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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 29729 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 29618 | . . 3 β’ (π β NrmCVec β π β π) |
4 | nmosetn0.4 | . . . . . 6 β’ π = (normCVβπ) | |
5 | 2, 4 | nvz0 29652 | . . . . 5 β’ (π β NrmCVec β (πβπ) = 0) |
6 | 0le1 11683 | . . . . 5 β’ 0 β€ 1 | |
7 | 5, 6 | eqbrtrdi 5145 | . . . 4 β’ (π β NrmCVec β (πβπ) β€ 1) |
8 | eqid 2733 | . . . 4 β’ (πβ(πβπ)) = (πβ(πβπ)) | |
9 | 7, 8 | jctir 522 | . . 3 β’ (π β NrmCVec β ((πβπ) β€ 1 β§ (πβ(πβπ)) = (πβ(πβπ)))) |
10 | fveq2 6843 | . . . . . 6 β’ (π¦ = π β (πβπ¦) = (πβπ)) | |
11 | 10 | breq1d 5116 | . . . . 5 β’ (π¦ = π β ((πβπ¦) β€ 1 β (πβπ) β€ 1)) |
12 | 2fveq3 6848 | . . . . . 6 β’ (π¦ = π β (πβ(πβπ¦)) = (πβ(πβπ))) | |
13 | 12 | eqeq2d 2744 | . . . . 5 β’ (π¦ = π β ((πβ(πβπ)) = (πβ(πβπ¦)) β (πβ(πβπ)) = (πβ(πβπ)))) |
14 | 11, 13 | anbi12d 632 | . . . 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 3580 | . . 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 585 | . 2 β’ (π β NrmCVec β βπ¦ β π ((πβπ¦) β€ 1 β§ (πβ(πβπ)) = (πβ(πβπ¦)))) |
17 | fvex 6856 | . . 3 β’ (πβ(πβπ)) β V | |
18 | eqeq1 2737 | . . . . 5 β’ (π₯ = (πβ(πβπ)) β (π₯ = (πβ(πβπ¦)) β (πβ(πβπ)) = (πβ(πβπ¦)))) | |
19 | 18 | anbi2d 630 | . . . 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 3631 | . 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 397 = wceq 1542 β wcel 2107 {cab 2710 βwrex 3070 class class class wbr 5106 βcfv 6497 0cc0 11056 1c1 11057 β€ cle 11195 NrmCVeccnv 29568 BaseSetcba 29570 0veccn0v 29572 normCVcnmcv 29574 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
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 3446 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 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-sup 9383 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-n0 12419 df-z 12505 df-uz 12769 df-rp 12921 df-seq 13913 df-exp 13974 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-grpo 29477 df-gid 29478 df-ginv 29479 df-ablo 29529 df-vc 29543 df-nv 29576 df-va 29579 df-ba 29580 df-sm 29581 df-0v 29582 df-nmcv 29584 |
This theorem is referenced by: nmooge0 29751 nmorepnf 29752 |
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