Theorem deceq1 9452

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 5926	. . 3 |- ( A = B -> ( ( 9 + 1 ) x. A ) = ( ( 9 + 1 ) x. B ) )
2	1	oveq1d 5933	. 2 |- ( A = B -> ( ( ( 9 + 1 ) x. A ) + C ) = ( ( ( 9 + 1 ) x. B ) + C ) )
3		df-dec 9449	. 2 |- ; A C = ( ( ( 9 + 1 ) x. A ) + C )
4		df-dec 9449	. 2 |- ; B C = ( ( ( 9 + 1 ) x. B ) + C )
5	2, 3, 4	3eqtr4g 2251	1 |- ( A = B -> ; A C = ; B C )

Syntax hints:   -> wi 4   = wceq 1364  (class class class)co 5918   1c1 7873   + caddc 7875   x. cmul 7877   9c9 9040  ;cdc 9448

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5215  df-fv 5262  df-ov 5921  df-dec 9449

This theorem is referenced by:  deceq1i  9454