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Mirrors > Home > MPE Home > Th. List > ishl2 | Structured version Visualization version Unicode version |
Description: A Hilbert space is a complete subcomplex pre-Hilbert space over or . (Contributed by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
hlress.f | Scalar |
hlress.k |
Ref | Expression |
---|---|
ishl2 | CMetSp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ishl 23158 | . 2 Ban | |
2 | df-3an 1039 | . . 3 CMetSp CMetSp | |
3 | 3ancomb 1047 | . . 3 CMetSp CMetSp | |
4 | cphnvc 22976 | . . . . . 6 NrmVec | |
5 | hlress.f | . . . . . . . . 9 Scalar | |
6 | 5 | isbn 23135 | . . . . . . . 8 Ban NrmVec CMetSp CMetSp |
7 | 3anass 1042 | . . . . . . . 8 NrmVec CMetSp CMetSp NrmVec CMetSp CMetSp | |
8 | 6, 7 | bitri 264 | . . . . . . 7 Ban NrmVec CMetSp CMetSp |
9 | 8 | baib 944 | . . . . . 6 NrmVec Ban CMetSp CMetSp |
10 | 4, 9 | syl 17 | . . . . 5 Ban CMetSp CMetSp |
11 | hlress.k | . . . . . . . . 9 | |
12 | 5, 11 | cphsca 22979 | . . . . . . . 8 ℂfld ↾s |
13 | 12 | eleq1d 2686 | . . . . . . 7 CMetSp ℂfld ↾s CMetSp |
14 | 5, 11 | cphsubrg 22980 | . . . . . . . . 9 SubRingℂfld |
15 | cphlvec 22975 | . . . . . . . . . . 11 | |
16 | 5 | lvecdrng 19105 | . . . . . . . . . . 11 |
17 | 15, 16 | syl 17 | . . . . . . . . . 10 |
18 | 12, 17 | eqeltrrd 2702 | . . . . . . . . 9 ℂfld ↾s |
19 | eqid 2622 | . . . . . . . . . . 11 ℂfld ↾s ℂfld ↾s | |
20 | 19 | cncdrg 23155 | . . . . . . . . . 10 SubRingℂfld ℂfld ↾s ℂfld ↾s CMetSp |
21 | 20 | 3expia 1267 | . . . . . . . . 9 SubRingℂfld ℂfld ↾s ℂfld ↾s CMetSp |
22 | 14, 18, 21 | syl2anc 693 | . . . . . . . 8 ℂfld ↾s CMetSp |
23 | elpri 4197 | . . . . . . . . 9 | |
24 | oveq2 6658 | . . . . . . . . . . 11 ℂfld ↾s ℂfld ↾s | |
25 | eqid 2622 | . . . . . . . . . . . . 13 ℂfld ℂfld | |
26 | 25 | recld2 22617 | . . . . . . . . . . . 12 ℂfld |
27 | cncms 23151 | . . . . . . . . . . . . 13 ℂfld CMetSp | |
28 | ax-resscn 9993 | . . . . . . . . . . . . 13 | |
29 | eqid 2622 | . . . . . . . . . . . . . 14 ℂfld ↾s ℂfld ↾s | |
30 | cnfldbas 19750 | . . . . . . . . . . . . . 14 ℂfld | |
31 | 29, 30, 25 | cmsss 23147 | . . . . . . . . . . . . 13 ℂfld CMetSp ℂfld ↾s CMetSp ℂfld |
32 | 27, 28, 31 | mp2an 708 | . . . . . . . . . . . 12 ℂfld ↾s CMetSp ℂfld |
33 | 26, 32 | mpbir 221 | . . . . . . . . . . 11 ℂfld ↾s CMetSp |
34 | 24, 33 | syl6eqel 2709 | . . . . . . . . . 10 ℂfld ↾s CMetSp |
35 | oveq2 6658 | . . . . . . . . . . 11 ℂfld ↾s ℂfld ↾s | |
36 | 30 | ressid 15935 | . . . . . . . . . . . . 13 ℂfld CMetSp ℂfld ↾s ℂfld |
37 | 27, 36 | ax-mp 5 | . . . . . . . . . . . 12 ℂfld ↾s ℂfld |
38 | 37, 27 | eqeltri 2697 | . . . . . . . . . . 11 ℂfld ↾s CMetSp |
39 | 35, 38 | syl6eqel 2709 | . . . . . . . . . 10 ℂfld ↾s CMetSp |
40 | 34, 39 | jaoi 394 | . . . . . . . . 9 ℂfld ↾s CMetSp |
41 | 23, 40 | syl 17 | . . . . . . . 8 ℂfld ↾s CMetSp |
42 | 22, 41 | impbid1 215 | . . . . . . 7 ℂfld ↾s CMetSp |
43 | 13, 42 | bitrd 268 | . . . . . 6 CMetSp |
44 | 43 | anbi2d 740 | . . . . 5 CMetSp CMetSp CMetSp |
45 | 10, 44 | bitrd 268 | . . . 4 Ban CMetSp |
46 | 45 | pm5.32ri 670 | . . 3 Ban CMetSp |
47 | 2, 3, 46 | 3bitr4ri 293 | . 2 Ban CMetSp |
48 | 1, 47 | bitri 264 | 1 CMetSp |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wo 383 wa 384 w3a 1037 wceq 1483 wcel 1990 wss 3574 cpr 4179 cfv 5888 (class class class)co 6650 cc 9934 cr 9935 cbs 15857 ↾s cress 15858 Scalarcsca 15944 ctopn 16082 cdr 18747 SubRingcsubrg 18776 clvec 19102 ℂfldccnfld 19746 ccld 20820 NrmVeccnvc 22386 ccph 22966 CMetSpccms 23129 Bancbn 23130 chl 23131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-mulg 17541 df-subg 17591 df-cntz 17750 df-cmn 18195 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-subrg 18778 df-lvec 19103 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-phl 19971 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-flim 21743 df-fcls 21745 df-xms 22125 df-ms 22126 df-tms 22127 df-nvc 22392 df-cncf 22681 df-cph 22968 df-cfil 23053 df-cmet 23055 df-cms 23132 df-bn 23133 df-hl 23134 |
This theorem is referenced by: (None) |
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