Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 14594 | . 2 ⊢ abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥)))) | |
2 | absval 14596 | . . 3 ⊢ (𝑥 ∈ ℂ → (abs‘𝑥) = (√‘(𝑥 · (∗‘𝑥)))) | |
3 | abscl 14637 | . . 3 ⊢ (𝑥 ∈ ℂ → (abs‘𝑥) ∈ ℝ) | |
4 | 2, 3 | eqeltrrd 2914 | . 2 ⊢ (𝑥 ∈ ℂ → (√‘(𝑥 · (∗‘𝑥))) ∈ ℝ) |
5 | 1, 4 | fmpti 6875 | 1 ⊢ abs:ℂ⟶ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ℂcc 10534 ℝcr 10535 · cmul 10541 ∗ccj 14454 √csqrt 14591 abscabs 14592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-sup 8905 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-z 11981 df-uz 12243 df-rp 12389 df-seq 13369 df-exp 13429 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 |
This theorem is referenced by: lo1o1 14888 lo1o12 14889 abscn2 14954 climabs 14959 rlimabs 14964 cnfldds 20554 cnfldfun 20556 cnfldfunALT 20557 absabv 20601 cnmet 23379 cnbl0 23381 cnblcld 23382 cnfldms 23383 cnfldnm 23386 abscncf 23508 cnfldcusp 23959 ovolfsf 24071 ovolctb 24090 iblabslem 24427 iblabs 24428 bddmulibl 24438 dvlip2 24591 c1liplem1 24592 pserulm 25009 psercn2 25010 psercnlem2 25011 psercnlem1 25012 psercn 25013 pserdvlem1 25014 pserdvlem2 25015 pserdv 25016 pserdv2 25017 abelth 25028 efif1olem3 25127 efif1olem4 25128 efifo 25130 eff1olem 25131 logcn 25229 efopnlem1 25238 logtayl 25242 cnnv 28453 cnnvg 28454 cnnvs 28456 cnnvnm 28457 cncph 28595 mblfinlem2 34929 ftc1anclem1 34966 ftc1anclem2 34967 ftc1anclem3 34968 ftc1anclem4 34969 ftc1anclem5 34970 ftc1anclem6 34971 ftc1anclem7 34972 ftc1anclem8 34973 ftc1anc 34974 extoimad 40513 imo72b2lem0 40514 imo72b2lem2 40516 imo72b2lem1 40519 imo72b2 40523 sblpnf 40640 binomcxplemdvbinom 40683 binomcxplemcvg 40684 binomcxplemdvsum 40685 binomcxplemnotnn0 40686 absfun 41616 cncficcgt0 42169 fourierdlem42 42433 hoicvr 42829 ovolval2lem 42924 ovolval3 42928 |
Copyright terms: Public domain | W3C validator |