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Mirrors > Home > MPE Home > Th. List > Mathboxes > dpval | Structured version Visualization version GIF version |
Description: Define the value of the decimal point operator. See df-dp 32853. (Contributed by David A. Wheeler, 15-May-2015.) |
Ref | Expression |
---|---|
dpval | ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dp2 32836 | . . 3 ⊢ _𝑥𝑦 = (𝑥 + (𝑦 / ;10)) | |
2 | oveq1 7455 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 + (𝑦 / ;10)) = (𝐴 + (𝑦 / ;10))) | |
3 | 1, 2 | eqtrid 2792 | . 2 ⊢ (𝑥 = 𝐴 → _𝑥𝑦 = (𝐴 + (𝑦 / ;10))) |
4 | oveq1 7455 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 / ;10) = (𝐵 / ;10)) | |
5 | 4 | oveq2d 7464 | . . 3 ⊢ (𝑦 = 𝐵 → (𝐴 + (𝑦 / ;10)) = (𝐴 + (𝐵 / ;10))) |
6 | df-dp2 32836 | . . 3 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
7 | 5, 6 | eqtr4di 2798 | . 2 ⊢ (𝑦 = 𝐵 → (𝐴 + (𝑦 / ;10)) = _𝐴𝐵) |
8 | df-dp 32853 | . 2 ⊢ . = (𝑥 ∈ ℕ0, 𝑦 ∈ ℝ ↦ _𝑥𝑦) | |
9 | 6 | ovexi 7482 | . 2 ⊢ _𝐴𝐵 ∈ V |
10 | 3, 7, 8, 9 | ovmpo 7610 | 1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 (class class class)co 7448 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 / cdiv 11947 ℕ0cn0 12553 ;cdc 12758 _cdp2 32835 .cdp 32852 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-dif 3979 df-un 3981 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-iota 6525 df-fun 6575 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-dp2 32836 df-dp 32853 |
This theorem is referenced by: dpcl 32855 dpfrac1 32856 dpval2 32857 dpmul1000 32863 dpadd2 32874 |
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