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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 6884 | . . . 4 โข (๐ฅ = ๐ด โ (โโ๐ฅ) = (โโ๐ด)) | |
2 | oveq12 7413 | . . . 4 โข ((๐ฅ = ๐ด โง (โโ๐ฅ) = (โโ๐ด)) โ (๐ฅ ยท (โโ๐ฅ)) = (๐ด ยท (โโ๐ด))) | |
3 | 1, 2 | mpdan 684 | . . 3 โข (๐ฅ = ๐ด โ (๐ฅ ยท (โโ๐ฅ)) = (๐ด ยท (โโ๐ด))) |
4 | 3 | fveq2d 6888 | . 2 โข (๐ฅ = ๐ด โ (โโ(๐ฅ ยท (โโ๐ฅ))) = (โโ(๐ด ยท (โโ๐ด)))) |
5 | df-abs 15187 | . 2 โข abs = (๐ฅ โ โ โฆ (โโ(๐ฅ ยท (โโ๐ฅ)))) | |
6 | fvex 6897 | . 2 โข (โโ(๐ด ยท (โโ๐ด))) โ V | |
7 | 4, 5, 6 | fvmpt 6991 | 1 โข (๐ด โ โ โ (absโ๐ด) = (โโ(๐ด ยท (โโ๐ด)))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 โcfv 6536 (class class class)co 7404 โcc 11107 ยท cmul 11114 โccj 15047 โcsqrt 15184 abscabs 15185 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fv 6544 df-ov 7407 df-abs 15187 |
This theorem is referenced by: absneg 15228 abscl 15229 abscj 15230 absvalsq 15231 absval2 15235 abs0 15236 absi 15237 absge0 15238 absrpcl 15239 absmul 15245 absid 15247 absre 15252 absf 15288 cphabscl 25064 cphipipcj 25079 tcphcphlem2 25115 siii 30611 norm-iii-i 30897 absfico 44470 |
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