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Mirrors > Home > MPE Home > Th. List > absabv | Structured version Visualization version GIF version |
Description: The regular absolute value function on the complex numbers is in fact an absolute value under our definition. (Contributed by Mario Carneiro, 4-Dec-2014.) |
Ref | Expression |
---|---|
absabv | β’ abs β (AbsValββfld) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2729 | . . 3 β’ (β€ β (AbsValββfld) = (AbsValββfld)) | |
2 | cnfldbas 21283 | . . . 4 β’ β = (Baseββfld) | |
3 | 2 | a1i 11 | . . 3 β’ (β€ β β = (Baseββfld)) |
4 | cnfldadd 21285 | . . . 4 β’ + = (+gββfld) | |
5 | 4 | a1i 11 | . . 3 β’ (β€ β + = (+gββfld)) |
6 | cnfldmul 21287 | . . . 4 β’ Β· = (.rββfld) | |
7 | 6 | a1i 11 | . . 3 β’ (β€ β Β· = (.rββfld)) |
8 | cnfld0 21320 | . . . 4 β’ 0 = (0gββfld) | |
9 | 8 | a1i 11 | . . 3 β’ (β€ β 0 = (0gββfld)) |
10 | cnring 21318 | . . . 4 β’ βfld β Ring | |
11 | 10 | a1i 11 | . . 3 β’ (β€ β βfld β Ring) |
12 | absf 15317 | . . . 4 β’ abs:ββΆβ | |
13 | 12 | a1i 11 | . . 3 β’ (β€ β abs:ββΆβ) |
14 | abs0 15265 | . . . 4 β’ (absβ0) = 0 | |
15 | 14 | a1i 11 | . . 3 β’ (β€ β (absβ0) = 0) |
16 | absgt0 15304 | . . . . 5 β’ (π₯ β β β (π₯ β 0 β 0 < (absβπ₯))) | |
17 | 16 | biimpa 476 | . . . 4 β’ ((π₯ β β β§ π₯ β 0) β 0 < (absβπ₯)) |
18 | 17 | 3adant1 1128 | . . 3 β’ ((β€ β§ π₯ β β β§ π₯ β 0) β 0 < (absβπ₯)) |
19 | absmul 15274 | . . . . 5 β’ ((π₯ β β β§ π¦ β β) β (absβ(π₯ Β· π¦)) = ((absβπ₯) Β· (absβπ¦))) | |
20 | 19 | ad2ant2r 746 | . . . 4 β’ (((π₯ β β β§ π₯ β 0) β§ (π¦ β β β§ π¦ β 0)) β (absβ(π₯ Β· π¦)) = ((absβπ₯) Β· (absβπ¦))) |
21 | 20 | 3adant1 1128 | . . 3 β’ ((β€ β§ (π₯ β β β§ π₯ β 0) β§ (π¦ β β β§ π¦ β 0)) β (absβ(π₯ Β· π¦)) = ((absβπ₯) Β· (absβπ¦))) |
22 | abstri 15310 | . . . . 5 β’ ((π₯ β β β§ π¦ β β) β (absβ(π₯ + π¦)) β€ ((absβπ₯) + (absβπ¦))) | |
23 | 22 | ad2ant2r 746 | . . . 4 β’ (((π₯ β β β§ π₯ β 0) β§ (π¦ β β β§ π¦ β 0)) β (absβ(π₯ + π¦)) β€ ((absβπ₯) + (absβπ¦))) |
24 | 23 | 3adant1 1128 | . . 3 β’ ((β€ β§ (π₯ β β β§ π₯ β 0) β§ (π¦ β β β§ π¦ β 0)) β (absβ(π₯ + π¦)) β€ ((absβπ₯) + (absβπ¦))) |
25 | 1, 3, 5, 7, 9, 11, 13, 15, 18, 21, 24 | isabvd 20700 | . 2 β’ (β€ β abs β (AbsValββfld)) |
26 | 25 | mptru 1541 | 1 β’ abs β (AbsValββfld) |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 395 = wceq 1534 β€wtru 1535 β wcel 2099 β wne 2937 class class class wbr 5148 βΆwf 6544 βcfv 6548 (class class class)co 7420 βcc 11137 βcr 11138 0cc0 11139 + caddc 11142 Β· cmul 11144 < clt 11279 β€ cle 11280 abscabs 15214 Basecbs 17180 +gcplusg 17233 .rcmulr 17234 0gc0g 17421 Ringcrg 20173 AbsValcabv 20696 βfldccnfld 21279 |
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-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 ax-addf 11218 ax-mulf 11219 |
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 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9466 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-rp 13008 df-ico 13363 df-fz 13518 df-seq 14000 df-exp 14060 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-plusg 17246 df-mulr 17247 df-starv 17248 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-0g 17423 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18893 df-minusg 18894 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-cring 20176 df-abv 20697 df-cnfld 21280 |
This theorem is referenced by: cnnrg 24710 cnindmet 25103 qabsabv 27575 |
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