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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 30507 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 30396 | . . 3 β’ (π β NrmCVec β π β π) |
4 | nmosetn0.4 | . . . . . 6 β’ π = (normCVβπ) | |
5 | 2, 4 | nvz0 30430 | . . . . 5 β’ (π β NrmCVec β (πβπ) = 0) |
6 | 0le1 11741 | . . . . 5 β’ 0 β€ 1 | |
7 | 5, 6 | eqbrtrdi 5180 | . . . 4 β’ (π β NrmCVec β (πβπ) β€ 1) |
8 | eqid 2726 | . . . 4 β’ (πβ(πβπ)) = (πβ(πβπ)) | |
9 | 7, 8 | jctir 520 | . . 3 β’ (π β NrmCVec β ((πβπ) β€ 1 β§ (πβ(πβπ)) = (πβ(πβπ)))) |
10 | fveq2 6885 | . . . . . 6 β’ (π¦ = π β (πβπ¦) = (πβπ)) | |
11 | 10 | breq1d 5151 | . . . . 5 β’ (π¦ = π β ((πβπ¦) β€ 1 β (πβπ) β€ 1)) |
12 | 2fveq3 6890 | . . . . . 6 β’ (π¦ = π β (πβ(πβπ¦)) = (πβ(πβπ))) | |
13 | 12 | eqeq2d 2737 | . . . . 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 3606 | . . 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 583 | . 2 β’ (π β NrmCVec β βπ¦ β π ((πβπ¦) β€ 1 β§ (πβ(πβπ)) = (πβ(πβπ¦)))) |
17 | fvex 6898 | . . 3 β’ (πβ(πβπ)) β V | |
18 | eqeq1 2730 | . . . . 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 3172 | . . 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 3663 | . 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 395 = wceq 1533 β wcel 2098 {cab 2703 βwrex 3064 class class class wbr 5141 βcfv 6537 0cc0 11112 1c1 11113 β€ cle 11253 NrmCVeccnv 30346 BaseSetcba 30348 0veccn0v 30350 normCVcnmcv 30352 |
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-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
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-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 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-seq 13973 df-exp 14033 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-grpo 30255 df-gid 30256 df-ginv 30257 df-ablo 30307 df-vc 30321 df-nv 30354 df-va 30357 df-ba 30358 df-sm 30359 df-0v 30360 df-nmcv 30362 |
This theorem is referenced by: nmooge0 30529 nmorepnf 30530 |
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