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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 32873 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵) | |
| 4 | 1, 2, 3 | mp2an 692 | . 2 ⊢ (𝐴.𝐵) = _𝐴𝐵 | 
| 5 | df-dp2 32855 | . 2 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
| 6 | 4, 5 | eqtri 2764 | 1 ⊢ (𝐴.𝐵) = (𝐴 + (𝐵 / ;10)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∈ wcel 2107 (class class class)co 7432 ℝcr 11155 0cc0 11156 1c1 11157 + caddc 11159 / cdiv 11921 ℕ0cn0 12528 ;cdc 12735 _cdp2 32854 .cdp 32871 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-dp2 32855 df-dp 32872 | 
| This theorem is referenced by: dpval3 32877 dpmul10 32878 dpmul100 32880 dp3mul10 32881 dplti 32888 dpgti 32889 dpadd2 32893 | 
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