Theorem deceq1 12682

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 7417	. . 3 ⊢ (𝐴 = 𝐵 → ((9 + 1) · 𝐴) = ((9 + 1) · 𝐵))
2	1	oveq1d 7424	. 2 ⊢ (𝐴 = 𝐵 → (((9 + 1) · 𝐴) + 𝐶) = (((9 + 1) · 𝐵) + 𝐶))
3		df-dec 12678	. 2 ⊢ ;𝐴𝐶 = (((9 + 1) · 𝐴) + 𝐶)
4		df-dec 12678	. 2 ⊢ ;𝐵𝐶 = (((9 + 1) · 𝐵) + 𝐶)
5	2, 3, 4	3eqtr4g 2798	1 ⊢ (𝐴 = 𝐵 → ;𝐴𝐶 = ;𝐵𝐶)

Syntax hints:   → wi 4   = wceq 1542  (class class class)co 7409  1c1 11111   + caddc 11113   · cmul 11115  9c9 12274  ;cdc 12677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-dec 12678

This theorem is referenced by:  deceq1i  12684