Theorem dp2eq1 32786

Description: Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)

Assertion

Ref	Expression
dp2eq1	⊢ (𝐴 = 𝐵 → _𝐴𝐶 = _𝐵𝐶)

Proof of Theorem dp2eq1

Step	Hyp	Ref	Expression
1		oveq1 7419	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 + (𝐶 / ;10)) = (𝐵 + (𝐶 / ;10)))
2		df-dp2 32785	. 2 ⊢ _𝐴𝐶 = (𝐴 + (𝐶 / ;10))
3		df-dp2 32785	. 2 ⊢ _𝐵𝐶 = (𝐵 + (𝐶 / ;10))
4	1, 2, 3	3eqtr4g 2794	1 ⊢ (𝐴 = 𝐵 → _𝐴𝐶 = _𝐵𝐶)

Syntax hints:   → wi 4   = wceq 1539  (class class class)co 7412  0cc0 11136  1c1 11137   + caddc 11139   / cdiv 11901  ;cdc 12715  _cdp2 32784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-iota 6493  df-fv 6548  df-ov 7415  df-dp2 32785

This theorem is referenced by:  dp2eq1i  32788