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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 12395 | . 2 ⊢ 2 ∈ ℝ+ | |
2 | rpdivcl 12415 | . 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 2114 (class class class)co 7156 / cdiv 11297 2c2 11693 ℝ+crp 12390 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-2 11701 df-rp 12391 |
This theorem is referenced by: rphalfcld 12444 rpltrp 12735 cau3lem 14714 2clim 14929 addcn2 14950 mulcn2 14952 climcau 15027 metcnpi3 23156 ngptgp 23245 iccntr 23429 reconnlem2 23435 opnreen 23439 xmetdcn2 23445 cnllycmp 23560 iscfil3 23876 cfilfcls 23877 iscmet3lem3 23893 iscmet3lem1 23894 iscmet3lem2 23895 iscmet3 23896 lmcau 23916 bcthlem5 23931 ivthlem2 24053 uniioombl 24190 dvcnvre 24616 aaliou 24927 ulmcaulem 24982 ulmcau 24983 ulmcn 24987 ulmdvlem3 24990 tanregt0 25123 argregt0 25193 argrege0 25194 logimul 25197 resqrtcn 25330 asin1 25472 reasinsin 25474 atanbnd 25504 atan1 25506 sqrtlim 25550 basellem4 25661 chpchtlim 26055 mulog2sumlem2 26111 pntlem3 26185 vacn 28471 ubthlem1 28647 nmcexi 29803 poimirlem29 34936 heicant 34942 ftc1anclem6 34987 ftc1anclem7 34988 ftc1anc 34990 heibor1lem 35102 heiborlem8 35111 bfplem2 35116 supxrge 41626 suplesup 41627 infleinflem1 41658 infleinf 41660 addlimc 41949 fourierdlem103 42514 fourierdlem104 42515 sge0xaddlem2 42736 smflimlem4 43070 |
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