Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  deceq1 GIF version

Theorem deceq1 8370
 Description: Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
Assertion
Ref Expression
deceq1 (𝐴 = 𝐵𝐴𝐶 = 𝐵𝐶)

Proof of Theorem deceq1
StepHypRef Expression
1 oveq2 5520 . . 3 (𝐴 = 𝐵 → (10 · 𝐴) = (10 · 𝐵))
21oveq1d 5527 . 2 (𝐴 = 𝐵 → ((10 · 𝐴) + 𝐶) = ((10 · 𝐵) + 𝐶))
3 df-dec 8369 . 2 𝐴𝐶 = ((10 · 𝐴) + 𝐶)
4 df-dec 8369 . 2 𝐵𝐶 = ((10 · 𝐵) + 𝐶)
52, 3, 43eqtr4g 2097 1 (𝐴 = 𝐵𝐴𝐶 = 𝐵𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243  (class class class)co 5512   + caddc 6892   · cmul 6894  10c10 7972  ;cdc 8368 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515  df-dec 8369 This theorem is referenced by:  deceq1i  8372
 Copyright terms: Public domain W3C validator