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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 30568 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵) | |
4 | 1, 2, 3 | mp2an 690 | . 2 ⊢ (𝐴.𝐵) = _𝐴𝐵 |
5 | df-dp2 30550 | . 2 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
6 | 4, 5 | eqtri 2846 | 1 ⊢ (𝐴.𝐵) = (𝐴 + (𝐵 / ;10)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 (class class class)co 7158 ℝcr 10538 0cc0 10539 1c1 10540 + caddc 10542 / cdiv 11299 ℕ0cn0 11900 ;cdc 12101 _cdp2 30549 .cdp 30566 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-dp2 30550 df-dp 30567 |
This theorem is referenced by: dpval3 30572 dpmul10 30573 dpmul100 30575 dp3mul10 30576 dplti 30583 dpgti 30584 dpadd2 30588 |
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