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Mirrors > Home > MPE Home > Th. List > dipcl | Structured version Visualization version GIF version |
Description: An inner product is a complex number. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 5-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ipcl.1 | β’ π = (BaseSetβπ) |
ipcl.7 | β’ π = (Β·πOLDβπ) |
Ref | Expression |
---|---|
dipcl | β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (π΄ππ΅) β β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ipcl.1 | . . 3 β’ π = (BaseSetβπ) | |
2 | eqid 2733 | . . 3 β’ ( +π£ βπ) = ( +π£ βπ) | |
3 | eqid 2733 | . . 3 β’ ( Β·π OLD βπ) = ( Β·π OLD βπ) | |
4 | eqid 2733 | . . 3 β’ (normCVβπ) = (normCVβπ) | |
5 | ipcl.7 | . . 3 β’ π = (Β·πOLDβπ) | |
6 | 1, 2, 3, 4, 5 | ipval 29687 | . 2 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (π΄ππ΅) = (Ξ£π β (1...4)((iβπ) Β· (((normCVβπ)β(π΄( +π£ βπ)((iβπ)( Β·π OLD βπ)π΅)))β2)) / 4)) |
7 | fzfid 13884 | . . . 4 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (1...4) β Fin) | |
8 | ax-icn 11115 | . . . . . . 7 β’ i β β | |
9 | elfznn 13476 | . . . . . . . 8 β’ (π β (1...4) β π β β) | |
10 | 9 | nnnn0d 12478 | . . . . . . 7 β’ (π β (1...4) β π β β0) |
11 | expcl 13991 | . . . . . . 7 β’ ((i β β β§ π β β0) β (iβπ) β β) | |
12 | 8, 10, 11 | sylancr 588 | . . . . . 6 β’ (π β (1...4) β (iβπ) β β) |
13 | 12 | adantl 483 | . . . . 5 β’ (((π β NrmCVec β§ π΄ β π β§ π΅ β π) β§ π β (1...4)) β (iβπ) β β) |
14 | 1, 2, 3, 4, 5 | ipval2lem4 29690 | . . . . . 6 β’ (((π β NrmCVec β§ π΄ β π β§ π΅ β π) β§ (iβπ) β β) β (((normCVβπ)β(π΄( +π£ βπ)((iβπ)( Β·π OLD βπ)π΅)))β2) β β) |
15 | 12, 14 | sylan2 594 | . . . . 5 β’ (((π β NrmCVec β§ π΄ β π β§ π΅ β π) β§ π β (1...4)) β (((normCVβπ)β(π΄( +π£ βπ)((iβπ)( Β·π OLD βπ)π΅)))β2) β β) |
16 | 13, 15 | mulcld 11180 | . . . 4 β’ (((π β NrmCVec β§ π΄ β π β§ π΅ β π) β§ π β (1...4)) β ((iβπ) Β· (((normCVβπ)β(π΄( +π£ βπ)((iβπ)( Β·π OLD βπ)π΅)))β2)) β β) |
17 | 7, 16 | fsumcl 15623 | . . 3 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β Ξ£π β (1...4)((iβπ) Β· (((normCVβπ)β(π΄( +π£ βπ)((iβπ)( Β·π OLD βπ)π΅)))β2)) β β) |
18 | 4cn 12243 | . . . 4 β’ 4 β β | |
19 | 4ne0 12266 | . . . 4 β’ 4 β 0 | |
20 | divcl 11824 | . . . 4 β’ ((Ξ£π β (1...4)((iβπ) Β· (((normCVβπ)β(π΄( +π£ βπ)((iβπ)( Β·π OLD βπ)π΅)))β2)) β β β§ 4 β β β§ 4 β 0) β (Ξ£π β (1...4)((iβπ) Β· (((normCVβπ)β(π΄( +π£ βπ)((iβπ)( Β·π OLD βπ)π΅)))β2)) / 4) β β) | |
21 | 18, 19, 20 | mp3an23 1454 | . . 3 β’ (Ξ£π β (1...4)((iβπ) Β· (((normCVβπ)β(π΄( +π£ βπ)((iβπ)( Β·π OLD βπ)π΅)))β2)) β β β (Ξ£π β (1...4)((iβπ) Β· (((normCVβπ)β(π΄( +π£ βπ)((iβπ)( Β·π OLD βπ)π΅)))β2)) / 4) β β) |
22 | 17, 21 | syl 17 | . 2 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (Ξ£π β (1...4)((iβπ) Β· (((normCVβπ)β(π΄( +π£ βπ)((iβπ)( Β·π OLD βπ)π΅)))β2)) / 4) β β) |
23 | 6, 22 | eqeltrd 2834 | 1 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (π΄ππ΅) β β) |
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 βcfv 6497 (class class class)co 7358 βcc 11054 0cc0 11056 1c1 11057 ici 11058 Β· cmul 11061 / cdiv 11817 2c2 12213 4c4 12215 β0cn0 12418 ...cfz 13430 βcexp 13973 Ξ£csu 15576 NrmCVeccnv 29568 +π£ cpv 29569 BaseSetcba 29570 Β·π OLD cns 29571 normCVcnmcv 29574 Β·πOLDcdip 29684 |
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-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 |
This theorem is referenced by: ipf 29697 ipipcj 29699 ip1ilem 29810 ip2i 29812 ipasslem1 29815 ipasslem2 29816 ipasslem4 29818 ipasslem5 29819 ipasslem7 29820 ipasslem8 29821 ipasslem9 29822 ipasslem10 29823 ipasslem11 29824 dipdi 29827 ip2dii 29828 dipassr 29830 dipsubdir 29832 dipsubdi 29833 pythi 29834 siilem1 29835 siilem2 29836 siii 29837 ipblnfi 29839 ip2eqi 29840 htthlem 29901 |
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