![]() |
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 30504 (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 30695 | . . . 4 β’ +β β AbelOp | |
2 | ablogrpo 30082 | . . . 4 β’ ( +β β AbelOp β +β β GrpOp) | |
3 | 1, 2 | ax-mp 5 | . . 3 β’ +β β GrpOp |
4 | ax-hfvadd 30535 | . . . 4 β’ +β :( β Γ β)βΆ β | |
5 | 4 | fdmi 6729 | . . 3 β’ dom +β = ( β Γ β) |
6 | 3, 5 | grporn 30056 | . 2 β’ β = ran +β |
7 | eqid 2731 | . . 3 β’ (BaseSetβπ) = (BaseSetβπ) | |
8 | hhnv.1 | . . . 4 β’ π = β¨β¨ +β , Β·β β©, normββ© | |
9 | 8 | hhva 30701 | . . 3 β’ +β = ( +π£ βπ) |
10 | 7, 9 | bafval 30139 | . 2 β’ (BaseSetβπ) = ran +β |
11 | 6, 10 | eqtr4i 2762 | 1 β’ β = (BaseSetβπ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 β wcel 2105 β¨cop 4634 Γ cxp 5674 ran crn 5677 βcfv 6543 GrpOpcgr 30024 AbelOpcablo 30079 BaseSetcba 30121 βchba 30454 +β cva 30455 Β·β csm 30456 normβcno 30458 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-hilex 30534 ax-hfvadd 30535 ax-hvcom 30536 ax-hvass 30537 ax-hv0cl 30538 ax-hvaddid 30539 ax-hfvmul 30540 ax-hvmulid 30541 ax-hvmulass 30542 ax-hvdistr1 30543 ax-hvdistr2 30544 ax-hvmul0 30545 ax-hfi 30614 ax-his1 30617 ax-his2 30618 ax-his3 30619 ax-his4 30620 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-sup 9443 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-n0 12480 df-z 12566 df-uz 12830 df-rp 12982 df-seq 13974 df-exp 14035 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-grpo 30028 df-gid 30029 df-ablo 30080 df-vc 30094 df-nv 30127 df-va 30130 df-ba 30131 df-hnorm 30503 df-hvsub 30506 |
This theorem is referenced by: hhvs 30705 hhmet 30709 hhmetdval 30711 hhip 30712 hhcau 30733 hhlm 30734 hhhl 30739 hhlnoi 31435 hhnmoi 31436 hh0oi 31438 |
Copyright terms: Public domain | W3C validator |