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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 30962 | . . . 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 30968 | . . . . 5 β’ π β NrmCVec |
4 | 2 | hhba 30970 | . . . . . 6 β’ β = (BaseSetβπ) |
5 | 2 | hhva 30969 | . . . . . 6 β’ +β = ( +π£ βπ) |
6 | 2 | hhsm 30972 | . . . . . 6 β’ Β·β = ( Β·π OLD βπ) |
7 | 2 | hhnm 30974 | . . . . . 6 β’ normβ = (normCVβπ) |
8 | eqid 2728 | . . . . . 6 β’ (Β·πOLDβπ) = (Β·πOLDβπ) | |
9 | 2 | hhvs 30973 | . . . . . 6 β’ ββ = ( βπ£ βπ) |
10 | 4, 5, 6, 7, 8, 9 | ipval3 30512 | . . . . 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 1445 | . . . 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 2771 | . . 3 β’ ((π₯ β β β§ π¦ β β) β (π₯ Β·ih π¦) = (π₯(Β·πOLDβπ)π¦)) |
13 | 12 | rgen2 3193 | . 2 β’ βπ₯ β β βπ¦ β β (π₯ Β·ih π¦) = (π₯(Β·πOLDβπ)π¦) |
14 | ax-hfi 30882 | . . 3 β’ Β·ih :( β Γ β)βΆβ | |
15 | 4, 8 | ipf 30516 | . . . 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 6716 | . . . 4 β’ ( Β·ih :( β Γ β)βΆβ β Β·ih Fn ( β Γ β)) | |
18 | ffn 6716 | . . . 4 β’ ((Β·πOLDβπ):( β Γ β)βΆβ β (Β·πOLDβπ) Fn ( β Γ β)) | |
19 | eqfnov2 7545 | . . . 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 595 | . . 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 395 = wceq 1534 β wcel 2099 βwral 3057 β¨cop 4630 Γ cxp 5670 Fn wfn 6537 βΆwf 6538 βcfv 6542 (class class class)co 7414 βcc 11130 ici 11134 + caddc 11135 Β· cmul 11137 β cmin 11468 / cdiv 11895 2c2 12291 4c4 12293 βcexp 14052 NrmCVeccnv 30387 Β·πOLDcdip 30503 βchba 30722 +β cva 30723 Β·β csm 30724 Β·ih csp 30725 normβcno 30726 ββ cmv 30728 |
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 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9658 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 ax-hilex 30802 ax-hfvadd 30803 ax-hvcom 30804 ax-hvass 30805 ax-hv0cl 30806 ax-hvaddid 30807 ax-hfvmul 30808 ax-hvmulid 30809 ax-hvmulass 30810 ax-hvdistr1 30811 ax-hvdistr2 30812 ax-hvmul0 30813 ax-hfi 30882 ax-his1 30885 ax-his2 30886 ax-his3 30887 ax-his4 30888 |
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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 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-se 5628 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-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9459 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-n0 12497 df-z 12583 df-uz 12847 df-rp 13001 df-fz 13511 df-fzo 13654 df-seq 13993 df-exp 14053 df-hash 14316 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15458 df-sum 15659 df-grpo 30296 df-gid 30297 df-ginv 30298 df-gdiv 30299 df-ablo 30348 df-vc 30362 df-nv 30395 df-va 30398 df-ba 30399 df-sm 30400 df-0v 30401 df-vs 30402 df-nmcv 30403 df-dip 30504 df-hnorm 30771 df-hvsub 30774 |
This theorem is referenced by: bcsiHIL 30983 occllem 31106 hmopbdoptHIL 31791 |
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