Theorem deceq1 12630

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 7377	. . 3 ⊢ (𝐴 = 𝐵 → ((9 + 1) · 𝐴) = ((9 + 1) · 𝐵))
2	1	oveq1d 7384	. 2 ⊢ (𝐴 = 𝐵 → (((9 + 1) · 𝐴) + 𝐶) = (((9 + 1) · 𝐵) + 𝐶))
3		df-dec 12626	. 2 ⊢ ;𝐴𝐶 = (((9 + 1) · 𝐴) + 𝐶)
4		df-dec 12626	. 2 ⊢ ;𝐵𝐶 = (((9 + 1) · 𝐵) + 𝐶)
5	2, 3, 4	3eqtr4g 2789	1 ⊢ (𝐴 = 𝐵 → ;𝐴𝐶 = ;𝐵𝐶)

Syntax hints:   → wi 4   = wceq 1540  (class class class)co 7369  1c1 11045   + caddc 11047   · cmul 11049  9c9 12224  ;cdc 12625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-iota 6452  df-fv 6507  df-ov 7372  df-dec 12626

This theorem is referenced by:  deceq1i  12632