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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 32856. (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 32839 | . . 3 ⊢ _𝑥𝑦 = (𝑥 + (𝑦 / ;10)) | |
2 | oveq1 7438 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 + (𝑦 / ;10)) = (𝐴 + (𝑦 / ;10))) | |
3 | 1, 2 | eqtrid 2787 | . 2 ⊢ (𝑥 = 𝐴 → _𝑥𝑦 = (𝐴 + (𝑦 / ;10))) |
4 | oveq1 7438 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 / ;10) = (𝐵 / ;10)) | |
5 | 4 | oveq2d 7447 | . . 3 ⊢ (𝑦 = 𝐵 → (𝐴 + (𝑦 / ;10)) = (𝐴 + (𝐵 / ;10))) |
6 | df-dp2 32839 | . . 3 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
7 | 5, 6 | eqtr4di 2793 | . 2 ⊢ (𝑦 = 𝐵 → (𝐴 + (𝑦 / ;10)) = _𝐴𝐵) |
8 | df-dp 32856 | . 2 ⊢ . = (𝑥 ∈ ℕ0, 𝑦 ∈ ℝ ↦ _𝑥𝑦) | |
9 | 6 | ovexi 7465 | . 2 ⊢ _𝐴𝐵 ∈ V |
10 | 3, 7, 8, 9 | ovmpo 7593 | 1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 / cdiv 11918 ℕ0cn0 12524 ;cdc 12731 _cdp2 32838 .cdp 32855 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-dp2 32839 df-dp 32856 |
This theorem is referenced by: dpcl 32858 dpfrac1 32859 dpval2 32860 dpmul1000 32866 dpadd2 32877 |
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