absf - Metamath Proof Explorer

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Theorem absf 15310

Description: Mapping domain and codomain of the absolute value function. (Contributed by NM, 30-Aug-2007.) (Revised by Mario Carneiro, 7-Nov-2013.)

**Assertion**
Ref	Expression
absf	⊢ abs:ℂ⟶ℝ

Proof of Theorem absf Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-abs 15209	. 2 ⊢ abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥))))
2		absval 15211	. . 3 ⊢ (𝑥 ∈ ℂ → (abs‘𝑥) = (√‘(𝑥 · (∗‘𝑥))))
3		abscl 15251	. . 3 ⊢ (𝑥 ∈ ℂ → (abs‘𝑥) ∈ ℝ)
4	2, 3	eqeltrrd 2830	. 2 ⊢ (𝑥 ∈ ℂ → (√‘(𝑥 · (∗‘𝑥))) ∈ ℝ)
5	1, 4	fmpti 7116	1 ⊢ abs:ℂ⟶ℝ

Colors of variables: wff setvar class

Syntax hints: ∈ wcel 2099 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ℂcc 11130 ℝcr 11131 · cmul 11137 ∗ccj 15069 √csqrt 15206 abscabs 15207

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-sup 9459 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-n0 12497 df-z 12583 df-uz 12847 df-rp 13001 df-seq 13993 df-exp 14053 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209

This theorem is referenced by: lo1o1 15502 lo1o12 15503 abscn2 15569 climabs 15574 rlimabs 15579 cnfldds 21284 cnfldfun 21286 cnfldfunALT 21287 cnflddsOLD 21297 cnfldfunOLD 21299 cnfldfunALTOLD 21300 cnfldfunALTOLDOLD 21301 absabv 21350 cnmet 24681 cnbl0 24683 cnblcld 24684 cnfldms 24685 cnfldnm 24688 abscncf 24814 cnfldcusp 25278 ovolfsf 25393 ovolctb 25412 iblabslem 25750 iblabs 25751 bddmulibl 25761 dvlip2 25921 c1liplem1 25922 pserulm 26351 psercn2 26352 psercn2OLD 26353 psercnlem2 26354 psercnlem1 26355 psercn 26356 pserdvlem1 26357 pserdvlem2 26358 pserdv 26359 pserdv2 26360 abelth 26371 efif1olem3 26471 efif1olem4 26472 efifo 26474 eff1olem 26475 logcn 26574 efopnlem1 26583 logtayl 26587 cnnv 30480 cnnvg 30481 cnnvs 30483 cnnvnm 30484 cncph 30622 mblfinlem2 37125 ftc1anclem1 37160 ftc1anclem2 37161 ftc1anclem3 37162 ftc1anclem4 37163 ftc1anclem5 37164 ftc1anclem6 37165 ftc1anclem7 37166 ftc1anclem8 37167 ftc1anc 37168 extoimad 43588 imo72b2lem0 43589 imo72b2lem2 43591 imo72b2lem1 43593 imo72b2 43596 sblpnf 43741 binomcxplemdvbinom 43784 binomcxplemcvg 43785 binomcxplemdvsum 43786 binomcxplemnotnn0 43787 absfun 44726 cncficcgt0 45270 fourierdlem42 45531 hoicvr 45930 ovolval2lem 46025 ovolval3 46029

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