MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  deceq2 Structured version   Visualization version   GIF version

Theorem deceq2 12683
Description: Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
Assertion
Ref Expression
deceq2 (๐ด = ๐ต โ†’ ๐ถ๐ด = ๐ถ๐ต)

Proof of Theorem deceq2
StepHypRef Expression
1 oveq2 7417 . 2 (๐ด = ๐ต โ†’ (((9 + 1) ยท ๐ถ) + ๐ด) = (((9 + 1) ยท ๐ถ) + ๐ต))
2 df-dec 12678 . 2 ๐ถ๐ด = (((9 + 1) ยท ๐ถ) + ๐ด)
3 df-dec 12678 . 2 ๐ถ๐ต = (((9 + 1) ยท ๐ถ) + ๐ต)
41, 2, 33eqtr4g 2798 1 (๐ด = ๐ต โ†’ ๐ถ๐ด = ๐ถ๐ต)
Colors of variables: wff setvar class
Syntax hints:   โ†’ wi 4   = wceq 1542  (class class class)co 7409  1c1 11111   + caddc 11113   ยท cmul 11115  9c9 12274  cdc 12677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-dec 12678
This theorem is referenced by:  deceq2i  12685
  Copyright terms: Public domain W3C validator