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

Theorem deceq2 10388
Description: Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
Assertion
Ref Expression
deceq2  |-  ( A  =  B  -> ; C A  = ; C B )

Proof of Theorem deceq2
StepHypRef Expression
1 oveq2 6091 . 2  |-  ( A  =  B  ->  (
( 10  x.  C
)  +  A )  =  ( ( 10  x.  C )  +  B ) )
2 df-dec 10385 . 2  |- ; C A  =  ( ( 10  x.  C
)  +  A )
3 df-dec 10385 . 2  |- ; C B  =  ( ( 10  x.  C
)  +  B )
41, 2, 33eqtr4g 2495 1  |-  ( A  =  B  -> ; C A  = ; C B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653  (class class class)co 6083    + caddc 8995    x. cmul 8997   10c10 10059  ;cdc 10384
This theorem is referenced by:  deceq2i  10390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086  df-dec 10385
  Copyright terms: Public domain W3C validator