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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 30565. (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 30548 | . . 3 ⊢ _𝑥𝑦 = (𝑥 + (𝑦 / ;10)) | |
2 | oveq1 7162 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 + (𝑦 / ;10)) = (𝐴 + (𝑦 / ;10))) | |
3 | 1, 2 | syl5eq 2868 | . 2 ⊢ (𝑥 = 𝐴 → _𝑥𝑦 = (𝐴 + (𝑦 / ;10))) |
4 | oveq1 7162 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 / ;10) = (𝐵 / ;10)) | |
5 | 4 | oveq2d 7171 | . . 3 ⊢ (𝑦 = 𝐵 → (𝐴 + (𝑦 / ;10)) = (𝐴 + (𝐵 / ;10))) |
6 | df-dp2 30548 | . . 3 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
7 | 5, 6 | syl6eqr 2874 | . 2 ⊢ (𝑦 = 𝐵 → (𝐴 + (𝑦 / ;10)) = _𝐴𝐵) |
8 | df-dp 30565 | . 2 ⊢ . = (𝑥 ∈ ℕ0, 𝑦 ∈ ℝ ↦ _𝑥𝑦) | |
9 | 6 | ovexi 7189 | . 2 ⊢ _𝐴𝐵 ∈ V |
10 | 3, 7, 8, 9 | ovmpo 7309 | 1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 (class class class)co 7155 ℝcr 10535 0cc0 10536 1c1 10537 + caddc 10539 / cdiv 11296 ℕ0cn0 11896 ;cdc 12097 _cdp2 30547 .cdp 30564 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-iota 6313 df-fun 6356 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-dp2 30548 df-dp 30565 |
This theorem is referenced by: dpcl 30567 dpfrac1 30568 dpval2 30569 dpmul1000 30575 dpadd2 30586 |
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