Theorem dp2eq1 33358

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 7416	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 + (𝐶 / ;10)) = (𝐵 + (𝐶 / ;10)))
2		df-dp2 33357	. 2 ⊢ _𝐴𝐶 = (𝐴 + (𝐶 / ;10))
3		df-dp2 33357	. 2 ⊢ _𝐵𝐶 = (𝐵 + (𝐶 / ;10))
4	1, 2, 3	3eqtr4g 2820	1 ⊢ (𝐴 = 𝐵 → _𝐴𝐶 = _𝐵𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  (class class class)co 7409  0cc0 11157  1c1 11158   + caddc 11160   / cdiv 11928  ;cdc 12769  _cdp2 33356

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 2147  ax-9 2155  ax-ext 2732

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 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6484  df-fv 6536  df-ov 7412  df-dp2 33357

This theorem is used by:  dp2eq1i  33360