Theorem dpval 32873

Description: Define the value of the decimal point operator. See df-dp 32872. (Contributed by David A. Wheeler, 15-May-2015.)

Assertion

Ref	Expression
dpval	⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵)

Proof of Theorem dpval

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-dp2 32855	. . 3 ⊢ _𝑥𝑦 = (𝑥 + (𝑦 / ;10))
2		oveq1 7439	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 + (𝑦 / ;10)) = (𝐴 + (𝑦 / ;10)))
3	1, 2	eqtrid 2788	. 2 ⊢ (𝑥 = 𝐴 → _𝑥𝑦 = (𝐴 + (𝑦 / ;10)))
4		oveq1 7439	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 / ;10) = (𝐵 / ;10))
5	4	oveq2d 7448	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 + (𝑦 / ;10)) = (𝐴 + (𝐵 / ;10)))
6		df-dp2 32855	. . 3 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10))
7	5, 6	eqtr4di 2794	. 2 ⊢ (𝑦 = 𝐵 → (𝐴 + (𝑦 / ;10)) = _𝐴𝐵)
8		df-dp 32872	. 2 ⊢ . = (𝑥 ∈ ℕ0, 𝑦 ∈ ℝ ↦ _𝑥𝑦)
9	6	ovexi 7466	. 2 ⊢ _𝐴𝐵 ∈ V
10	3, 7, 8, 9	ovmpo 7594	1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  (class class class)co 7432  ℝcr 11155  0cc0 11156  1c1 11157   + caddc 11159   / cdiv 11921  ℕ₀cn0 12528  ;cdc 12735  _cdp2 32854  .cdp 32871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-dp2 32855  df-dp 32872

This theorem is referenced by:  dpcl  32874  dpfrac1  32875  dpval2  32876  dpmul1000  32882  dpadd2  32893