Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rpdp2cl | Structured version Visualization version GIF version |
Description: Closure for a decimal fraction in the positive real numbers. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
Ref | Expression |
---|---|
rpdp2cl.a | ⊢ 𝐴 ∈ ℕ0 |
rpdp2cl.b | ⊢ 𝐵 ∈ ℝ+ |
Ref | Expression |
---|---|
rpdp2cl | ⊢ _𝐴𝐵 ∈ ℝ+ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dp2 30721 | . 2 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
2 | rpdp2cl.a | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
3 | 2 | nn0rei 11987 | . . . 4 ⊢ 𝐴 ∈ ℝ |
4 | rpssre 12479 | . . . . 5 ⊢ ℝ+ ⊆ ℝ | |
5 | rpdp2cl.b | . . . . . 6 ⊢ 𝐵 ∈ ℝ+ | |
6 | 10nn 12195 | . . . . . . 7 ⊢ ;10 ∈ ℕ | |
7 | nnrp 12483 | . . . . . . 7 ⊢ (;10 ∈ ℕ → ;10 ∈ ℝ+) | |
8 | 6, 7 | ax-mp 5 | . . . . . 6 ⊢ ;10 ∈ ℝ+ |
9 | rpdivcl 12497 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ+ ∧ ;10 ∈ ℝ+) → (𝐵 / ;10) ∈ ℝ+) | |
10 | 5, 8, 9 | mp2an 692 | . . . . 5 ⊢ (𝐵 / ;10) ∈ ℝ+ |
11 | 4, 10 | sselii 3874 | . . . 4 ⊢ (𝐵 / ;10) ∈ ℝ |
12 | readdcl 10698 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 / ;10) ∈ ℝ) → (𝐴 + (𝐵 / ;10)) ∈ ℝ) | |
13 | 3, 11, 12 | mp2an 692 | . . 3 ⊢ (𝐴 + (𝐵 / ;10)) ∈ ℝ |
14 | 3, 11 | pm3.2i 474 | . . . 4 ⊢ (𝐴 ∈ ℝ ∧ (𝐵 / ;10) ∈ ℝ) |
15 | 2 | nn0ge0i 12003 | . . . . 5 ⊢ 0 ≤ 𝐴 |
16 | rpgt0 12484 | . . . . . 6 ⊢ ((𝐵 / ;10) ∈ ℝ+ → 0 < (𝐵 / ;10)) | |
17 | 10, 16 | ax-mp 5 | . . . . 5 ⊢ 0 < (𝐵 / ;10) |
18 | 15, 17 | pm3.2i 474 | . . . 4 ⊢ (0 ≤ 𝐴 ∧ 0 < (𝐵 / ;10)) |
19 | addgegt0 11205 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ (𝐵 / ;10) ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 < (𝐵 / ;10))) → 0 < (𝐴 + (𝐵 / ;10))) | |
20 | 14, 18, 19 | mp2an 692 | . . 3 ⊢ 0 < (𝐴 + (𝐵 / ;10)) |
21 | elrp 12474 | . . 3 ⊢ ((𝐴 + (𝐵 / ;10)) ∈ ℝ+ ↔ ((𝐴 + (𝐵 / ;10)) ∈ ℝ ∧ 0 < (𝐴 + (𝐵 / ;10)))) | |
22 | 13, 20, 21 | mpbir2an 711 | . 2 ⊢ (𝐴 + (𝐵 / ;10)) ∈ ℝ+ |
23 | 1, 22 | eqeltri 2829 | 1 ⊢ _𝐴𝐵 ∈ ℝ+ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∈ wcel 2114 class class class wbr 5030 (class class class)co 7170 ℝcr 10614 0cc0 10615 1c1 10616 + caddc 10618 < clt 10753 ≤ cle 10754 / cdiv 11375 ℕcn 11716 ℕ0cn0 11976 ;cdc 12179 ℝ+crp 12472 _cdp2 30720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-dec 12180 df-rp 12473 df-dp2 30721 |
This theorem is referenced by: rpdpcl 30752 dpexpp1 30757 hgt750lemd 32198 hgt750lem 32201 hgt750lem2 32202 hgt750leme 32208 |
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