Theorem dp2eq2 33230

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 7430	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 / ;10) = (𝐵 / ;10))
2	1	oveq2d 7439	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 + (𝐴 / ;10)) = (𝐶 + (𝐵 / ;10)))
3		df-dp2 33228	. 2 ⊢ _𝐶𝐴 = (𝐶 + (𝐴 / ;10))
4		df-dp2 33228	. 2 ⊢ _𝐶𝐵 = (𝐶 + (𝐵 / ;10))
5	2, 3, 4	3eqtr4g 2826	1 ⊢ (𝐴 = 𝐵 → _𝐶𝐴 = _𝐶𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  (class class class)co 7423  0cc0 11118  1c1 11119   + caddc 11121   / cdiv 11889  ;cdc 12729  _cdp2 33227

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-iota 6499  df-fv 6551  df-ov 7426  df-dp2 33228

This theorem is used by:  dp2eq2i  33232