![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2727 | . . 3 β’ (β€ β (AbsValββfld) = (AbsValββfld)) | |
2 | cnfldbas 21240 | . . . 4 β’ β = (Baseββfld) | |
3 | 2 | a1i 11 | . . 3 β’ (β€ β β = (Baseββfld)) |
4 | cnfldadd 21242 | . . . 4 β’ + = (+gββfld) | |
5 | 4 | a1i 11 | . . 3 β’ (β€ β + = (+gββfld)) |
6 | cnfldmul 21244 | . . . 4 β’ Β· = (.rββfld) | |
7 | 6 | a1i 11 | . . 3 β’ (β€ β Β· = (.rββfld)) |
8 | cnfld0 21277 | . . . 4 β’ 0 = (0gββfld) | |
9 | 8 | a1i 11 | . . 3 β’ (β€ β 0 = (0gββfld)) |
10 | cnring 21275 | . . . 4 β’ βfld β Ring | |
11 | 10 | a1i 11 | . . 3 β’ (β€ β βfld β Ring) |
12 | absf 15288 | . . . 4 β’ abs:ββΆβ | |
13 | 12 | a1i 11 | . . 3 β’ (β€ β abs:ββΆβ) |
14 | abs0 15236 | . . . 4 β’ (absβ0) = 0 | |
15 | 14 | a1i 11 | . . 3 β’ (β€ β (absβ0) = 0) |
16 | absgt0 15275 | . . . . 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 1127 | . . 3 β’ ((β€ β§ π₯ β β β§ π₯ β 0) β 0 < (absβπ₯)) |
19 | absmul 15245 | . . . . 5 β’ ((π₯ β β β§ π¦ β β) β (absβ(π₯ Β· π¦)) = ((absβπ₯) Β· (absβπ¦))) | |
20 | 19 | ad2ant2r 744 | . . . 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 1127 | . . 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 15281 | . . . . 5 β’ ((π₯ β β β§ π¦ β β) β (absβ(π₯ + π¦)) β€ ((absβπ₯) + (absβπ¦))) | |
23 | 22 | ad2ant2r 744 | . . . 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 1127 | . . 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 20661 | . 2 β’ (β€ β abs β (AbsValββfld)) |
26 | 25 | mptru 1540 | 1 β’ abs β (AbsValββfld) |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 395 = wceq 1533 β€wtru 1534 β wcel 2098 β wne 2934 class class class wbr 5141 βΆwf 6532 βcfv 6536 (class class class)co 7404 βcc 11107 βcr 11108 0cc0 11109 + caddc 11112 Β· cmul 11114 < clt 11249 β€ cle 11250 abscabs 15185 Basecbs 17151 +gcplusg 17204 .rcmulr 17205 0gc0g 17392 Ringcrg 20136 AbsValcabv 20657 βfldccnfld 21236 |
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-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 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-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-rp 12978 df-ico 13333 df-fz 13488 df-seq 13970 df-exp 14031 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-mulr 17218 df-starv 17219 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-minusg 18865 df-cmn 19700 df-abl 19701 df-mgp 20038 df-rng 20056 df-ur 20085 df-ring 20138 df-cring 20139 df-abv 20658 df-cnfld 21237 |
This theorem is referenced by: cnnrg 24648 cnindmet 25041 qabsabv 27513 |
Copyright terms: Public domain | W3C validator |