![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 10328 | . 2 ⊢ 1 ∈ ℝ | |
2 | 0le1 10843 | . 2 ⊢ 0 ≤ 1 | |
3 | absid 14377 | . 2 ⊢ ((1 ∈ ℝ ∧ 0 ≤ 1) → (abs‘1) = 1) | |
4 | 1, 2, 3 | mp2an 684 | 1 ⊢ (abs‘1) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 ∈ wcel 2157 class class class wbr 4843 ‘cfv 6101 ℝcr 10223 0cc0 10224 1c1 10225 ≤ cle 10364 abscabs 14315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-sup 8590 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-2 11376 df-3 11377 df-n0 11581 df-z 11667 df-uz 11931 df-rp 12075 df-seq 13056 df-exp 13115 df-cj 14180 df-re 14181 df-im 14182 df-sqrt 14316 df-abs 14317 |
This theorem is referenced by: absexp 14385 absexpz 14386 iseraltlem3 14755 geolim 14939 geolim2 14940 georeclim 14941 geoisum1c 14949 efieq1re 15265 eirrlem 15268 3lcm2e6woprm 15663 4sqlem13 15994 4sqlem19 16000 gzrngunit 20134 ncvsm1 23281 dvlipcn 24098 dvfsumabs 24127 geolim3 24435 abelthlem1 24526 abelthlem2 24527 coskpi 24614 sineq0 24615 logtayl 24747 abscxpbnd 24838 root1cj 24841 bndatandm 25008 lgamgulmlem2 25108 lgamgulmlem5 25111 mule1 25226 logfacbnd3 25300 dchrabs 25337 zabsle1 25373 lgslem2 25375 lgsfcl2 25380 lgseisen 25456 2sqlem9 25504 2sqlem10 25505 nvm1 28045 nvmtri 28051 normlem7tALT 28501 norm-ii-i 28519 normsubi 28523 qqhval2lem 30541 qqh0 30544 subfaclim 31687 binomcxplemrat 39331 sineq0ALT 39933 fprodabs2 40571 |
Copyright terms: Public domain | W3C validator |