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Theorem deceq1 8401
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 = 𝐵𝐴𝐶 = 𝐵𝐶)

Proof of Theorem deceq1
StepHypRef Expression
1 oveq2 5545 . . 3 (𝐴 = 𝐵 → ((9 + 1) · 𝐴) = ((9 + 1) · 𝐵))
21oveq1d 5552 . 2 (𝐴 = 𝐵 → (((9 + 1) · 𝐴) + 𝐶) = (((9 + 1) · 𝐵) + 𝐶))
3 df-dec 8398 . 2 𝐴𝐶 = (((9 + 1) · 𝐴) + 𝐶)
4 df-dec 8398 . 2 𝐵𝐶 = (((9 + 1) · 𝐵) + 𝐶)
52, 3, 43eqtr4g 2111 1 (𝐴 = 𝐵𝐴𝐶 = 𝐵𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1257  (class class class)co 5537  1c1 6918   + caddc 6920   · cmul 6922  9c9 8017  cdc 8397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-rex 2327  df-v 2574  df-un 2947  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-br 3790  df-iota 4892  df-fv 4935  df-ov 5540  df-dec 8398
This theorem is referenced by:  deceq1i  8403
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