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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 31746 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵) | |
4 | 1, 2, 3 | mp2an 690 | . 2 ⊢ (𝐴.𝐵) = _𝐴𝐵 |
5 | df-dp2 31728 | . 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 1541 ∈ wcel 2106 (class class class)co 7357 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 / cdiv 11812 ℕ0cn0 12413 ;cdc 12618 _cdp2 31727 .cdp 31744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-iota 6448 df-fun 6498 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-dp2 31728 df-dp 31745 |
This theorem is referenced by: dpval3 31750 dpmul10 31751 dpmul100 31753 dp3mul10 31754 dplti 31761 dpgti 31762 dpadd2 31766 |
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