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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 15275 | . 2 ⊢ abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥)))) | |
| 2 | absval 15277 | . . 3 ⊢ (𝑥 ∈ ℂ → (abs‘𝑥) = (√‘(𝑥 · (∗‘𝑥)))) | |
| 3 | abscl 15317 | . . 3 ⊢ (𝑥 ∈ ℂ → (abs‘𝑥) ∈ ℝ) | |
| 4 | 2, 3 | eqeltrrd 2842 | . 2 ⊢ (𝑥 ∈ ℂ → (√‘(𝑥 · (∗‘𝑥))) ∈ ℝ) |
| 5 | 1, 4 | fmpti 7132 | 1 ⊢ abs:ℂ⟶ℝ |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 · cmul 11160 ∗ccj 15135 √csqrt 15272 abscabs 15273 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 |
| This theorem is referenced by: lo1o1 15568 lo1o12 15569 abscn2 15635 climabs 15640 rlimabs 15645 cnfldds 21376 cnfldfun 21378 cnfldfunALT 21379 cnflddsOLD 21389 cnfldfunOLD 21391 cnfldfunALTOLD 21392 cnfldfunALTOLDOLD 21393 absabv 21442 cnmet 24792 cnbl0 24794 cnblcld 24795 cnfldms 24796 cnfldnm 24799 abscncf 24927 cnfldcusp 25391 ovolfsf 25506 ovolctb 25525 iblabslem 25863 iblabs 25864 bddmulibl 25874 dvlip2 26034 c1liplem1 26035 pserulm 26465 psercn2 26466 psercn2OLD 26467 psercnlem2 26468 psercnlem1 26469 psercn 26470 pserdvlem1 26471 pserdvlem2 26472 pserdv 26473 pserdv2 26474 abelth 26485 efif1olem3 26586 efif1olem4 26587 efifo 26589 eff1olem 26590 logcn 26689 efopnlem1 26698 logtayl 26702 cnnv 30696 cnnvg 30697 cnnvs 30699 cnnvnm 30700 cncph 30838 mblfinlem2 37665 ftc1anclem1 37700 ftc1anclem2 37701 ftc1anclem3 37702 ftc1anclem4 37703 ftc1anclem5 37704 ftc1anclem6 37705 ftc1anclem7 37706 ftc1anclem8 37707 ftc1anc 37708 absex 42289 extoimad 44177 imo72b2lem0 44178 imo72b2lem2 44180 imo72b2lem1 44182 imo72b2 44185 sblpnf 44329 binomcxplemdvbinom 44372 binomcxplemcvg 44373 binomcxplemdvsum 44374 binomcxplemnotnn0 44375 absfun 45361 cncficcgt0 45903 fourierdlem42 46164 hoicvr 46563 ovolval2lem 46658 ovolval3 46662 |
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