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Theorem deceq1 8614
Description: Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
Assertion
Ref Expression
deceq1  |-  ( A  =  B  -> ; A C  = ; B C )

Proof of Theorem deceq1
StepHypRef Expression
1 oveq2 5571 . . 3  |-  ( A  =  B  ->  (
( 9  +  1 )  x.  A )  =  ( ( 9  +  1 )  x.  B ) )
21oveq1d 5578 . 2  |-  ( A  =  B  ->  (
( ( 9  +  1 )  x.  A
)  +  C )  =  ( ( ( 9  +  1 )  x.  B )  +  C ) )
3 df-dec 8611 . 2  |- ; A C  =  ( ( ( 9  +  1 )  x.  A
)  +  C )
4 df-dec 8611 . 2  |- ; B C  =  ( ( ( 9  +  1 )  x.  B
)  +  C )
52, 3, 43eqtr4g 2140 1  |-  ( A  =  B  -> ; A C  = ; B C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285  (class class class)co 5563   1c1 7096    + caddc 7098    x. cmul 7100   9c9 8215  ;cdc 8610
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-v 2612  df-un 2986  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-iota 4917  df-fv 4960  df-ov 5566  df-dec 8611
This theorem is referenced by:  deceq1i  8616
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