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Mirrors > Home > MPE Home > Th. List > rphalfcl | Structured version Visualization version GIF version |
Description: Closure law for half of a positive real. (Contributed by Mario Carneiro, 31-Jan-2014.) |
Ref | Expression |
---|---|
rphalfcl | ⊢ (𝐴 ∈ ℝ+ → (𝐴 / 2) ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2rp 12961 | . 2 ⊢ 2 ∈ ℝ+ | |
2 | rpdivcl 12981 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 2 ∈ ℝ+) → (𝐴 / 2) ∈ ℝ+) | |
3 | 1, 2 | mpan2 689 | 1 ⊢ (𝐴 ∈ ℝ+ → (𝐴 / 2) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 (class class class)co 7393 / cdiv 11853 2c2 12249 ℝ+crp 12956 |
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 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-div 11854 df-2 12257 df-rp 12957 |
This theorem is referenced by: rphalfcld 13010 rpltrp 13302 cau3lem 15283 2clim 15498 addcn2 15520 mulcn2 15522 climcau 15599 metcnpi3 23984 ngptgp 24074 iccntr 24266 reconnlem2 24272 opnreen 24276 xmetdcn2 24282 cnllycmp 24401 iscfil3 24719 cfilfcls 24720 iscmet3lem3 24736 iscmet3lem1 24737 iscmet3lem2 24738 iscmet3 24739 lmcau 24759 bcthlem5 24774 ivthlem2 24898 uniioombl 25035 dvcnvre 25465 aaliou 25780 ulmcaulem 25835 ulmcau 25836 ulmcn 25840 ulmdvlem3 25843 tanregt0 25977 argregt0 26047 argrege0 26048 logimul 26051 resqrtcn 26184 asin1 26326 reasinsin 26328 atanbnd 26358 atan1 26360 sqrtlim 26404 basellem4 26515 chpchtlim 26909 mulog2sumlem2 26965 pntlem3 27039 vacn 29810 ubthlem1 29986 nmcexi 31142 poimirlem29 36319 heicant 36325 ftc1anclem6 36368 ftc1anclem7 36369 ftc1anc 36371 heibor1lem 36480 heiborlem8 36489 bfplem2 36494 supxrge 43819 suplesup 43820 infleinflem1 43851 infleinf 43853 addlimc 44135 fourierdlem103 44696 fourierdlem104 44697 sge0xaddlem2 44921 smflimlem4 45261 |
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