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Mirrors > Home > HSE Home > Th. List > hhip | Structured version Visualization version GIF version |
Description: The inner product operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hhnv.1 | β’ π = β¨β¨ +β , Β·β β©, normββ© |
Ref | Expression |
---|---|
hhip | β’ Β·ih = (Β·πOLDβπ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | polid 30143 | . . . 4 β’ ((π₯ β β β§ π¦ β β) β (π₯ Β·ih π¦) = (((((normββ(π₯ +β π¦))β2) β ((normββ(π₯ ββ π¦))β2)) + (i Β· (((normββ(π₯ +β (i Β·β π¦)))β2) β ((normββ(π₯ ββ (i Β·β π¦)))β2)))) / 4)) | |
2 | hhnv.1 | . . . . . 6 β’ π = β¨β¨ +β , Β·β β©, normββ© | |
3 | 2 | hhnv 30149 | . . . . 5 β’ π β NrmCVec |
4 | 2 | hhba 30151 | . . . . . 6 β’ β = (BaseSetβπ) |
5 | 2 | hhva 30150 | . . . . . 6 β’ +β = ( +π£ βπ) |
6 | 2 | hhsm 30153 | . . . . . 6 β’ Β·β = ( Β·π OLD βπ) |
7 | 2 | hhnm 30155 | . . . . . 6 β’ normβ = (normCVβπ) |
8 | eqid 2737 | . . . . . 6 β’ (Β·πOLDβπ) = (Β·πOLDβπ) | |
9 | 2 | hhvs 30154 | . . . . . 6 β’ ββ = ( βπ£ βπ) |
10 | 4, 5, 6, 7, 8, 9 | ipval3 29693 | . . . . 5 β’ ((π β NrmCVec β§ π₯ β β β§ π¦ β β) β (π₯(Β·πOLDβπ)π¦) = (((((normββ(π₯ +β π¦))β2) β ((normββ(π₯ ββ π¦))β2)) + (i Β· (((normββ(π₯ +β (i Β·β π¦)))β2) β ((normββ(π₯ ββ (i Β·β π¦)))β2)))) / 4)) |
11 | 3, 10 | mp3an1 1449 | . . . 4 β’ ((π₯ β β β§ π¦ β β) β (π₯(Β·πOLDβπ)π¦) = (((((normββ(π₯ +β π¦))β2) β ((normββ(π₯ ββ π¦))β2)) + (i Β· (((normββ(π₯ +β (i Β·β π¦)))β2) β ((normββ(π₯ ββ (i Β·β π¦)))β2)))) / 4)) |
12 | 1, 11 | eqtr4d 2780 | . . 3 β’ ((π₯ β β β§ π¦ β β) β (π₯ Β·ih π¦) = (π₯(Β·πOLDβπ)π¦)) |
13 | 12 | rgen2 3195 | . 2 β’ βπ₯ β β βπ¦ β β (π₯ Β·ih π¦) = (π₯(Β·πOLDβπ)π¦) |
14 | ax-hfi 30063 | . . 3 β’ Β·ih :( β Γ β)βΆβ | |
15 | 4, 8 | ipf 29697 | . . . 4 β’ (π β NrmCVec β (Β·πOLDβπ):( β Γ β)βΆβ) |
16 | 3, 15 | ax-mp 5 | . . 3 β’ (Β·πOLDβπ):( β Γ β)βΆβ |
17 | ffn 6673 | . . . 4 β’ ( Β·ih :( β Γ β)βΆβ β Β·ih Fn ( β Γ β)) | |
18 | ffn 6673 | . . . 4 β’ ((Β·πOLDβπ):( β Γ β)βΆβ β (Β·πOLDβπ) Fn ( β Γ β)) | |
19 | eqfnov2 7491 | . . . 4 β’ (( Β·ih Fn ( β Γ β) β§ (Β·πOLDβπ) Fn ( β Γ β)) β ( Β·ih = (Β·πOLDβπ) β βπ₯ β β βπ¦ β β (π₯ Β·ih π¦) = (π₯(Β·πOLDβπ)π¦))) | |
20 | 17, 18, 19 | syl2an 597 | . . 3 β’ (( Β·ih :( β Γ β)βΆβ β§ (Β·πOLDβπ):( β Γ β)βΆβ) β ( Β·ih = (Β·πOLDβπ) β βπ₯ β β βπ¦ β β (π₯ Β·ih π¦) = (π₯(Β·πOLDβπ)π¦))) |
21 | 14, 16, 20 | mp2an 691 | . 2 β’ ( Β·ih = (Β·πOLDβπ) β βπ₯ β β βπ¦ β β (π₯ Β·ih π¦) = (π₯(Β·πOLDβπ)π¦)) |
22 | 13, 21 | mpbir 230 | 1 β’ Β·ih = (Β·πOLDβπ) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3065 β¨cop 4597 Γ cxp 5636 Fn wfn 6496 βΆwf 6497 βcfv 6501 (class class class)co 7362 βcc 11056 ici 11060 + caddc 11061 Β· cmul 11063 β cmin 11392 / cdiv 11819 2c2 12215 4c4 12217 βcexp 13974 NrmCVeccnv 29568 Β·πOLDcdip 29684 βchba 29903 +β cva 29904 Β·β csm 29905 Β·ih csp 29906 normβcno 29907 ββ cmv 29909 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 ax-hilex 29983 ax-hfvadd 29984 ax-hvcom 29985 ax-hvass 29986 ax-hv0cl 29987 ax-hvaddid 29988 ax-hfvmul 29989 ax-hvmulid 29990 ax-hvmulass 29991 ax-hvdistr1 29992 ax-hvdistr2 29993 ax-hvmul0 29994 ax-hfi 30063 ax-his1 30066 ax-his2 30067 ax-his3 30068 ax-his4 30069 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-n0 12421 df-z 12507 df-uz 12771 df-rp 12923 df-fz 13432 df-fzo 13575 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-sum 15578 df-grpo 29477 df-gid 29478 df-ginv 29479 df-gdiv 29480 df-ablo 29529 df-vc 29543 df-nv 29576 df-va 29579 df-ba 29580 df-sm 29581 df-0v 29582 df-vs 29583 df-nmcv 29584 df-dip 29685 df-hnorm 29952 df-hvsub 29955 |
This theorem is referenced by: bcsiHIL 30164 occllem 30287 hmopbdoptHIL 30972 |
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