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Mirrors > Home > MPE Home > Th. List > abs1 | Structured version Visualization version GIF version |
Description: The absolute value of one is one. (Contributed by David A. Wheeler, 16-Jul-2016.) |
Ref | Expression |
---|---|
abs1 | ⊢ (abs‘1) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 10630 | . 2 ⊢ 1 ∈ ℝ | |
2 | 0le1 11152 | . 2 ⊢ 0 ≤ 1 | |
3 | absid 14648 | . 2 ⊢ ((1 ∈ ℝ ∧ 0 ≤ 1) → (abs‘1) = 1) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ (abs‘1) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 ℝcr 10525 0cc0 10526 1c1 10527 ≤ cle 10665 abscabs 14585 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 |
This theorem is referenced by: absexp 14656 absexpz 14657 iseraltlem3 15032 geolim 15218 geolim2 15219 georeclim 15220 geoisum1c 15228 efieq1re 15544 eirrlem 15549 3lcm2e6woprm 15949 4sqlem13 16283 4sqlem19 16289 gzrngunit 20157 ncvsm1 23759 dvlipcn 24597 dvfsumabs 24626 geolim3 24935 abelthlem1 25026 abelthlem2 25027 coskpi 25115 sineq0 25116 logtayl 25251 abscxpbnd 25342 root1cj 25345 bndatandm 25515 lgamgulmlem2 25615 lgamgulmlem5 25618 mule1 25733 logfacbnd3 25807 dchrabs 25844 zabsle1 25880 lgslem2 25882 lgsfcl2 25887 lgseisen 25963 2sqlem9 26011 2sqlem10 26012 nvm1 28448 nvmtri 28454 normlem7tALT 28902 norm-ii-i 28920 normsubi 28924 qqhval2lem 31332 qqh0 31335 subfaclim 32548 lcm1un 39301 nn0rppwr 39490 binomcxplemrat 41054 sineq0ALT 41643 fprodabs2 42237 |
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