Theorem deceq1 9383

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 5879	. . 3 ⊢ (𝐴 = 𝐵 → ((9 + 1) · 𝐴) = ((9 + 1) · 𝐵))
2	1	oveq1d 5886	. 2 ⊢ (𝐴 = 𝐵 → (((9 + 1) · 𝐴) + 𝐶) = (((9 + 1) · 𝐵) + 𝐶))
3		df-dec 9380	. 2 ⊢ ;𝐴𝐶 = (((9 + 1) · 𝐴) + 𝐶)
4		df-dec 9380	. 2 ⊢ ;𝐵𝐶 = (((9 + 1) · 𝐵) + 𝐶)
5	2, 3, 4	3eqtr4g 2235	1 ⊢ (𝐴 = 𝐵 → ;𝐴𝐶 = ;𝐵𝐶)

Syntax hints:   → wi 4   = wceq 1353  (class class class)co 5871  1c1 7808   + caddc 7810   · cmul 7812  9c9 8972  ;cdc 9379

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-iota 5176  df-fv 5222  df-ov 5874  df-dec 9380

This theorem is referenced by:  deceq1i  9385