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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 32589 | . 2 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
2 | rpdp2cl.a | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
3 | 2 | nn0rei 12507 | . . . 4 ⊢ 𝐴 ∈ ℝ |
4 | rpssre 13007 | . . . . 5 ⊢ ℝ+ ⊆ ℝ | |
5 | rpdp2cl.b | . . . . . 6 ⊢ 𝐵 ∈ ℝ+ | |
6 | 10nn 12717 | . . . . . . 7 ⊢ ;10 ∈ ℕ | |
7 | nnrp 13011 | . . . . . . 7 ⊢ (;10 ∈ ℕ → ;10 ∈ ℝ+) | |
8 | 6, 7 | ax-mp 5 | . . . . . 6 ⊢ ;10 ∈ ℝ+ |
9 | rpdivcl 13025 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ+ ∧ ;10 ∈ ℝ+) → (𝐵 / ;10) ∈ ℝ+) | |
10 | 5, 8, 9 | mp2an 691 | . . . . 5 ⊢ (𝐵 / ;10) ∈ ℝ+ |
11 | 4, 10 | sselii 3975 | . . . 4 ⊢ (𝐵 / ;10) ∈ ℝ |
12 | readdcl 11215 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 / ;10) ∈ ℝ) → (𝐴 + (𝐵 / ;10)) ∈ ℝ) | |
13 | 3, 11, 12 | mp2an 691 | . . 3 ⊢ (𝐴 + (𝐵 / ;10)) ∈ ℝ |
14 | 3, 11 | pm3.2i 470 | . . . 4 ⊢ (𝐴 ∈ ℝ ∧ (𝐵 / ;10) ∈ ℝ) |
15 | 2 | nn0ge0i 12523 | . . . . 5 ⊢ 0 ≤ 𝐴 |
16 | rpgt0 13012 | . . . . . 6 ⊢ ((𝐵 / ;10) ∈ ℝ+ → 0 < (𝐵 / ;10)) | |
17 | 10, 16 | ax-mp 5 | . . . . 5 ⊢ 0 < (𝐵 / ;10) |
18 | 15, 17 | pm3.2i 470 | . . . 4 ⊢ (0 ≤ 𝐴 ∧ 0 < (𝐵 / ;10)) |
19 | addgegt0 11725 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ (𝐵 / ;10) ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 < (𝐵 / ;10))) → 0 < (𝐴 + (𝐵 / ;10))) | |
20 | 14, 18, 19 | mp2an 691 | . . 3 ⊢ 0 < (𝐴 + (𝐵 / ;10)) |
21 | elrp 13002 | . . 3 ⊢ ((𝐴 + (𝐵 / ;10)) ∈ ℝ+ ↔ ((𝐴 + (𝐵 / ;10)) ∈ ℝ ∧ 0 < (𝐴 + (𝐵 / ;10)))) | |
22 | 13, 20, 21 | mpbir2an 710 | . 2 ⊢ (𝐴 + (𝐵 / ;10)) ∈ ℝ+ |
23 | 1, 22 | eqeltri 2824 | 1 ⊢ _𝐴𝐵 ∈ ℝ+ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∈ wcel 2099 class class class wbr 5142 (class class class)co 7414 ℝcr 11131 0cc0 11132 1c1 11133 + caddc 11135 < clt 11272 ≤ cle 11273 / cdiv 11895 ℕcn 12236 ℕ0cn0 12496 ;cdc 12701 ℝ+crp 13000 _cdp2 32588 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-dec 12702 df-rp 13001 df-dp2 32589 |
This theorem is referenced by: rpdpcl 32620 dpexpp1 32625 hgt750lemd 34270 hgt750lem 34273 hgt750lem2 34274 hgt750leme 34280 |
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