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Mirrors > Home > MPE Home > Th. List > absf | Structured version Visualization version GIF version |
Description: Mapping domain and codomain of the absolute value function. (Contributed by NM, 30-Aug-2007.) (Revised by Mario Carneiro, 7-Nov-2013.) |
Ref | Expression |
---|---|
absf | ⊢ abs:ℂ⟶ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-abs 14353 | . 2 ⊢ abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥)))) | |
2 | absval 14355 | . . 3 ⊢ (𝑥 ∈ ℂ → (abs‘𝑥) = (√‘(𝑥 · (∗‘𝑥)))) | |
3 | abscl 14395 | . . 3 ⊢ (𝑥 ∈ ℂ → (abs‘𝑥) ∈ ℝ) | |
4 | 2, 3 | eqeltrrd 2907 | . 2 ⊢ (𝑥 ∈ ℂ → (√‘(𝑥 · (∗‘𝑥))) ∈ ℝ) |
5 | 1, 4 | fmpti 6631 | 1 ⊢ abs:ℂ⟶ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2164 ⟶wf 6119 ‘cfv 6123 (class class class)co 6905 ℂcc 10250 ℝcr 10251 · cmul 10257 ∗ccj 14213 √csqrt 14350 abscabs 14351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-sup 8617 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-n0 11619 df-z 11705 df-uz 11969 df-rp 12113 df-seq 13096 df-exp 13155 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 |
This theorem is referenced by: lo1o1 14640 lo1o12 14641 abscn2 14706 climabs 14711 rlimabs 14716 cnfldds 20116 cnfldfun 20118 cnfldfunALT 20119 absabv 20163 cnmet 22945 cnbl0 22947 cnblcld 22948 cnfldms 22949 cnfldnm 22952 abscncf 23074 cnfldcusp 23525 ovolfsf 23637 ovolctb 23656 iblabslem 23993 iblabs 23994 bddmulibl 24004 dvlip2 24157 c1liplem1 24158 pserulm 24575 psercn2 24576 psercnlem2 24577 psercnlem1 24578 psercn 24579 pserdvlem1 24580 pserdvlem2 24581 pserdv 24582 pserdv2 24583 abelth 24594 efif1olem3 24690 efif1olem4 24691 efifo 24693 eff1olem 24694 logcn 24792 efopnlem1 24801 logtayl 24805 cnnv 28076 cnnvg 28077 cnnvs 28079 cnnvnm 28080 cncph 28218 mblfinlem2 33984 ftc1anclem1 34021 ftc1anclem2 34022 ftc1anclem3 34023 ftc1anclem4 34024 ftc1anclem5 34025 ftc1anclem6 34026 ftc1anclem7 34027 ftc1anclem8 34028 ftc1anc 34029 extoimad 39297 imo72b2lem0 39298 imo72b2lem2 39300 imo72b2lem1 39304 imo72b2 39308 sblpnf 39342 binomcxplemdvbinom 39385 binomcxplemcvg 39386 binomcxplemdvsum 39387 binomcxplemnotnn0 39388 absfun 40356 cncficcgt0 40889 fourierdlem42 41153 hoicvr 41549 ovolval2lem 41644 ovolval3 41648 |
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