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Mirrors > Home > MPE Home > Th. List > ishil2 | Structured version Visualization version GIF version |
Description: The predicate "is a Hilbert space" (over a *-division ring). (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 22-Jun-2014.) |
Ref | Expression |
---|---|
ishil2.v | β’ π = (Baseβπ») |
ishil2.s | β’ β = (LSSumβπ») |
ishil2.o | β’ β₯ = (ocvβπ») |
ishil2.c | β’ πΆ = (ClSubSpβπ») |
Ref | Expression |
---|---|
ishil2 | β’ (π» β Hil β (π» β PreHil β§ βπ β πΆ (π β ( β₯ βπ )) = π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . 3 β’ (projβπ») = (projβπ») | |
2 | ishil2.c | . . 3 β’ πΆ = (ClSubSpβπ») | |
3 | 1, 2 | ishil 21140 | . 2 β’ (π» β Hil β (π» β PreHil β§ dom (projβπ») = πΆ)) |
4 | 1, 2 | pjcss 21138 | . . . . . 6 β’ (π» β PreHil β dom (projβπ») β πΆ) |
5 | eqss 3960 | . . . . . . 7 β’ (dom (projβπ») = πΆ β (dom (projβπ») β πΆ β§ πΆ β dom (projβπ»))) | |
6 | 5 | baib 537 | . . . . . 6 β’ (dom (projβπ») β πΆ β (dom (projβπ») = πΆ β πΆ β dom (projβπ»))) |
7 | 4, 6 | syl 17 | . . . . 5 β’ (π» β PreHil β (dom (projβπ») = πΆ β πΆ β dom (projβπ»))) |
8 | dfss3 3933 | . . . . 5 β’ (πΆ β dom (projβπ») β βπ β πΆ π β dom (projβπ»)) | |
9 | 7, 8 | bitrdi 287 | . . . 4 β’ (π» β PreHil β (dom (projβπ») = πΆ β βπ β πΆ π β dom (projβπ»))) |
10 | eqid 2733 | . . . . . . 7 β’ (LSubSpβπ») = (LSubSpβπ») | |
11 | 2, 10 | csslss 21111 | . . . . . 6 β’ ((π» β PreHil β§ π β πΆ) β π β (LSubSpβπ»)) |
12 | ishil2.v | . . . . . . . 8 β’ π = (Baseβπ») | |
13 | ishil2.o | . . . . . . . 8 β’ β₯ = (ocvβπ») | |
14 | ishil2.s | . . . . . . . 8 β’ β = (LSSumβπ») | |
15 | 12, 10, 13, 14, 1 | pjdm2 21133 | . . . . . . 7 β’ (π» β PreHil β (π β dom (projβπ») β (π β (LSubSpβπ») β§ (π β ( β₯ βπ )) = π))) |
16 | 15 | baibd 541 | . . . . . 6 β’ ((π» β PreHil β§ π β (LSubSpβπ»)) β (π β dom (projβπ») β (π β ( β₯ βπ )) = π)) |
17 | 11, 16 | syldan 592 | . . . . 5 β’ ((π» β PreHil β§ π β πΆ) β (π β dom (projβπ») β (π β ( β₯ βπ )) = π)) |
18 | 17 | ralbidva 3169 | . . . 4 β’ (π» β PreHil β (βπ β πΆ π β dom (projβπ») β βπ β πΆ (π β ( β₯ βπ )) = π)) |
19 | 9, 18 | bitrd 279 | . . 3 β’ (π» β PreHil β (dom (projβπ») = πΆ β βπ β πΆ (π β ( β₯ βπ )) = π)) |
20 | 19 | pm5.32i 576 | . 2 β’ ((π» β PreHil β§ dom (projβπ») = πΆ) β (π» β PreHil β§ βπ β πΆ (π β ( β₯ βπ )) = π)) |
21 | 3, 20 | bitri 275 | 1 β’ (π» β Hil β (π» β PreHil β§ βπ β πΆ (π β ( β₯ βπ )) = π)) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3061 β wss 3911 dom cdm 5634 βcfv 6497 (class class class)co 7358 Basecbs 17088 LSSumclsm 19421 LSubSpclss 20407 PreHilcphl 21044 ocvcocv 21080 ClSubSpccss 21081 projcpj 21122 Hilchil 21123 |
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-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 |
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-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-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-sca 17154 df-vsca 17155 df-ip 17156 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-mhm 18606 df-grp 18756 df-minusg 18757 df-sbg 18758 df-subg 18930 df-ghm 19011 df-cntz 19102 df-lsm 19423 df-pj1 19424 df-cmn 19569 df-abl 19570 df-mgp 19902 df-ur 19919 df-ring 19971 df-oppr 20054 df-rnghom 20153 df-staf 20318 df-srng 20319 df-lmod 20338 df-lss 20408 df-lmhm 20498 df-lvec 20579 df-sra 20649 df-rgmod 20650 df-phl 21046 df-ocv 21083 df-css 21084 df-pj 21125 df-hil 21126 |
This theorem is referenced by: hlhilhillem 40473 |
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