Theorem deceq1 9326

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

Step	Hyp	Ref	Expression
1		oveq2 5850	. . 3 |- ( A = B -> ( ( 9 + 1 ) x. A ) = ( ( 9 + 1 ) x. B ) )
2	1	oveq1d 5857	. 2 |- ( A = B -> ( ( ( 9 + 1 ) x. A ) + C ) = ( ( ( 9 + 1 ) x. B ) + C ) )
3		df-dec 9323	. 2 |- ; A C = ( ( ( 9 + 1 ) x. A ) + C )
4		df-dec 9323	. 2 |- ; B C = ( ( ( 9 + 1 ) x. B ) + C )
5	2, 3, 4	3eqtr4g 2224	1 |- ( A = B -> ; A C = ; B C )

Syntax hints:   -> wi 4   = wceq 1343  (class class class)co 5842   1c1 7754   + caddc 7756   x. cmul 7758   9c9 8915  ;cdc 9322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-iota 5153  df-fv 5196  df-ov 5845  df-dec 9323

This theorem is referenced by:  deceq1i  9328