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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 14595 | . 2 ⊢ abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥)))) | |
2 | absval 14597 | . . 3 ⊢ (𝑥 ∈ ℂ → (abs‘𝑥) = (√‘(𝑥 · (∗‘𝑥)))) | |
3 | abscl 14638 | . . 3 ⊢ (𝑥 ∈ ℂ → (abs‘𝑥) ∈ ℝ) | |
4 | 2, 3 | eqeltrrd 2914 | . 2 ⊢ (𝑥 ∈ ℂ → (√‘(𝑥 · (∗‘𝑥))) ∈ ℝ) |
5 | 1, 4 | fmpti 6876 | 1 ⊢ abs:ℂ⟶ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 · cmul 10542 ∗ccj 14455 √csqrt 14592 abscabs 14593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 |
This theorem is referenced by: lo1o1 14889 lo1o12 14890 abscn2 14955 climabs 14960 rlimabs 14965 cnfldds 20555 cnfldfun 20557 cnfldfunALT 20558 absabv 20602 cnmet 23380 cnbl0 23382 cnblcld 23383 cnfldms 23384 cnfldnm 23387 abscncf 23509 cnfldcusp 23960 ovolfsf 24072 ovolctb 24091 iblabslem 24428 iblabs 24429 bddmulibl 24439 dvlip2 24592 c1liplem1 24593 pserulm 25010 psercn2 25011 psercnlem2 25012 psercnlem1 25013 psercn 25014 pserdvlem1 25015 pserdvlem2 25016 pserdv 25017 pserdv2 25018 abelth 25029 efif1olem3 25128 efif1olem4 25129 efifo 25131 eff1olem 25132 logcn 25230 efopnlem1 25239 logtayl 25243 cnnv 28454 cnnvg 28455 cnnvs 28457 cnnvnm 28458 cncph 28596 mblfinlem2 34945 ftc1anclem1 34982 ftc1anclem2 34983 ftc1anclem3 34984 ftc1anclem4 34985 ftc1anclem5 34986 ftc1anclem6 34987 ftc1anclem7 34988 ftc1anclem8 34989 ftc1anc 34990 extoimad 40564 imo72b2lem0 40565 imo72b2lem2 40567 imo72b2lem1 40570 imo72b2 40574 sblpnf 40691 binomcxplemdvbinom 40734 binomcxplemcvg 40735 binomcxplemdvsum 40736 binomcxplemnotnn0 40737 absfun 41667 cncficcgt0 42220 fourierdlem42 42483 hoicvr 42879 ovolval2lem 42974 ovolval3 42978 |
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