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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 6847 | . . . 4 โข (๐ฅ = ๐ด โ (โโ๐ฅ) = (โโ๐ด)) | |
2 | oveq12 7371 | . . . 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 6851 | . 2 โข (๐ฅ = ๐ด โ (โโ(๐ฅ ยท (โโ๐ฅ))) = (โโ(๐ด ยท (โโ๐ด)))) |
5 | df-abs 15128 | . 2 โข abs = (๐ฅ โ โ โฆ (โโ(๐ฅ ยท (โโ๐ฅ)))) | |
6 | fvex 6860 | . 2 โข (โโ(๐ด ยท (โโ๐ด))) โ V | |
7 | 4, 5, 6 | fvmpt 6953 | 1 โข (๐ด โ โ โ (absโ๐ด) = (โโ(๐ด ยท (โโ๐ด)))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1542 โ wcel 2107 โcfv 6501 (class class class)co 7362 โcc 11056 ยท cmul 11063 โccj 14988 โcsqrt 15125 abscabs 15126 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6453 df-fun 6503 df-fv 6509 df-ov 7365 df-abs 15128 |
This theorem is referenced by: absneg 15169 abscl 15170 abscj 15171 absvalsq 15172 absval2 15176 abs0 15177 absi 15178 absge0 15179 absrpcl 15180 absmul 15186 absid 15188 absre 15193 absf 15229 cphabscl 24565 cphipipcj 24580 tcphcphlem2 24616 siii 29837 norm-iii-i 30123 absfico 43513 |
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