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Mirrors > Home > MPE Home > Th. List > Mathboxes > dpval2 | Structured version Visualization version GIF version |
Description: Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
Ref | Expression |
---|---|
dpval2.a | ⊢ 𝐴 ∈ ℕ0 |
dpval2.b | ⊢ 𝐵 ∈ ℝ |
Ref | Expression |
---|---|
dpval2 | ⊢ (𝐴.𝐵) = (𝐴 + (𝐵 / ;10)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dpval2.a | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
2 | dpval2.b | . . 3 ⊢ 𝐵 ∈ ℝ | |
3 | dpval 30566 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵) | |
4 | 1, 2, 3 | mp2an 690 | . 2 ⊢ (𝐴.𝐵) = _𝐴𝐵 |
5 | df-dp2 30548 | . 2 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
6 | 4, 5 | eqtri 2844 | 1 ⊢ (𝐴.𝐵) = (𝐴 + (𝐵 / ;10)) |
Colors of variables: wff setvar class |
Syntax hints: = 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: dpval3 30570 dpmul10 30571 dpmul100 30573 dp3mul10 30574 dplti 30581 dpgti 30582 dpadd2 30586 |
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