Theorem deceq1 9347

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 5861	. . 3 |- ( A = B -> ( ( 9 + 1 ) x. A ) = ( ( 9 + 1 ) x. B ) )
2	1	oveq1d 5868	. 2 |- ( A = B -> ( ( ( 9 + 1 ) x. A ) + C ) = ( ( ( 9 + 1 ) x. B ) + C ) )
3		df-dec 9344	. 2 |- ; A C = ( ( ( 9 + 1 ) x. A ) + C )
4		df-dec 9344	. 2 |- ; B C = ( ( ( 9 + 1 ) x. B ) + C )
5	2, 3, 4	3eqtr4g 2228	1 |- ( A = B -> ; A C = ; B C )

Syntax hints:   -> wi 4   = wceq 1348  (class class class)co 5853   1c1 7775   + caddc 7777   x. cmul 7779   9c9 8936  ;cdc 9343

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856  df-dec 9344

This theorem is referenced by:  deceq1i  9349