Theorem deceq1 8370

Description: Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)

Assertion

Ref	Expression
deceq1	|- ( A = B -> ; A C = ; B C )

Proof of Theorem deceq1

Step	Hyp	Ref	Expression
1		oveq2 5520	. . 3 |- ( A = B -> ( 10 x. A ) = ( 10 x. B ) )
2	1	oveq1d 5527	. 2 |- ( A = B -> ( ( 10 x. A ) + C ) = ( ( 10 x. B ) + C ) )
3		df-dec 8369	. 2 |- ; A C = ( ( 10 x. A ) + C )
4		df-dec 8369	. 2 |- ; B C = ( ( 10 x. B ) + C )
5	2, 3, 4	3eqtr4g 2097	1 |- ( A = B -> ; A C = ; B C )

Syntax hints:   -> wi 4   = wceq 1243  (class class class)co 5512   + caddc 6892   x. 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