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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 15190 | . 2 ⊢ abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥)))) | |
2 | absval 15192 | . . 3 ⊢ (𝑥 ∈ ℂ → (abs‘𝑥) = (√‘(𝑥 · (∗‘𝑥)))) | |
3 | abscl 15232 | . . 3 ⊢ (𝑥 ∈ ℂ → (abs‘𝑥) ∈ ℝ) | |
4 | 2, 3 | eqeltrrd 2833 | . 2 ⊢ (𝑥 ∈ ℂ → (√‘(𝑥 · (∗‘𝑥))) ∈ ℝ) |
5 | 1, 4 | fmpti 7113 | 1 ⊢ abs:ℂ⟶ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ℂcc 11114 ℝcr 11115 · cmul 11121 ∗ccj 15050 √csqrt 15187 abscabs 15188 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-sup 9443 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-z 12566 df-uz 12830 df-rp 12982 df-seq 13974 df-exp 14035 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 |
This theorem is referenced by: lo1o1 15483 lo1o12 15484 abscn2 15550 climabs 15555 rlimabs 15560 cnfldds 21158 cnfldfun 21160 cnfldfunALT 21161 cnfldfunALTOLD 21162 absabv 21206 cnmet 24521 cnbl0 24523 cnblcld 24524 cnfldms 24525 cnfldnm 24528 abscncf 24654 cnfldcusp 25118 ovolfsf 25233 ovolctb 25252 iblabslem 25590 iblabs 25591 bddmulibl 25601 dvlip2 25761 c1liplem1 25762 pserulm 26184 psercn2 26185 psercn2OLD 26186 psercnlem2 26187 psercnlem1 26188 psercn 26189 pserdvlem1 26190 pserdvlem2 26191 pserdv 26192 pserdv2 26193 abelth 26204 efif1olem3 26304 efif1olem4 26305 efifo 26307 eff1olem 26308 logcn 26406 efopnlem1 26415 logtayl 26419 cnnv 30212 cnnvg 30213 cnnvs 30215 cnnvnm 30216 cncph 30354 gg-cnfldds 35494 gg-cnfldfun 35496 gg-cnfldfunALT 35497 mblfinlem2 36842 ftc1anclem1 36877 ftc1anclem2 36878 ftc1anclem3 36879 ftc1anclem4 36880 ftc1anclem5 36881 ftc1anclem6 36882 ftc1anclem7 36883 ftc1anclem8 36884 ftc1anc 36885 extoimad 43231 imo72b2lem0 43232 imo72b2lem2 43234 imo72b2lem1 43236 imo72b2 43239 sblpnf 43384 binomcxplemdvbinom 43427 binomcxplemcvg 43428 binomcxplemdvsum 43429 binomcxplemnotnn0 43430 absfun 44371 cncficcgt0 44915 fourierdlem42 45176 hoicvr 45575 ovolval2lem 45670 ovolval3 45674 |
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