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Mirrors > Home > MPE Home > Th. List > nmosetn0 | Structured version Visualization version GIF version |
Description: The set in the supremum of the operator norm definition df-nmoo 30583 is nonempty. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmosetn0.1 | β’ π = (BaseSetβπ) |
nmosetn0.5 | β’ π = (0vecβπ) |
nmosetn0.4 | β’ π = (normCVβπ) |
Ref | Expression |
---|---|
nmosetn0 | β’ (π β NrmCVec β (πβ(πβπ)) β {π₯ β£ βπ¦ β π ((πβπ¦) β€ 1 β§ π₯ = (πβ(πβπ¦)))}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmosetn0.1 | . . . 4 β’ π = (BaseSetβπ) | |
2 | nmosetn0.5 | . . . 4 β’ π = (0vecβπ) | |
3 | 1, 2 | nvzcl 30472 | . . 3 β’ (π β NrmCVec β π β π) |
4 | nmosetn0.4 | . . . . . 6 β’ π = (normCVβπ) | |
5 | 2, 4 | nvz0 30506 | . . . . 5 β’ (π β NrmCVec β (πβπ) = 0) |
6 | 0le1 11777 | . . . . 5 β’ 0 β€ 1 | |
7 | 5, 6 | eqbrtrdi 5191 | . . . 4 β’ (π β NrmCVec β (πβπ) β€ 1) |
8 | eqid 2728 | . . . 4 β’ (πβ(πβπ)) = (πβ(πβπ)) | |
9 | 7, 8 | jctir 519 | . . 3 β’ (π β NrmCVec β ((πβπ) β€ 1 β§ (πβ(πβπ)) = (πβ(πβπ)))) |
10 | fveq2 6902 | . . . . . 6 β’ (π¦ = π β (πβπ¦) = (πβπ)) | |
11 | 10 | breq1d 5162 | . . . . 5 β’ (π¦ = π β ((πβπ¦) β€ 1 β (πβπ) β€ 1)) |
12 | 2fveq3 6907 | . . . . . 6 β’ (π¦ = π β (πβ(πβπ¦)) = (πβ(πβπ))) | |
13 | 12 | eqeq2d 2739 | . . . . 5 β’ (π¦ = π β ((πβ(πβπ)) = (πβ(πβπ¦)) β (πβ(πβπ)) = (πβ(πβπ)))) |
14 | 11, 13 | anbi12d 630 | . . . 4 β’ (π¦ = π β (((πβπ¦) β€ 1 β§ (πβ(πβπ)) = (πβ(πβπ¦))) β ((πβπ) β€ 1 β§ (πβ(πβπ)) = (πβ(πβπ))))) |
15 | 14 | rspcev 3611 | . . 3 β’ ((π β π β§ ((πβπ) β€ 1 β§ (πβ(πβπ)) = (πβ(πβπ)))) β βπ¦ β π ((πβπ¦) β€ 1 β§ (πβ(πβπ)) = (πβ(πβπ¦)))) |
16 | 3, 9, 15 | syl2anc 582 | . 2 β’ (π β NrmCVec β βπ¦ β π ((πβπ¦) β€ 1 β§ (πβ(πβπ)) = (πβ(πβπ¦)))) |
17 | fvex 6915 | . . 3 β’ (πβ(πβπ)) β V | |
18 | eqeq1 2732 | . . . . 5 β’ (π₯ = (πβ(πβπ)) β (π₯ = (πβ(πβπ¦)) β (πβ(πβπ)) = (πβ(πβπ¦)))) | |
19 | 18 | anbi2d 628 | . . . 4 β’ (π₯ = (πβ(πβπ)) β (((πβπ¦) β€ 1 β§ π₯ = (πβ(πβπ¦))) β ((πβπ¦) β€ 1 β§ (πβ(πβπ)) = (πβ(πβπ¦))))) |
20 | 19 | rexbidv 3176 | . . 3 β’ (π₯ = (πβ(πβπ)) β (βπ¦ β π ((πβπ¦) β€ 1 β§ π₯ = (πβ(πβπ¦))) β βπ¦ β π ((πβπ¦) β€ 1 β§ (πβ(πβπ)) = (πβ(πβπ¦))))) |
21 | 17, 20 | elab 3669 | . 2 β’ ((πβ(πβπ)) β {π₯ β£ βπ¦ β π ((πβπ¦) β€ 1 β§ π₯ = (πβ(πβπ¦)))} β βπ¦ β π ((πβπ¦) β€ 1 β§ (πβ(πβπ)) = (πβ(πβπ¦)))) |
22 | 16, 21 | sylibr 233 | 1 β’ (π β NrmCVec β (πβ(πβπ)) β {π₯ β£ βπ¦ β π ((πβπ¦) β€ 1 β§ π₯ = (πβ(πβπ¦)))}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 {cab 2705 βwrex 3067 class class class wbr 5152 βcfv 6553 0cc0 11148 1c1 11149 β€ cle 11289 NrmCVeccnv 30422 BaseSetcba 30424 0veccn0v 30426 normCVcnmcv 30428 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 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 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-sup 9475 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-3 12316 df-n0 12513 df-z 12599 df-uz 12863 df-rp 13017 df-seq 14009 df-exp 14069 df-cj 15088 df-re 15089 df-im 15090 df-sqrt 15224 df-abs 15225 df-grpo 30331 df-gid 30332 df-ginv 30333 df-ablo 30383 df-vc 30397 df-nv 30430 df-va 30433 df-ba 30434 df-sm 30435 df-0v 30436 df-nmcv 30438 |
This theorem is referenced by: nmooge0 30605 nmorepnf 30606 |
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