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Theorem deceq1 9209
 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 5789 . . 3 (𝐴 = 𝐵 → ((9 + 1) · 𝐴) = ((9 + 1) · 𝐵))
21oveq1d 5796 . 2 (𝐴 = 𝐵 → (((9 + 1) · 𝐴) + 𝐶) = (((9 + 1) · 𝐵) + 𝐶))
3 df-dec 9206 . 2 𝐴𝐶 = (((9 + 1) · 𝐴) + 𝐶)
4 df-dec 9206 . 2 𝐵𝐶 = (((9 + 1) · 𝐵) + 𝐶)
52, 3, 43eqtr4g 2198 1 (𝐴 = 𝐵𝐴𝐶 = 𝐵𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332  (class class class)co 5781  1c1 7644   + caddc 7646   · cmul 7648  9c9 8801  ;cdc 9205 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-un 3079  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-iota 5095  df-fv 5138  df-ov 5784  df-dec 9206 This theorem is referenced by:  deceq1i  9211
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