x (𝐶 + x) = 0)) (𝐶 + A) = (𝐶 + B)) → A ℂ)
26 addid2 6909 . . . . . . 7 (A ℂ → (0 + A) = A)
2725, 26syl 14 . . . . . 6 ((((A B 𝐶 ℝ) (x (𝐶 + x) = 0)) (𝐶 + A) = (𝐶 + B)) → (0 + A) = A)
2824, 27eqtrd 2069 . . . . 5 ((((A B 𝐶 ℝ) (x (𝐶 + x) = 0)) (𝐶 + A) = (𝐶 + B)) → ((x + 𝐶) + A) = A)
2923oveq1d 5470 . . . . . 6 ((((A B 𝐶 ℝ) (x (𝐶 + x) = 0)) (𝐶 + A) = (𝐶 + B)) → ((x + 𝐶) + B) = (0 + B))
3013adantr 261 . . . . . . 7 ((((A B 𝐶 ℝ) (x (𝐶 + x) = 0)) (𝐶 + A) = (𝐶 + B)) → B ℂ)
31 addid2 6909 . . . . . . 7 (B ℂ → (0 + B) = B)
3230, 31syl 14 . . . . . 6 ((((A B 𝐶 ℝ) (x (𝐶 + x) = 0)) (𝐶 + A) = (𝐶 + B)) → (0 + B) = B)
3329, 32eqtrd 2069 . . . . 5 ((((A B 𝐶 ℝ) (x (𝐶 + x) = 0)) (𝐶 + A) = (𝐶 + B)) → ((x + 𝐶) + B) = B)
3417, 28, 333eqtr3d 2077 . . . 4 ((((A B 𝐶 ℝ) (x (𝐶 + x) = 0)) (𝐶 + A) = (𝐶 + B)) → A = B)
3534ex 108 . . 3 (((A B 𝐶 ℝ) (x (𝐶 + x) = 0)) → ((𝐶 + A) = (𝐶 + B) → A = B))
362, 35rexlimddv 2431 . 2 ((A B 𝐶 ℝ) → ((𝐶 + A) = (𝐶 + B) → A = B))
37 oveq2 5463 . 2 (A = B → (𝐶 + A) = (𝐶 + B))
3836, 37impbid1 130 1 ((A B 𝐶 ℝ) → ((𝐶 + A) = (𝐶 + B) ↔ A = B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242   wcel 1390  wrex 2301  (class class class)co 5455  cc 6669  cr 6670  0cc0 6671   + caddc 6674
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-resscn 6735  ax-1cn 6736  ax-icn 6738  ax-addcl 6739  ax-mulcl 6741  ax-addcom 6743  ax-addass 6745  ax-i2m1 6748  ax-0id 6751  ax-rnegex 6752
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853  df-ov 5458
This theorem is referenced by: (None)
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