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Theorem deceq1 12187
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 7181 . . 3 (𝐴 = 𝐵 → ((9 + 1) · 𝐴) = ((9 + 1) · 𝐵))
21oveq1d 7188 . 2 (𝐴 = 𝐵 → (((9 + 1) · 𝐴) + 𝐶) = (((9 + 1) · 𝐵) + 𝐶))
3 df-dec 12183 . 2 𝐴𝐶 = (((9 + 1) · 𝐴) + 𝐶)
4 df-dec 12183 . 2 𝐵𝐶 = (((9 + 1) · 𝐵) + 𝐶)
52, 3, 43eqtr4g 2799 1 (𝐴 = 𝐵𝐴𝐶 = 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  (class class class)co 7173  1c1 10619   + caddc 10621   · cmul 10623  9c9 11781  cdc 12182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-v 3401  df-un 3849  df-in 3851  df-ss 3861  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-iota 6298  df-fv 6348  df-ov 7176  df-dec 12183
This theorem is referenced by:  deceq1i  12189
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