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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 15271 | . 2 ⊢ abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥)))) | |
2 | absval 15273 | . . 3 ⊢ (𝑥 ∈ ℂ → (abs‘𝑥) = (√‘(𝑥 · (∗‘𝑥)))) | |
3 | abscl 15313 | . . 3 ⊢ (𝑥 ∈ ℂ → (abs‘𝑥) ∈ ℝ) | |
4 | 2, 3 | eqeltrrd 2839 | . 2 ⊢ (𝑥 ∈ ℂ → (√‘(𝑥 · (∗‘𝑥))) ∈ ℝ) |
5 | 1, 4 | fmpti 7131 | 1 ⊢ abs:ℂ⟶ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 · cmul 11157 ∗ccj 15131 √csqrt 15268 abscabs 15269 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 |
This theorem is referenced by: lo1o1 15564 lo1o12 15565 abscn2 15631 climabs 15636 rlimabs 15641 cnfldds 21393 cnfldfun 21395 cnfldfunALT 21396 cnflddsOLD 21406 cnfldfunOLD 21408 cnfldfunALTOLD 21409 cnfldfunALTOLDOLD 21410 absabv 21459 cnmet 24807 cnbl0 24809 cnblcld 24810 cnfldms 24811 cnfldnm 24814 abscncf 24940 cnfldcusp 25404 ovolfsf 25519 ovolctb 25538 iblabslem 25877 iblabs 25878 bddmulibl 25888 dvlip2 26048 c1liplem1 26049 pserulm 26479 psercn2 26480 psercn2OLD 26481 psercnlem2 26482 psercnlem1 26483 psercn 26484 pserdvlem1 26485 pserdvlem2 26486 pserdv 26487 pserdv2 26488 abelth 26499 efif1olem3 26600 efif1olem4 26601 efifo 26603 eff1olem 26604 logcn 26703 efopnlem1 26712 logtayl 26716 cnnv 30705 cnnvg 30706 cnnvs 30708 cnnvnm 30709 cncph 30847 mblfinlem2 37644 ftc1anclem1 37679 ftc1anclem2 37680 ftc1anclem3 37681 ftc1anclem4 37682 ftc1anclem5 37683 ftc1anclem6 37684 ftc1anclem7 37685 ftc1anclem8 37686 ftc1anc 37687 absex 42267 extoimad 44153 imo72b2lem0 44154 imo72b2lem2 44156 imo72b2lem1 44158 imo72b2 44161 sblpnf 44305 binomcxplemdvbinom 44348 binomcxplemcvg 44349 binomcxplemdvsum 44350 binomcxplemnotnn0 44351 absfun 45299 cncficcgt0 45843 fourierdlem42 46104 hoicvr 46503 ovolval2lem 46598 ovolval3 46602 |
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