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Mirrors > Home > MPE Home > Th. List > rehalfcl | Structured version Visualization version GIF version |
Description: Real closure of half. (Contributed by NM, 1-Jan-2006.) |
Ref | Expression |
---|---|
rehalfcl | ⊢ (𝐴 ∈ ℝ → (𝐴 / 2) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 11507 | . 2 ⊢ 2 ∈ ℝ | |
2 | 2ne0 11544 | . 2 ⊢ 2 ≠ 0 | |
3 | redivcl 11152 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 2 ∈ ℝ ∧ 2 ≠ 0) → (𝐴 / 2) ∈ ℝ) | |
4 | 1, 2, 3 | mp3an23 1432 | 1 ⊢ (𝐴 ∈ ℝ → (𝐴 / 2) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2048 ≠ wne 2961 (class class class)co 6970 ℝcr 10326 0cc0 10327 / cdiv 11090 2c2 11488 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-po 5319 df-so 5320 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-2 11496 |
This theorem is referenced by: halfpos 11670 lt2halves 11675 nominpos 11677 avgle1 11680 avgle2 11681 rehalfcld 11687 rehalfcli 11689 fldiv4lem1div2uz2 13014 efgt0 15306 sin02gt0 15395 tangtx 24784 sinq12gt0 24786 cosordlem 24806 cxpcn3lem 25019 basellem1 25350 gausslemma2dlem1a 25633 chebbnd1lem2 25738 chebbnd1lem3 25739 sin2h 34271 cos2h 34272 tan2h 34273 itg2addnclem 34332 infleinflem1 41013 fourierdlem57 41825 fourierdlem66 41834 smflimlem4 42427 |
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