![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ipeq0 | Structured version Visualization version GIF version |
Description: The inner product of a vector with itself is zero iff the vector is zero. Part of Definition 3.1-1 of [Kreyszig] p. 129. (Contributed by NM, 24-Jan-2008.) (Revised by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
phlsrng.f | β’ πΉ = (Scalarβπ) |
phllmhm.h | β’ , = (Β·πβπ) |
phllmhm.v | β’ π = (Baseβπ) |
ip0l.z | β’ π = (0gβπΉ) |
ip0l.o | β’ 0 = (0gβπ) |
Ref | Expression |
---|---|
ipeq0 | β’ ((π β PreHil β§ π΄ β π) β ((π΄ , π΄) = π β π΄ = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | phllmhm.v | . . . . . 6 β’ π = (Baseβπ) | |
2 | phlsrng.f | . . . . . 6 β’ πΉ = (Scalarβπ) | |
3 | phllmhm.h | . . . . . 6 β’ , = (Β·πβπ) | |
4 | ip0l.o | . . . . . 6 β’ 0 = (0gβπ) | |
5 | eqid 2724 | . . . . . 6 β’ (*πβπΉ) = (*πβπΉ) | |
6 | ip0l.z | . . . . . 6 β’ π = (0gβπΉ) | |
7 | 1, 2, 3, 4, 5, 6 | isphl 21489 | . . . . 5 β’ (π β PreHil β (π β LVec β§ πΉ β *-Ring β§ βπ₯ β π ((π¦ β π β¦ (π¦ , π₯)) β (π LMHom (ringLModβπΉ)) β§ ((π₯ , π₯) = π β π₯ = 0 ) β§ βπ¦ β π ((*πβπΉ)β(π₯ , π¦)) = (π¦ , π₯)))) |
8 | 7 | simp3bi 1144 | . . . 4 β’ (π β PreHil β βπ₯ β π ((π¦ β π β¦ (π¦ , π₯)) β (π LMHom (ringLModβπΉ)) β§ ((π₯ , π₯) = π β π₯ = 0 ) β§ βπ¦ β π ((*πβπΉ)β(π₯ , π¦)) = (π¦ , π₯))) |
9 | simp2 1134 | . . . . 5 β’ (((π¦ β π β¦ (π¦ , π₯)) β (π LMHom (ringLModβπΉ)) β§ ((π₯ , π₯) = π β π₯ = 0 ) β§ βπ¦ β π ((*πβπΉ)β(π₯ , π¦)) = (π¦ , π₯)) β ((π₯ , π₯) = π β π₯ = 0 )) | |
10 | 9 | ralimi 3075 | . . . 4 β’ (βπ₯ β π ((π¦ β π β¦ (π¦ , π₯)) β (π LMHom (ringLModβπΉ)) β§ ((π₯ , π₯) = π β π₯ = 0 ) β§ βπ¦ β π ((*πβπΉ)β(π₯ , π¦)) = (π¦ , π₯)) β βπ₯ β π ((π₯ , π₯) = π β π₯ = 0 )) |
11 | 8, 10 | syl 17 | . . 3 β’ (π β PreHil β βπ₯ β π ((π₯ , π₯) = π β π₯ = 0 )) |
12 | oveq12 7410 | . . . . . . 7 β’ ((π₯ = π΄ β§ π₯ = π΄) β (π₯ , π₯) = (π΄ , π΄)) | |
13 | 12 | anidms 566 | . . . . . 6 β’ (π₯ = π΄ β (π₯ , π₯) = (π΄ , π΄)) |
14 | 13 | eqeq1d 2726 | . . . . 5 β’ (π₯ = π΄ β ((π₯ , π₯) = π β (π΄ , π΄) = π)) |
15 | eqeq1 2728 | . . . . 5 β’ (π₯ = π΄ β (π₯ = 0 β π΄ = 0 )) | |
16 | 14, 15 | imbi12d 344 | . . . 4 β’ (π₯ = π΄ β (((π₯ , π₯) = π β π₯ = 0 ) β ((π΄ , π΄) = π β π΄ = 0 ))) |
17 | 16 | rspccva 3603 | . . 3 β’ ((βπ₯ β π ((π₯ , π₯) = π β π₯ = 0 ) β§ π΄ β π) β ((π΄ , π΄) = π β π΄ = 0 )) |
18 | 11, 17 | sylan 579 | . 2 β’ ((π β PreHil β§ π΄ β π) β ((π΄ , π΄) = π β π΄ = 0 )) |
19 | 2, 3, 1, 6, 4 | ip0l 21497 | . . 3 β’ ((π β PreHil β§ π΄ β π) β ( 0 , π΄) = π) |
20 | oveq1 7408 | . . . 4 β’ (π΄ = 0 β (π΄ , π΄) = ( 0 , π΄)) | |
21 | 20 | eqeq1d 2726 | . . 3 β’ (π΄ = 0 β ((π΄ , π΄) = π β ( 0 , π΄) = π)) |
22 | 19, 21 | syl5ibrcom 246 | . 2 β’ ((π β PreHil β§ π΄ β π) β (π΄ = 0 β (π΄ , π΄) = π)) |
23 | 18, 22 | impbid 211 | 1 β’ ((π β PreHil β§ π΄ β π) β ((π΄ , π΄) = π β π΄ = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3053 β¦ cmpt 5221 βcfv 6533 (class class class)co 7401 Basecbs 17143 *πcstv 17198 Scalarcsca 17199 Β·πcip 17201 0gc0g 17384 *-Ringcsr 20677 LMHom clmhm 20857 LVecclvec 20940 ringLModcrglmod 21010 PreHilcphl 21485 |
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 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-plusg 17209 df-sca 17212 df-vsca 17213 df-ip 17214 df-0g 17386 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-grp 18856 df-ghm 19129 df-lmod 20698 df-lmhm 20860 df-lvec 20941 df-sra 21011 df-rgmod 21012 df-phl 21487 |
This theorem is referenced by: ip2eq 21514 phlssphl 21520 ocvin 21535 lsmcss 21553 obsne0 21588 cphipeq0 25054 ipcau2 25084 tcphcph 25087 |
Copyright terms: Public domain | W3C validator |