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Mirrors > Home > MPE Home > Th. List > hlpr | Structured version Visualization version GIF version |
Description: The scalar field of a subcomplex Hilbert space is either β or β. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
hlress.f | β’ πΉ = (Scalarβπ) |
hlress.k | β’ πΎ = (BaseβπΉ) |
Ref | Expression |
---|---|
hlpr | β’ (π β βHil β πΎ β {β, β}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlress.f | . . 3 β’ πΉ = (Scalarβπ) | |
2 | hlress.k | . . 3 β’ πΎ = (BaseβπΉ) | |
3 | 1, 2 | hlprlem 24576 | . 2 β’ (π β βHil β (πΎ β (SubRingββfld) β§ (βfld βΎs πΎ) β DivRing β§ (βfld βΎs πΎ) β CMetSp)) |
4 | eqid 2736 | . . 3 β’ (βfld βΎs πΎ) = (βfld βΎs πΎ) | |
5 | 4 | cncdrg 24568 | . 2 β’ ((πΎ β (SubRingββfld) β§ (βfld βΎs πΎ) β DivRing β§ (βfld βΎs πΎ) β CMetSp) β πΎ β {β, β}) |
6 | 3, 5 | syl 17 | 1 β’ (π β βHil β πΎ β {β, β}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1087 = wceq 1539 β wcel 2104 {cpr 4567 βcfv 6458 (class class class)co 7307 βcc 10915 βcr 10916 Basecbs 16957 βΎs cress 16986 Scalarcsca 17010 DivRingcdr 20036 SubRingcsubrg 20065 βfldccnfld 20642 CMetSpccms 24541 βHilchl 24543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 ax-pre-sup 10995 ax-addf 10996 ax-mulf 10997 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-tp 4570 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-iin 4934 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-se 5556 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-isom 6467 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-of 7565 df-om 7745 df-1st 7863 df-2nd 7864 df-supp 8009 df-tpos 8073 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-2o 8329 df-er 8529 df-map 8648 df-ixp 8717 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-fsupp 9173 df-fi 9214 df-sup 9245 df-inf 9246 df-oi 9313 df-card 9741 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-div 11679 df-nn 12020 df-2 12082 df-3 12083 df-4 12084 df-5 12085 df-6 12086 df-7 12087 df-8 12088 df-9 12089 df-n0 12280 df-z 12366 df-dec 12484 df-uz 12629 df-q 12735 df-rp 12777 df-xneg 12894 df-xadd 12895 df-xmul 12896 df-ioo 13129 df-ico 13131 df-icc 13132 df-fz 13286 df-fzo 13429 df-seq 13768 df-exp 13829 df-hash 14091 df-cj 14855 df-re 14856 df-im 14857 df-sqrt 14991 df-abs 14992 df-struct 16893 df-sets 16910 df-slot 16928 df-ndx 16940 df-base 16958 df-ress 16987 df-plusg 17020 df-mulr 17021 df-starv 17022 df-sca 17023 df-vsca 17024 df-ip 17025 df-tset 17026 df-ple 17027 df-ds 17029 df-unif 17030 df-hom 17031 df-cco 17032 df-rest 17178 df-topn 17179 df-0g 17197 df-gsum 17198 df-topgen 17199 df-pt 17200 df-prds 17203 df-xrs 17258 df-qtop 17263 df-imas 17264 df-xps 17266 df-mre 17340 df-mrc 17341 df-acs 17343 df-mgm 18371 df-sgrp 18420 df-mnd 18431 df-submnd 18476 df-grp 18625 df-minusg 18626 df-mulg 18746 df-subg 18797 df-cntz 18968 df-cmn 19433 df-mgp 19766 df-ur 19783 df-ring 19830 df-cring 19831 df-oppr 19907 df-dvdsr 19928 df-unit 19929 df-invr 19959 df-dvr 19970 df-drng 20038 df-subrg 20067 df-lvec 20410 df-psmet 20634 df-xmet 20635 df-met 20636 df-bl 20637 df-mopn 20638 df-fbas 20639 df-fg 20640 df-cnfld 20643 df-phl 20876 df-top 22088 df-topon 22105 df-topsp 22127 df-bases 22141 df-cld 22215 df-ntr 22216 df-cls 22217 df-nei 22294 df-cn 22423 df-cnp 22424 df-haus 22511 df-cmp 22583 df-tx 22758 df-hmeo 22951 df-fil 23042 df-flim 23135 df-fcls 23137 df-xms 23518 df-ms 23519 df-tms 23520 df-cncf 24086 df-cph 24377 df-cfil 24464 df-cmet 24466 df-cms 24544 df-bn 24545 df-hl 24546 |
This theorem is referenced by: (None) |
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