Theorem dp2eq2 32840

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 7457	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 / ;10) = (𝐵 / ;10))
2	1	oveq2d 7466	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 + (𝐴 / ;10)) = (𝐶 + (𝐵 / ;10)))
3		df-dp2 32838	. 2 ⊢ _𝐶𝐴 = (𝐶 + (𝐴 / ;10))
4		df-dp2 32838	. 2 ⊢ _𝐶𝐵 = (𝐶 + (𝐵 / ;10))
5	2, 3, 4	3eqtr4g 2805	1 ⊢ (𝐴 = 𝐵 → _𝐶𝐴 = _𝐶𝐵)

Syntax hints:   → wi 4   = wceq 1537  (class class class)co 7450  0cc0 11186  1c1 11187   + caddc 11189   / cdiv 11949  ;cdc 12760  _cdp2 32837

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6527  df-fv 6583  df-ov 7453  df-dp2 32838

This theorem is referenced by:  dp2eq2i  32842