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Mirrors > Home > ILE Home > Th. List > absval | Unicode version |
Description: The absolute value (modulus) of a complex number. Proposition 10-3.7(a) of [Gleason] p. 133. (Contributed by NM, 27-Jul-1999.) (Revised by Mario Carneiro, 7-Nov-2013.) |
Ref | Expression |
---|---|
absval |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rsqrt 10770 | . . . 4 | |
2 | reex 7754 | . . . . 5 | |
3 | 2 | mptex 5646 | . . . 4 |
4 | 1, 3 | eqeltri 2212 | . . 3 |
5 | id 19 | . . . 4 | |
6 | cjcl 10620 | . . . 4 | |
7 | 5, 6 | mulcld 7786 | . . 3 |
8 | fvexg 5440 | . . 3 | |
9 | 4, 7, 8 | sylancr 410 | . 2 |
10 | fveq2 5421 | . . . . 5 | |
11 | oveq12 5783 | . . . . 5 | |
12 | 10, 11 | mpdan 417 | . . . 4 |
13 | 12 | fveq2d 5425 | . . 3 |
14 | df-abs 10771 | . . 3 | |
15 | 13, 14 | fvmptg 5497 | . 2 |
16 | 9, 15 | mpdan 417 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cvv 2686 class class class wbr 3929 cmpt 3989 cfv 5123 crio 5729 (class class class)co 5774 cc 7618 cr 7619 cc0 7620 cmul 7625 cle 7801 c2 8771 cexp 10292 ccj 10611 csqrt 10768 cabs 10769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-ltxr 7805 df-sub 7935 df-neg 7936 df-reap 8337 df-cj 10614 df-rsqrt 10770 df-abs 10771 |
This theorem is referenced by: absneg 10822 abscl 10823 abscj 10824 absvalsq 10825 absval2 10829 abs0 10830 absi 10831 absge0 10832 absrpclap 10833 absmul 10841 absid 10843 absre 10849 absf 10882 |
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