Theorem deceq1 9186

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 5782	. . 3 |- ( A = B -> ( ( 9 + 1 ) x. A ) = ( ( 9 + 1 ) x. B ) )
2	1	oveq1d 5789	. 2 |- ( A = B -> ( ( ( 9 + 1 ) x. A ) + C ) = ( ( ( 9 + 1 ) x. B ) + C ) )
3		df-dec 9183	. 2 |- ; A C = ( ( ( 9 + 1 ) x. A ) + C )
4		df-dec 9183	. 2 |- ; B C = ( ( ( 9 + 1 ) x. B ) + C )
5	2, 3, 4	3eqtr4g 2197	1 |- ( A = B -> ; A C = ; B C )

Syntax hints:   -> wi 4   = wceq 1331  (class class class)co 5774   1c1 7621   + caddc 7623   x. cmul 7625   9c9 8778  ;cdc 9182

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777  df-dec 9183

This theorem is referenced by:  deceq1i  9188