Theorem dp2eq2 32957

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

Assertion

Ref	Expression
dp2eq2	⊢ (𝐴 = 𝐵 → _𝐶𝐴 = _𝐶𝐵)

Proof of Theorem dp2eq2

Step	Hyp	Ref	Expression
1		oveq1 7365	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 / ;10) = (𝐵 / ;10))
2	1	oveq2d 7374	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 + (𝐴 / ;10)) = (𝐶 + (𝐵 / ;10)))
3		df-dp2 32955	. 2 ⊢ _𝐶𝐴 = (𝐶 + (𝐴 / ;10))
4		df-dp2 32955	. 2 ⊢ _𝐶𝐵 = (𝐶 + (𝐵 / ;10))
5	2, 3, 4	3eqtr4g 2796	1 ⊢ (𝐴 = 𝐵 → _𝐶𝐴 = _𝐶𝐵)

Syntax hints:   → wi 4   = wceq 1541  (class class class)co 7358  0cc0 11028  1c1 11029   + caddc 11031   / cdiv 11796  ;cdc 12609  _cdp2 32954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-ov 7361  df-dp2 32955

This theorem is referenced by:  dp2eq2i  32959