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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 2727 | . . . . . 6 β’ (*πβπΉ) = (*πβπΉ) | |
6 | ip0l.z | . . . . . 6 β’ π = (0gβπΉ) | |
7 | 1, 2, 3, 4, 5, 6 | isphl 21547 | . . . . 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 1145 | . . . 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 1135 | . . . . 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 3078 | . . . 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 7423 | . . . . . . 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 2729 | . . . . 5 β’ (π₯ = π΄ β ((π₯ , π₯) = π β (π΄ , π΄) = π)) |
15 | eqeq1 2731 | . . . . 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 3606 | . . 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 21555 | . . 3 β’ ((π β PreHil β§ π΄ β π) β ( 0 , π΄) = π) |
20 | oveq1 7421 | . . . 4 β’ (π΄ = 0 β (π΄ , π΄) = ( 0 , π΄)) | |
21 | 20 | eqeq1d 2729 | . . 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 1085 = wceq 1534 β wcel 2099 βwral 3056 β¦ cmpt 5225 βcfv 6542 (class class class)co 7414 Basecbs 17171 *πcstv 17226 Scalarcsca 17227 Β·πcip 17229 0gc0g 17412 *-Ringcsr 20713 LMHom clmhm 20893 LVecclvec 20976 ringLModcrglmod 21046 PreHilcphl 21543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-plusg 17237 df-sca 17240 df-vsca 17241 df-ip 17242 df-0g 17414 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-grp 18884 df-ghm 19159 df-lmod 20734 df-lmhm 20896 df-lvec 20977 df-sra 21047 df-rgmod 21048 df-phl 21545 |
This theorem is referenced by: ip2eq 21572 phlssphl 21578 ocvin 21593 lsmcss 21611 obsne0 21646 cphipeq0 25119 ipcau2 25149 tcphcph 25152 |
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