Theorem deceq1 9508

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 5952	. . 3 |- ( A = B -> ( ( 9 + 1 ) x. A ) = ( ( 9 + 1 ) x. B ) )
2	1	oveq1d 5959	. 2 |- ( A = B -> ( ( ( 9 + 1 ) x. A ) + C ) = ( ( ( 9 + 1 ) x. B ) + C ) )
3		df-dec 9505	. 2 |- ; A C = ( ( ( 9 + 1 ) x. A ) + C )
4		df-dec 9505	. 2 |- ; B C = ( ( ( 9 + 1 ) x. B ) + C )
5	2, 3, 4	3eqtr4g 2263	1 |- ( A = B -> ; A C = ; B C )

Syntax hints:   -> wi 4   = wceq 1373  (class class class)co 5944   1c1 7926   + caddc 7928   x. cmul 7930   9c9 9094  ;cdc 9504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-iota 5232  df-fv 5279  df-ov 5947  df-dec 9505

This theorem is referenced by:  deceq1i  9510