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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlhilbase | Structured version Visualization version GIF version |
Description: The base set of the final constructed Hilbert space. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
Ref | Expression |
---|---|
hlhilbase.h | β’ π» = (LHypβπΎ) |
hlhilbase.u | β’ π = ((HLHilβπΎ)βπ) |
hlhilbase.k | β’ (π β (πΎ β HL β§ π β π»)) |
hlhilbase.l | β’ πΏ = ((DVecHβπΎ)βπ) |
hlhilbase.m | β’ π = (BaseβπΏ) |
Ref | Expression |
---|---|
hlhilbase | β’ (π β π = (Baseβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlhilbase.m | . . . 4 β’ π = (BaseβπΏ) | |
2 | 1 | fvexi 6903 | . . 3 β’ π β V |
3 | eqid 2733 | . . . 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), (π₯ β π, π¦ β π β¦ ((((HDMapβπΎ)βπ)βπ¦)βπ₯))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), (π₯ β π, π¦ β π β¦ ((((HDMapβπΎ)βπ)βπ¦)βπ₯))β©}) | |
4 | 3 | phlbase 17289 | . . 3 β’ (π β V β π = (Baseβ({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), (π₯ β π, π¦ β π β¦ ((((HDMapβπΎ)βπ)βπ¦)βπ₯))β©}))) |
5 | 2, 4 | ax-mp 5 | . 2 β’ π = (Baseβ({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), (π₯ β π, π¦ β π β¦ ((((HDMapβπΎ)βπ)βπ¦)βπ₯))β©})) |
6 | hlhilbase.h | . . . 4 β’ π» = (LHypβπΎ) | |
7 | hlhilbase.u | . . . 4 β’ π = ((HLHilβπΎ)βπ) | |
8 | hlhilbase.l | . . . 4 β’ πΏ = ((DVecHβπΎ)βπ) | |
9 | eqid 2733 | . . . 4 β’ (+gβπΏ) = (+gβπΏ) | |
10 | eqid 2733 | . . . 4 β’ ((EDRingβπΎ)βπ) = ((EDRingβπΎ)βπ) | |
11 | eqid 2733 | . . . 4 β’ ((HGMapβπΎ)βπ) = ((HGMapβπΎ)βπ) | |
12 | eqid 2733 | . . . 4 β’ (((EDRingβπΎ)βπ) sSet β¨(*πβndx), ((HGMapβπΎ)βπ)β©) = (((EDRingβπΎ)βπ) sSet β¨(*πβndx), ((HGMapβπΎ)βπ)β©) | |
13 | eqid 2733 | . . . 4 β’ ( Β·π βπΏ) = ( Β·π βπΏ) | |
14 | eqid 2733 | . . . 4 β’ ((HDMapβπΎ)βπ) = ((HDMapβπΎ)βπ) | |
15 | eqid 2733 | . . . 4 β’ (π₯ β π, π¦ β π β¦ ((((HDMapβπΎ)βπ)βπ¦)βπ₯)) = (π₯ β π, π¦ β π β¦ ((((HDMapβπΎ)βπ)βπ¦)βπ₯)) | |
16 | hlhilbase.k | . . . 4 β’ (π β (πΎ β HL β§ π β π»)) | |
17 | 6, 7, 8, 1, 9, 10, 11, 12, 13, 14, 15, 16 | hlhilset 40794 | . . 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), (π₯ β π, π¦ β π β¦ ((((HDMapβπΎ)βπ)βπ¦)βπ₯))β©})) |
18 | 17 | fveq2d 6893 | . 2 β’ (π β (Baseβπ) = (Baseβ({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), (π₯ β π, π¦ β π β¦ ((((HDMapβπΎ)βπ)βπ¦)βπ₯))β©}))) |
19 | 5, 18 | eqtr4id 2792 | 1 β’ (π β π = (Baseβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3475 βͺ cun 3946 {cpr 4630 {ctp 4632 β¨cop 4634 βcfv 6541 (class class class)co 7406 β cmpo 7408 sSet csts 17093 ndxcnx 17123 Basecbs 17141 +gcplusg 17194 *πcstv 17196 Scalarcsca 17197 Β·π cvsca 17198 Β·πcip 17199 HLchlt 38209 LHypclh 38844 EDRingcedring 39613 DVecHcdvh 39938 HDMapchdma 40652 HGMapchg 40743 HLHilchlh 40792 |
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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 df-struct 17077 df-slot 17112 df-ndx 17124 df-base 17142 df-plusg 17207 df-sca 17210 df-vsca 17211 df-ip 17212 df-hlhil 40793 |
This theorem is referenced by: hlhillvec 40815 hlhil0 40819 hlhillsm 40820 hlhilocv 40821 hlhillcs 40822 hlhilphllem 40823 hlhilhillem 40824 |
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