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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlhillsm | Structured version Visualization version GIF version |
Description: The vector sum operation for the final constructed Hilbert space. (Contributed by NM, 23-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.) |
Ref | Expression |
---|---|
hlhil0.h | β’ π» = (LHypβπΎ) |
hlhil0.l | β’ πΏ = ((DVecHβπΎ)βπ) |
hlhil0.u | β’ π = ((HLHilβπΎ)βπ) |
hlhil0.k | β’ (π β (πΎ β HL β§ π β π»)) |
hlhillsm.a | β’ β = (LSSumβπΏ) |
Ref | Expression |
---|---|
hlhillsm | β’ (π β β = (LSSumβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlhillsm.a | . 2 β’ β = (LSSumβπΏ) | |
2 | eqidd 2727 | . . 3 β’ (π β (BaseβπΏ) = (BaseβπΏ)) | |
3 | hlhil0.h | . . . 4 β’ π» = (LHypβπΎ) | |
4 | hlhil0.u | . . . 4 β’ π = ((HLHilβπΎ)βπ) | |
5 | hlhil0.k | . . . 4 β’ (π β (πΎ β HL β§ π β π»)) | |
6 | hlhil0.l | . . . 4 β’ πΏ = ((DVecHβπΎ)βπ) | |
7 | eqid 2726 | . . . 4 β’ (BaseβπΏ) = (BaseβπΏ) | |
8 | 3, 4, 5, 6, 7 | hlhilbase 41319 | . . 3 β’ (π β (BaseβπΏ) = (Baseβπ)) |
9 | eqid 2726 | . . . . 5 β’ (+gβπΏ) = (+gβπΏ) | |
10 | 3, 4, 5, 6, 9 | hlhilplus 41320 | . . . 4 β’ (π β (+gβπΏ) = (+gβπ)) |
11 | 10 | oveqdr 7432 | . . 3 β’ ((π β§ (π₯ β (BaseβπΏ) β§ π¦ β (BaseβπΏ))) β (π₯(+gβπΏ)π¦) = (π₯(+gβπ)π¦)) |
12 | 6 | fvexi 6898 | . . . 4 β’ πΏ β V |
13 | 12 | a1i 11 | . . 3 β’ (π β πΏ β V) |
14 | 4 | fvexi 6898 | . . . 4 β’ π β V |
15 | 14 | a1i 11 | . . 3 β’ (π β π β V) |
16 | 2, 8, 11, 13, 15 | lsmpropd 19594 | . 2 β’ (π β (LSSumβπΏ) = (LSSumβπ)) |
17 | 1, 16 | eqtrid 2778 | 1 β’ (π β β = (LSSumβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 Vcvv 3468 βcfv 6536 Basecbs 17150 +gcplusg 17203 LSSumclsm 19551 HLchlt 38732 LHypclh 39367 DVecHcdvh 40461 HLHilchlh 41315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-struct 17086 df-slot 17121 df-ndx 17133 df-base 17151 df-plusg 17216 df-sca 17219 df-vsca 17220 df-ip 17221 df-lsm 19553 df-hlhil 41316 |
This theorem is referenced by: hlhilhillem 41347 |
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