Theorem dp2eq2 33174

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 7419	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 / ;10) = (𝐵 / ;10))
2	1	oveq2d 7428	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 + (𝐴 / ;10)) = (𝐶 + (𝐵 / ;10)))
3		df-dp2 33172	. 2 ⊢ _𝐶𝐴 = (𝐶 + (𝐴 / ;10))
4		df-dp2 33172	. 2 ⊢ _𝐶𝐵 = (𝐶 + (𝐵 / ;10))
5	2, 3, 4	3eqtr4g 2823	1 ⊢ (𝐴 = 𝐵 → _𝐶𝐴 = _𝐶𝐵)

Syntax hints:   → wi 4   = wceq 1570  (class class class)co 7412  0cc0 11101  1c1 11102   + caddc 11104   / cdiv 11872  ;cdc 12712  _cdp2 33171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-iota 6494  df-fv 6546  df-ov 7415  df-dp2 33172

This theorem is referenced by:  dp2eq2i  33176