Theorem dp2eq2 33106

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 7407	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 / ;10) = (𝐵 / ;10))
2	1	oveq2d 7416	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 + (𝐴 / ;10)) = (𝐶 + (𝐵 / ;10)))
3		df-dp2 33104	. 2 ⊢ _𝐶𝐴 = (𝐶 + (𝐴 / ;10))
4		df-dp2 33104	. 2 ⊢ _𝐶𝐵 = (𝐶 + (𝐵 / ;10))
5	2, 3, 4	3eqtr4g 2825	1 ⊢ (𝐴 = 𝐵 → _𝐶𝐴 = _𝐶𝐵)

Syntax hints:   → wi 4   = wceq 1563  (class class class)co 7400  0cc0 11088  1c1 11089   + caddc 11091   / cdiv 11859  ;cdc 12702  _cdp2 33103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-iota 6481  df-fv 6533  df-ov 7403  df-dp2 33104

This theorem is referenced by:  dp2eq2i  33108