Theorem dp2eq1 33439

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 7427	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 + (𝐶 / ;10)) = (𝐵 + (𝐶 / ;10)))
2		df-dp2 33438	. 2 ⊢ _𝐴𝐶 = (𝐴 + (𝐶 / ;10))
3		df-dp2 33438	. 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 1569  (class class class)co 7420  0cc0 11200  1c1 11201   + caddc 11203   / cdiv 11973  ;cdc 12814  _cdp2 33437

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6546  df-ov 7423  df-dp2 33438

This theorem is used by:  dp2eq1i  33441