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Mirrors > Home > MPE Home > Th. List > hlipgt0 | Structured version Visualization version GIF version |
Description: The inner product of a Hilbert space vector by itself is positive. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hlipgt0.1 | β’ π = (BaseSetβπ) |
hlipgt0.5 | β’ π = (0vecβπ) |
hlipgt0.7 | β’ π = (Β·πOLDβπ) |
Ref | Expression |
---|---|
hlipgt0 | β’ ((π β CHilOLD β§ π΄ β π β§ π΄ β π) β 0 < (π΄ππ΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlnv 29875 | . 2 β’ (π β CHilOLD β π β NrmCVec) | |
2 | hlipgt0.1 | . . . . . 6 β’ π = (BaseSetβπ) | |
3 | eqid 2733 | . . . . . 6 β’ (normCVβπ) = (normCVβπ) | |
4 | 2, 3 | nvcl 29645 | . . . . 5 β’ ((π β NrmCVec β§ π΄ β π) β ((normCVβπ)βπ΄) β β) |
5 | 4 | 3adant3 1133 | . . . 4 β’ ((π β NrmCVec β§ π΄ β π β§ π΄ β π) β ((normCVβπ)βπ΄) β β) |
6 | hlipgt0.5 | . . . . . . . 8 β’ π = (0vecβπ) | |
7 | 2, 6, 3 | nvz 29653 | . . . . . . 7 β’ ((π β NrmCVec β§ π΄ β π) β (((normCVβπ)βπ΄) = 0 β π΄ = π)) |
8 | 7 | biimpd 228 | . . . . . 6 β’ ((π β NrmCVec β§ π΄ β π) β (((normCVβπ)βπ΄) = 0 β π΄ = π)) |
9 | 8 | necon3d 2961 | . . . . 5 β’ ((π β NrmCVec β§ π΄ β π) β (π΄ β π β ((normCVβπ)βπ΄) β 0)) |
10 | 9 | 3impia 1118 | . . . 4 β’ ((π β NrmCVec β§ π΄ β π β§ π΄ β π) β ((normCVβπ)βπ΄) β 0) |
11 | 5, 10 | sqgt0d 14159 | . . 3 β’ ((π β NrmCVec β§ π΄ β π β§ π΄ β π) β 0 < (((normCVβπ)βπ΄)β2)) |
12 | hlipgt0.7 | . . . . 5 β’ π = (Β·πOLDβπ) | |
13 | 2, 3, 12 | ipidsq 29694 | . . . 4 β’ ((π β NrmCVec β§ π΄ β π) β (π΄ππ΄) = (((normCVβπ)βπ΄)β2)) |
14 | 13 | 3adant3 1133 | . . 3 β’ ((π β NrmCVec β§ π΄ β π β§ π΄ β π) β (π΄ππ΄) = (((normCVβπ)βπ΄)β2)) |
15 | 11, 14 | breqtrrd 5134 | . 2 β’ ((π β NrmCVec β§ π΄ β π β§ π΄ β π) β 0 < (π΄ππ΄)) |
16 | 1, 15 | syl3an1 1164 | 1 β’ ((π β CHilOLD β§ π΄ β π β§ π΄ β π) β 0 < (π΄ππ΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2940 class class class wbr 5106 βcfv 6497 (class class class)co 7358 βcr 11055 0cc0 11056 < clt 11194 2c2 12213 βcexp 13973 NrmCVeccnv 29568 BaseSetcba 29570 0veccn0v 29572 normCVcnmcv 29574 Β·πOLDcdip 29684 CHilOLDchlo 29869 |
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-inf2 9582 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-int 4909 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-se 5590 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-isom 6506 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-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-oi 9451 df-card 9880 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-4 12223 df-n0 12419 df-z 12505 df-uz 12769 df-rp 12921 df-fz 13431 df-fzo 13574 df-seq 13913 df-exp 13974 df-hash 14237 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-clim 15376 df-sum 15577 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 df-dip 29685 df-cbn 29847 df-hlo 29870 |
This theorem is referenced by: axhis4-zf 29981 |
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