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Mirrors > Home > MPE Home > Th. List > absval | Structured version Visualization version GIF 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 | โข (๐ด โ โ โ (absโ๐ด) = (โโ(๐ด ยท (โโ๐ด)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6897 | . . . 4 โข (๐ฅ = ๐ด โ (โโ๐ฅ) = (โโ๐ด)) | |
2 | oveq12 7429 | . . . 4 โข ((๐ฅ = ๐ด โง (โโ๐ฅ) = (โโ๐ด)) โ (๐ฅ ยท (โโ๐ฅ)) = (๐ด ยท (โโ๐ด))) | |
3 | 1, 2 | mpdan 686 | . . 3 โข (๐ฅ = ๐ด โ (๐ฅ ยท (โโ๐ฅ)) = (๐ด ยท (โโ๐ด))) |
4 | 3 | fveq2d 6901 | . 2 โข (๐ฅ = ๐ด โ (โโ(๐ฅ ยท (โโ๐ฅ))) = (โโ(๐ด ยท (โโ๐ด)))) |
5 | df-abs 15216 | . 2 โข abs = (๐ฅ โ โ โฆ (โโ(๐ฅ ยท (โโ๐ฅ)))) | |
6 | fvex 6910 | . 2 โข (โโ(๐ด ยท (โโ๐ด))) โ V | |
7 | 4, 5, 6 | fvmpt 7005 | 1 โข (๐ด โ โ โ (absโ๐ด) = (โโ(๐ด ยท (โโ๐ด)))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1534 โ wcel 2099 โcfv 6548 (class class class)co 7420 โcc 11137 ยท cmul 11144 โccj 15076 โcsqrt 15213 abscabs 15214 |
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 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 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 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6500 df-fun 6550 df-fv 6556 df-ov 7423 df-abs 15216 |
This theorem is referenced by: absneg 15257 abscl 15258 abscj 15259 absvalsq 15260 absval2 15264 abs0 15265 absi 15266 absge0 15267 absrpcl 15268 absmul 15274 absid 15276 absre 15281 absf 15317 cphabscl 25126 cphipipcj 25141 tcphcphlem2 25177 siii 30676 norm-iii-i 30962 absfico 44591 |
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