Theorem dpval 33246

Description: Define the value of the decimal point operator. See df-dp 33245. (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 33228	. . 3 ⊢ _𝑥𝑦 = (𝑥 + (𝑦 / ;10))
2		oveq1 7430	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 + (𝑦 / ;10)) = (𝐴 + (𝑦 / ;10)))
3	1, 2	eqtrid 2813	. 2 ⊢ (𝑥 = 𝐴 → _𝑥𝑦 = (𝐴 + (𝑦 / ;10)))
4		oveq1 7430	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 / ;10) = (𝐵 / ;10))
5	4	oveq2d 7439	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 + (𝑦 / ;10)) = (𝐴 + (𝐵 / ;10)))
6		df-dp2 33228	. . 3 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10))
7	5, 6	eqtr4di 2819	. 2 ⊢ (𝑦 = 𝐵 → (𝐴 + (𝑦 / ;10)) = _𝐴𝐵)
8		df-dp 33245	. 2 ⊢ . = (𝑥 ∈ ℕ0, 𝑦 ∈ ℝ ↦ _𝑥𝑦)
9	6	ovexi 7457	. 2 ⊢ _𝐴𝐵 ∈ V
10	3, 7, 8, 9	ovmpo 7583	1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  (class class class)co 7423  ℝcr 11117  0cc0 11118  1c1 11119   + caddc 11121   / cdiv 11889  ℕ₀cn0 12522  ;cdc 12729  _cdp2 33227  .cdp 33244

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

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-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6499  df-fun 6545  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-dp2 33228  df-dp 33245

This theorem is used by:  dpcl  33247  dpfrac1  33248  dpval2  33249  dpmul1000  33255  dpadd2  33266