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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 30837. (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 30820 | . . 3 ⊢ _𝑥𝑦 = (𝑥 + (𝑦 / ;10)) | |
2 | oveq1 7198 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 + (𝑦 / ;10)) = (𝐴 + (𝑦 / ;10))) | |
3 | 1, 2 | syl5eq 2783 | . 2 ⊢ (𝑥 = 𝐴 → _𝑥𝑦 = (𝐴 + (𝑦 / ;10))) |
4 | oveq1 7198 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 / ;10) = (𝐵 / ;10)) | |
5 | 4 | oveq2d 7207 | . . 3 ⊢ (𝑦 = 𝐵 → (𝐴 + (𝑦 / ;10)) = (𝐴 + (𝐵 / ;10))) |
6 | df-dp2 30820 | . . 3 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
7 | 5, 6 | eqtr4di 2789 | . 2 ⊢ (𝑦 = 𝐵 → (𝐴 + (𝑦 / ;10)) = _𝐴𝐵) |
8 | df-dp 30837 | . 2 ⊢ . = (𝑥 ∈ ℕ0, 𝑦 ∈ ℝ ↦ _𝑥𝑦) | |
9 | 6 | ovexi 7225 | . 2 ⊢ _𝐴𝐵 ∈ V |
10 | 3, 7, 8, 9 | ovmpo 7347 | 1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 (class class class)co 7191 ℝcr 10693 0cc0 10694 1c1 10695 + caddc 10697 / cdiv 11454 ℕ0cn0 12055 ;cdc 12258 _cdp2 30819 .cdp 30836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-iota 6316 df-fun 6360 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-dp2 30820 df-dp 30837 |
This theorem is referenced by: dpcl 30839 dpfrac1 30840 dpval2 30841 dpmul1000 30847 dpadd2 30858 |
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