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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlhilip | Structured version Visualization version GIF version |
Description: Inner product operation for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
Ref | Expression |
---|---|
hlhilip.h | β’ π» = (LHypβπΎ) |
hlhilip.l | β’ πΏ = ((DVecHβπΎ)βπ) |
hlhilip.v | β’ π = (BaseβπΏ) |
hlhilip.s | β’ π = ((HDMapβπΎ)βπ) |
hlhilip.u | β’ π = ((HLHilβπΎ)βπ) |
hlhilip.k | β’ (π β (πΎ β HL β§ π β π»)) |
hlhilip.p | β’ , = (π₯ β π, π¦ β π β¦ ((πβπ¦)βπ₯)) |
Ref | Expression |
---|---|
hlhilip | β’ (π β , = (Β·πβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlhilip.p | . . . 4 β’ , = (π₯ β π, π¦ β π β¦ ((πβπ¦)βπ₯)) | |
2 | hlhilip.v | . . . . . 6 β’ π = (BaseβπΏ) | |
3 | 2 | fvexi 6861 | . . . . 5 β’ π β V |
4 | 3, 3 | mpoex 8017 | . . . 4 β’ (π₯ β π, π¦ β π β¦ ((πβπ¦)βπ₯)) β V |
5 | 1, 4 | eqeltri 2834 | . . 3 β’ , β V |
6 | eqid 2737 | . . . 4 β’ ({β¨(Baseβndx), πβ©, β¨(+gβndx), (+gβπΏ)β©, β¨(Scalarβndx), (((EDRingβπΎ)βπ) sSet β¨(*πβndx), ((HGMapβπΎ)βπ)β©)β©} βͺ {β¨( Β·π βndx), ( Β·π βπΏ)β©, β¨(Β·πβndx), , β©}) = ({β¨(Baseβndx), πβ©, β¨(+gβndx), (+gβπΏ)β©, β¨(Scalarβndx), (((EDRingβπΎ)βπ) sSet β¨(*πβndx), ((HGMapβπΎ)βπ)β©)β©} βͺ {β¨( Β·π βndx), ( Β·π βπΏ)β©, β¨(Β·πβndx), , β©}) | |
7 | 6 | phlip 17239 | . . 3 β’ ( , β V β , = (Β·πβ({β¨(Baseβndx), πβ©, β¨(+gβndx), (+gβπΏ)β©, β¨(Scalarβndx), (((EDRingβπΎ)βπ) sSet β¨(*πβndx), ((HGMapβπΎ)βπ)β©)β©} βͺ {β¨( Β·π βndx), ( Β·π βπΏ)β©, β¨(Β·πβndx), , β©}))) |
8 | 5, 7 | ax-mp 5 | . 2 β’ , = (Β·πβ({β¨(Baseβndx), πβ©, β¨(+gβndx), (+gβπΏ)β©, β¨(Scalarβndx), (((EDRingβπΎ)βπ) sSet β¨(*πβndx), ((HGMapβπΎ)βπ)β©)β©} βͺ {β¨( Β·π βndx), ( Β·π βπΏ)β©, β¨(Β·πβndx), , β©})) |
9 | hlhilip.h | . . . 4 β’ π» = (LHypβπΎ) | |
10 | hlhilip.u | . . . 4 β’ π = ((HLHilβπΎ)βπ) | |
11 | hlhilip.l | . . . 4 β’ πΏ = ((DVecHβπΎ)βπ) | |
12 | eqid 2737 | . . . 4 β’ (+gβπΏ) = (+gβπΏ) | |
13 | eqid 2737 | . . . 4 β’ ((EDRingβπΎ)βπ) = ((EDRingβπΎ)βπ) | |
14 | eqid 2737 | . . . 4 β’ ((HGMapβπΎ)βπ) = ((HGMapβπΎ)βπ) | |
15 | eqid 2737 | . . . 4 β’ (((EDRingβπΎ)βπ) sSet β¨(*πβndx), ((HGMapβπΎ)βπ)β©) = (((EDRingβπΎ)βπ) sSet β¨(*πβndx), ((HGMapβπΎ)βπ)β©) | |
16 | eqid 2737 | . . . 4 β’ ( Β·π βπΏ) = ( Β·π βπΏ) | |
17 | hlhilip.s | . . . 4 β’ π = ((HDMapβπΎ)βπ) | |
18 | hlhilip.k | . . . 4 β’ (π β (πΎ β HL β§ π β π»)) | |
19 | 9, 10, 11, 2, 12, 13, 14, 15, 16, 17, 1, 18 | hlhilset 40426 | . . 3 β’ (π β π = ({β¨(Baseβndx), πβ©, β¨(+gβndx), (+gβπΏ)β©, β¨(Scalarβndx), (((EDRingβπΎ)βπ) sSet β¨(*πβndx), ((HGMapβπΎ)βπ)β©)β©} βͺ {β¨( Β·π βndx), ( Β·π βπΏ)β©, β¨(Β·πβndx), , β©})) |
20 | 19 | fveq2d 6851 | . 2 β’ (π β (Β·πβπ) = (Β·πβ({β¨(Baseβndx), πβ©, β¨(+gβndx), (+gβπΏ)β©, β¨(Scalarβndx), (((EDRingβπΎ)βπ) sSet β¨(*πβndx), ((HGMapβπΎ)βπ)β©)β©} βͺ {β¨( Β·π βndx), ( Β·π βπΏ)β©, β¨(Β·πβndx), , β©}))) |
21 | 8, 20 | eqtr4id 2796 | 1 β’ (π β , = (Β·πβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3448 βͺ cun 3913 {cpr 4593 {ctp 4595 β¨cop 4597 βcfv 6501 (class class class)co 7362 β cmpo 7364 sSet csts 17042 ndxcnx 17072 Basecbs 17090 +gcplusg 17140 *πcstv 17142 Scalarcsca 17143 Β·π cvsca 17144 Β·πcip 17145 HLchlt 37841 LHypclh 38476 EDRingcedring 39245 DVecHcdvh 39570 HDMapchdma 40284 HGMapchg 40375 HLHilchlh 40424 |
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-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 |
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-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-tp 4596 df-op 4598 df-uni 4871 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-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-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-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-n0 12421 df-z 12507 df-uz 12771 df-fz 13432 df-struct 17026 df-slot 17061 df-ndx 17073 df-base 17091 df-plusg 17153 df-sca 17156 df-vsca 17157 df-ip 17158 df-hlhil 40425 |
This theorem is referenced by: hlhilipval 40445 |
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