Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > HSE Home > Th. List > hhba | Structured version Visualization version GIF version |
Description: The base set of Hilbert space. This theorem provides an independent proof of df-hba 29380 (see comments in that definition). (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hhnv.1 | β’ π = β¨β¨ +β , Β·β β©, normββ© |
Ref | Expression |
---|---|
hhba | β’ β = (BaseSetβπ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hilablo 29571 | . . . 4 β’ +β β AbelOp | |
2 | ablogrpo 28958 | . . . 4 β’ ( +β β AbelOp β +β β GrpOp) | |
3 | 1, 2 | ax-mp 5 | . . 3 β’ +β β GrpOp |
4 | ax-hfvadd 29411 | . . . 4 β’ +β :( β Γ β)βΆ β | |
5 | 4 | fdmi 6642 | . . 3 β’ dom +β = ( β Γ β) |
6 | 3, 5 | grporn 28932 | . 2 β’ β = ran +β |
7 | eqid 2736 | . . 3 β’ (BaseSetβπ) = (BaseSetβπ) | |
8 | hhnv.1 | . . . 4 β’ π = β¨β¨ +β , Β·β β©, normββ© | |
9 | 8 | hhva 29577 | . . 3 β’ +β = ( +π£ βπ) |
10 | 7, 9 | bafval 29015 | . 2 β’ (BaseSetβπ) = ran +β |
11 | 6, 10 | eqtr4i 2767 | 1 β’ β = (BaseSetβπ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 β wcel 2104 β¨cop 4571 Γ cxp 5598 ran crn 5601 βcfv 6458 GrpOpcgr 28900 AbelOpcablo 28955 BaseSetcba 28997 βchba 29330 +β cva 29331 Β·β csm 29332 normβcno 29334 |
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 10977 ax-resscn 10978 ax-1cn 10979 ax-icn 10980 ax-addcl 10981 ax-addrcl 10982 ax-mulcl 10983 ax-mulrcl 10984 ax-mulcom 10985 ax-addass 10986 ax-mulass 10987 ax-distr 10988 ax-i2m1 10989 ax-1ne0 10990 ax-1rid 10991 ax-rnegex 10992 ax-rrecex 10993 ax-cnre 10994 ax-pre-lttri 10995 ax-pre-lttrn 10996 ax-pre-ltadd 10997 ax-pre-mulgt0 10998 ax-pre-sup 10999 ax-hilex 29410 ax-hfvadd 29411 ax-hvcom 29412 ax-hvass 29413 ax-hv0cl 29414 ax-hvaddid 29415 ax-hfvmul 29416 ax-hvmulid 29417 ax-hvmulass 29418 ax-hvdistr1 29419 ax-hvdistr2 29420 ax-hvmul0 29421 ax-hfi 29490 ax-his1 29493 ax-his2 29494 ax-his3 29495 ax-his4 29496 |
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 3304 df-reu 3305 df-rab 3306 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-op 4572 df-uni 4845 df-iun 4933 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-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-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-1st 7863 df-2nd 7864 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-er 8529 df-en 8765 df-dom 8766 df-sdom 8767 df-sup 9249 df-pnf 11061 df-mnf 11062 df-xr 11063 df-ltxr 11064 df-le 11065 df-sub 11257 df-neg 11258 df-div 11683 df-nn 12024 df-2 12086 df-3 12087 df-4 12088 df-n0 12284 df-z 12370 df-uz 12633 df-rp 12781 df-seq 13772 df-exp 13833 df-cj 14859 df-re 14860 df-im 14861 df-sqrt 14995 df-abs 14996 df-grpo 28904 df-gid 28905 df-ablo 28956 df-vc 28970 df-nv 29003 df-va 29006 df-ba 29007 df-hnorm 29379 df-hvsub 29382 |
This theorem is referenced by: hhvs 29581 hhmet 29585 hhmetdval 29587 hhip 29588 hhcau 29609 hhlm 29610 hhhl 29615 hhlnoi 30311 hhnmoi 30312 hh0oi 30314 |
Copyright terms: Public domain | W3C validator |