Theorem deceq1 9090

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 → ;𝐴𝐶 = ;𝐵𝐶)

Proof of Theorem deceq1

Step	Hyp	Ref	Expression
1		oveq2 5736	. . 3 ⊢ (𝐴 = 𝐵 → ((9 + 1) · 𝐴) = ((9 + 1) · 𝐵))
2	1	oveq1d 5743	. 2 ⊢ (𝐴 = 𝐵 → (((9 + 1) · 𝐴) + 𝐶) = (((9 + 1) · 𝐵) + 𝐶))
3		df-dec 9087	. 2 ⊢ ;𝐴𝐶 = (((9 + 1) · 𝐴) + 𝐶)
4		df-dec 9087	. 2 ⊢ ;𝐵𝐶 = (((9 + 1) · 𝐵) + 𝐶)
5	2, 3, 4	3eqtr4g 2172	1 ⊢ (𝐴 = 𝐵 → ;𝐴𝐶 = ;𝐵𝐶)

Syntax hints:   → wi 4   = wceq 1314  (class class class)co 5728  1c1 7548   + caddc 7550   · cmul 7552  9c9 8688  ;cdc 9086

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-rex 2396  df-v 2659  df-un 3041  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-iota 5046  df-fv 5089  df-ov 5731  df-dec 9087

This theorem is referenced by:  deceq1i  9092