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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iooabslt | Structured version Visualization version GIF version |
Description: An upper bound for the distance from the center of an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
iooabslt.1 | β’ (π β π΄ β β) |
iooabslt.2 | β’ (π β π΅ β β) |
iooabslt.3 | β’ (π β πΆ β ((π΄ β π΅)(,)(π΄ + π΅))) |
Ref | Expression |
---|---|
iooabslt | β’ (π β (absβ(π΄ β πΆ)) < π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iooabslt.1 | . . . 4 β’ (π β π΄ β β) | |
2 | 1 | recnd 11184 | . . 3 β’ (π β π΄ β β) |
3 | iooabslt.3 | . . . . 5 β’ (π β πΆ β ((π΄ β π΅)(,)(π΄ + π΅))) | |
4 | elioore 13295 | . . . . 5 β’ (πΆ β ((π΄ β π΅)(,)(π΄ + π΅)) β πΆ β β) | |
5 | 3, 4 | syl 17 | . . . 4 β’ (π β πΆ β β) |
6 | 5 | recnd 11184 | . . 3 β’ (π β πΆ β β) |
7 | eqid 2737 | . . . 4 β’ (abs β β ) = (abs β β ) | |
8 | 7 | cnmetdval 24137 | . . 3 β’ ((π΄ β β β§ πΆ β β) β (π΄(abs β β )πΆ) = (absβ(π΄ β πΆ))) |
9 | 2, 6, 8 | syl2anc 585 | . 2 β’ (π β (π΄(abs β β )πΆ) = (absβ(π΄ β πΆ))) |
10 | iooabslt.2 | . . . . . . . . 9 β’ (π β π΅ β β) | |
11 | eqid 2737 | . . . . . . . . . 10 β’ ((abs β β ) βΎ (β Γ β)) = ((abs β β ) βΎ (β Γ β)) | |
12 | 11 | bl2ioo 24158 | . . . . . . . . 9 β’ ((π΄ β β β§ π΅ β β) β (π΄(ballβ((abs β β ) βΎ (β Γ β)))π΅) = ((π΄ β π΅)(,)(π΄ + π΅))) |
13 | 1, 10, 12 | syl2anc 585 | . . . . . . . 8 β’ (π β (π΄(ballβ((abs β β ) βΎ (β Γ β)))π΅) = ((π΄ β π΅)(,)(π΄ + π΅))) |
14 | 3, 13 | eleqtrrd 2841 | . . . . . . 7 β’ (π β πΆ β (π΄(ballβ((abs β β ) βΎ (β Γ β)))π΅)) |
15 | cnxmet 24139 | . . . . . . . . 9 β’ (abs β β ) β (βMetββ) | |
16 | 15 | a1i 11 | . . . . . . . 8 β’ (π β (abs β β ) β (βMetββ)) |
17 | 2, 1 | elind 4155 | . . . . . . . 8 β’ (π β π΄ β (β β© β)) |
18 | 10 | rexrd 11206 | . . . . . . . 8 β’ (π β π΅ β β*) |
19 | 11 | blres 23787 | . . . . . . . 8 β’ (((abs β β ) β (βMetββ) β§ π΄ β (β β© β) β§ π΅ β β*) β (π΄(ballβ((abs β β ) βΎ (β Γ β)))π΅) = ((π΄(ballβ(abs β β ))π΅) β© β)) |
20 | 16, 17, 18, 19 | syl3anc 1372 | . . . . . . 7 β’ (π β (π΄(ballβ((abs β β ) βΎ (β Γ β)))π΅) = ((π΄(ballβ(abs β β ))π΅) β© β)) |
21 | 14, 20 | eleqtrd 2840 | . . . . . 6 β’ (π β πΆ β ((π΄(ballβ(abs β β ))π΅) β© β)) |
22 | elin 3927 | . . . . . 6 β’ (πΆ β ((π΄(ballβ(abs β β ))π΅) β© β) β (πΆ β (π΄(ballβ(abs β β ))π΅) β§ πΆ β β)) | |
23 | 21, 22 | sylib 217 | . . . . 5 β’ (π β (πΆ β (π΄(ballβ(abs β β ))π΅) β§ πΆ β β)) |
24 | 23 | simpld 496 | . . . 4 β’ (π β πΆ β (π΄(ballβ(abs β β ))π΅)) |
25 | elbl 23744 | . . . . 5 β’ (((abs β β ) β (βMetββ) β§ π΄ β β β§ π΅ β β*) β (πΆ β (π΄(ballβ(abs β β ))π΅) β (πΆ β β β§ (π΄(abs β β )πΆ) < π΅))) | |
26 | 16, 2, 18, 25 | syl3anc 1372 | . . . 4 β’ (π β (πΆ β (π΄(ballβ(abs β β ))π΅) β (πΆ β β β§ (π΄(abs β β )πΆ) < π΅))) |
27 | 24, 26 | mpbid 231 | . . 3 β’ (π β (πΆ β β β§ (π΄(abs β β )πΆ) < π΅)) |
28 | 27 | simprd 497 | . 2 β’ (π β (π΄(abs β β )πΆ) < π΅) |
29 | 9, 28 | eqbrtrrd 5130 | 1 β’ (π β (absβ(π΄ β πΆ)) < π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 β© cin 3910 class class class wbr 5106 Γ cxp 5632 βΎ cres 5636 β ccom 5638 βcfv 6497 (class class class)co 7358 βcc 11050 βcr 11051 + caddc 11055 β*cxr 11189 < clt 11190 β cmin 11386 (,)cioo 13265 abscabs 15120 βMetcxmet 20784 ballcbl 20786 |
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 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-pre-sup 11130 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8649 df-map 8768 df-en 8885 df-dom 8886 df-sdom 8887 df-sup 9379 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-2 12217 df-3 12218 df-n0 12415 df-z 12501 df-uz 12765 df-rp 12917 df-xadd 13035 df-ioo 13269 df-seq 13908 df-exp 13969 df-cj 14985 df-re 14986 df-im 14987 df-sqrt 15121 df-abs 15122 df-psmet 20791 df-xmet 20792 df-met 20793 df-bl 20794 |
This theorem is referenced by: lptre2pt 43888 |
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