Theorem dpval 33122

Description: Define the value of the decimal point operator. See df-dp 33121. (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 33104	. . 3 ⊢ _𝑥𝑦 = (𝑥 + (𝑦 / ;10))
2		oveq1 7407	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 + (𝑦 / ;10)) = (𝐴 + (𝑦 / ;10)))
3	1, 2	eqtrid 2812	. 2 ⊢ (𝑥 = 𝐴 → _𝑥𝑦 = (𝐴 + (𝑦 / ;10)))
4		oveq1 7407	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 / ;10) = (𝐵 / ;10))
5	4	oveq2d 7416	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 + (𝑦 / ;10)) = (𝐴 + (𝐵 / ;10)))
6		df-dp2 33104	. . 3 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10))
7	5, 6	eqtr4di 2818	. 2 ⊢ (𝑦 = 𝐵 → (𝐴 + (𝑦 / ;10)) = _𝐴𝐵)
8		df-dp 33121	. 2 ⊢ . = (𝑥 ∈ ℕ0, 𝑦 ∈ ℝ ↦ _𝑥𝑦)
9	6	ovexi 7434	. 2 ⊢ _𝐴𝐵 ∈ V
10	3, 7, 8, 9	ovmpo 7560	1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  (class class class)co 7400  ℝcr 11087  0cc0 11088  1c1 11089   + caddc 11091   / cdiv 11859  ℕ₀cn0 12495  ;cdc 12702  _cdp2 33103  .cdp 33120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-iota 6481  df-fun 6527  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-dp2 33104  df-dp 33121

This theorem is referenced by:  dpcl  33123  dpfrac1  33124  dpval2  33125  dpmul1000  33131  dpadd2  33142