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Mirrors > Home > MPE Home > Th. List > nmounbseqi | Structured version Visualization version GIF version |
Description: An unbounded operator determines an unbounded sequence. (Contributed by NM, 11-Jan-2008.) (Revised by Mario Carneiro, 7-Apr-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmoubi.1 | β’ π = (BaseSetβπ) |
nmoubi.y | β’ π = (BaseSetβπ) |
nmoubi.l | β’ πΏ = (normCVβπ) |
nmoubi.m | β’ π = (normCVβπ) |
nmoubi.3 | β’ π = (π normOpOLD π) |
nmoubi.u | β’ π β NrmCVec |
nmoubi.w | β’ π β NrmCVec |
Ref | Expression |
---|---|
nmounbseqi | β’ ((π:πβΆπ β§ (πβπ) = +β) β βπ(π:ββΆπ β§ βπ β β ((πΏβ(πβπ)) β€ 1 β§ π < (πβ(πβ(πβπ)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmoubi.1 | . . . 4 β’ π = (BaseSetβπ) | |
2 | nmoubi.y | . . . 4 β’ π = (BaseSetβπ) | |
3 | nmoubi.l | . . . 4 β’ πΏ = (normCVβπ) | |
4 | nmoubi.m | . . . 4 β’ π = (normCVβπ) | |
5 | nmoubi.3 | . . . 4 β’ π = (π normOpOLD π) | |
6 | nmoubi.u | . . . 4 β’ π β NrmCVec | |
7 | nmoubi.w | . . . 4 β’ π β NrmCVec | |
8 | 1, 2, 3, 4, 5, 6, 7 | nmounbi 29767 | . . 3 β’ (π:πβΆπ β ((πβπ) = +β β βπ β β βπ¦ β π ((πΏβπ¦) β€ 1 β§ π < (πβ(πβπ¦))))) |
9 | 8 | biimpa 478 | . 2 β’ ((π:πβΆπ β§ (πβπ) = +β) β βπ β β βπ¦ β π ((πΏβπ¦) β€ 1 β§ π < (πβ(πβπ¦)))) |
10 | nnre 12168 | . . . 4 β’ (π β β β π β β) | |
11 | 10 | imim1i 63 | . . 3 β’ ((π β β β βπ¦ β π ((πΏβπ¦) β€ 1 β§ π < (πβ(πβπ¦)))) β (π β β β βπ¦ β π ((πΏβπ¦) β€ 1 β§ π < (πβ(πβπ¦))))) |
12 | 11 | ralimi2 3078 | . 2 β’ (βπ β β βπ¦ β π ((πΏβπ¦) β€ 1 β§ π < (πβ(πβπ¦))) β βπ β β βπ¦ β π ((πΏβπ¦) β€ 1 β§ π < (πβ(πβπ¦)))) |
13 | 1 | fvexi 6860 | . . 3 β’ π β V |
14 | nnenom 13894 | . . 3 β’ β β Ο | |
15 | fveq2 6846 | . . . . 5 β’ (π¦ = (πβπ) β (πΏβπ¦) = (πΏβ(πβπ))) | |
16 | 15 | breq1d 5119 | . . . 4 β’ (π¦ = (πβπ) β ((πΏβπ¦) β€ 1 β (πΏβ(πβπ)) β€ 1)) |
17 | 2fveq3 6851 | . . . . 5 β’ (π¦ = (πβπ) β (πβ(πβπ¦)) = (πβ(πβ(πβπ)))) | |
18 | 17 | breq2d 5121 | . . . 4 β’ (π¦ = (πβπ) β (π < (πβ(πβπ¦)) β π < (πβ(πβ(πβπ))))) |
19 | 16, 18 | anbi12d 632 | . . 3 β’ (π¦ = (πβπ) β (((πΏβπ¦) β€ 1 β§ π < (πβ(πβπ¦))) β ((πΏβ(πβπ)) β€ 1 β§ π < (πβ(πβ(πβπ)))))) |
20 | 13, 14, 19 | axcc4 10383 | . 2 β’ (βπ β β βπ¦ β π ((πΏβπ¦) β€ 1 β§ π < (πβ(πβπ¦))) β βπ(π:ββΆπ β§ βπ β β ((πΏβ(πβπ)) β€ 1 β§ π < (πβ(πβ(πβπ)))))) |
21 | 9, 12, 20 | 3syl 18 | 1 β’ ((π:πβΆπ β§ (πβπ) = +β) β βπ(π:ββΆπ β§ βπ β β ((πΏβ(πβπ)) β€ 1 β§ π < (πβ(πβ(πβπ)))))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 βwex 1782 β wcel 2107 βwral 3061 βwrex 3070 class class class wbr 5109 βΆwf 6496 βcfv 6500 (class class class)co 7361 βcr 11058 1c1 11060 +βcpnf 11194 < clt 11197 β€ cle 11198 βcn 12161 NrmCVeccnv 29575 BaseSetcba 29577 normCVcnmcv 29581 normOpOLD cnmoo 29732 |
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 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-inf2 9585 ax-cc 10379 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-sup 9386 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-n0 12422 df-z 12508 df-uz 12772 df-rp 12924 df-seq 13916 df-exp 13977 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-grpo 29484 df-gid 29485 df-ginv 29486 df-ablo 29536 df-vc 29550 df-nv 29583 df-va 29586 df-ba 29587 df-sm 29588 df-0v 29589 df-nmcv 29591 df-nmoo 29736 |
This theorem is referenced by: (None) |
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