HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  hst1h Structured version   Unicode version

Theorem hst1h 25454
Description: The norm of a Hilbert-space-valued state equals one iff the state value equals the state value of the lattice unit. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
Assertion
Ref Expression
hst1h  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( normh `  ( S `  A ) )  =  1  <->  ( S `  A )  =  ( S `  ~H )
) )

Proof of Theorem hst1h
StepHypRef Expression
1 hstcl 25444 . . . . 5  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  ( S `  A )  e.  ~H )
2 ax-hvaddid 24229 . . . . 5  |-  ( ( S `  A )  e.  ~H  ->  (
( S `  A
)  +h  0h )  =  ( S `  A ) )
31, 2syl 16 . . . 4  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( S `  A
)  +h  0h )  =  ( S `  A ) )
43adantr 462 . . 3  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  (
( S `  A
)  +h  0h )  =  ( S `  A ) )
5 ax-1cn 9328 . . . . . . . . . . . 12  |-  1  e.  CC
6 choccl 24532 . . . . . . . . . . . . . . . 16  |-  ( A  e.  CH  ->  ( _|_ `  A )  e. 
CH )
7 hstcl 25444 . . . . . . . . . . . . . . . 16  |-  ( ( S  e.  CHStates  /\  ( _|_ `  A )  e. 
CH )  ->  ( S `  ( _|_ `  A ) )  e. 
~H )
86, 7sylan2 471 . . . . . . . . . . . . . . 15  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  ( S `  ( _|_ `  A ) )  e. 
~H )
9 normcl 24350 . . . . . . . . . . . . . . 15  |-  ( ( S `  ( _|_ `  A ) )  e. 
~H  ->  ( normh `  ( S `  ( _|_ `  A ) ) )  e.  RR )
108, 9syl 16 . . . . . . . . . . . . . 14  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  ( normh `  ( S `  ( _|_ `  A ) ) )  e.  RR )
1110resqcld 12018 . . . . . . . . . . . . 13  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 )  e.  RR )
1211recnd 9400 . . . . . . . . . . . 12  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 )  e.  CC )
13 pncan2 9605 . . . . . . . . . . . 12  |-  ( ( 1  e.  CC  /\  ( ( normh `  ( S `  ( _|_ `  A ) ) ) ^ 2 )  e.  CC )  ->  (
( 1  +  ( ( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 ) )  - 
1 )  =  ( ( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 ) )
145, 12, 13sylancr 656 . . . . . . . . . . 11  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( 1  +  ( ( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 ) )  - 
1 )  =  ( ( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 ) )
1514adantr 462 . . . . . . . . . 10  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  (
( 1  +  ( ( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 ) )  - 
1 )  =  ( ( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 ) )
16 oveq1 6087 . . . . . . . . . . . . . 14  |-  ( (
normh `  ( S `  A ) )  =  1  ->  ( ( normh `  ( S `  A ) ) ^
2 )  =  ( 1 ^ 2 ) )
17 sq1 11944 . . . . . . . . . . . . . 14  |-  ( 1 ^ 2 )  =  1
1816, 17syl6req 2482 . . . . . . . . . . . . 13  |-  ( (
normh `  ( S `  A ) )  =  1  ->  1  =  ( ( normh `  ( S `  A )
) ^ 2 ) )
1918oveq1d 6095 . . . . . . . . . . . 12  |-  ( (
normh `  ( S `  A ) )  =  1  ->  ( 1  +  ( ( normh `  ( S `  ( _|_ `  A ) ) ) ^ 2 ) )  =  ( ( ( normh `  ( S `  A ) ) ^
2 )  +  ( ( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 ) ) )
20 hstnmoc 25450 . . . . . . . . . . . 12  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( ( normh `  ( S `  A )
) ^ 2 )  +  ( ( normh `  ( S `  ( _|_ `  A ) ) ) ^ 2 ) )  =  1 )
2119, 20sylan9eqr 2487 . . . . . . . . . . 11  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  (
1  +  ( (
normh `  ( S `  ( _|_ `  A ) ) ) ^ 2 ) )  =  1 )
2221oveq1d 6095 . . . . . . . . . 10  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  (
( 1  +  ( ( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 ) )  - 
1 )  =  ( 1  -  1 ) )
2315, 22eqtr3d 2467 . . . . . . . . 9  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  (
( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 )  =  ( 1  -  1 ) )
24 1m1e0 10378 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
2523, 24syl6eq 2481 . . . . . . . 8  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  (
( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 )  =  0 )
2625ex 434 . . . . . . 7  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( normh `  ( S `  A ) )  =  1  ->  ( ( normh `  ( S `  ( _|_ `  A ) ) ) ^ 2 )  =  0 ) )
2710recnd 9400 . . . . . . . . 9  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  ( normh `  ( S `  ( _|_ `  A ) ) )  e.  CC )
28 sqeq0 11914 . . . . . . . . 9  |-  ( (
normh `  ( S `  ( _|_ `  A ) ) )  e.  CC  ->  ( ( ( normh `  ( S `  ( _|_ `  A ) ) ) ^ 2 )  =  0  <->  ( normh `  ( S `  ( _|_ `  A ) ) )  =  0 ) )
2927, 28syl 16 . . . . . . . 8  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( ( normh `  ( S `  ( _|_ `  A ) ) ) ^ 2 )  =  0  <->  ( normh `  ( S `  ( _|_ `  A ) ) )  =  0 ) )
30 norm-i 24354 . . . . . . . . 9  |-  ( ( S `  ( _|_ `  A ) )  e. 
~H  ->  ( ( normh `  ( S `  ( _|_ `  A ) ) )  =  0  <->  ( S `  ( _|_ `  A ) )  =  0h ) )
318, 30syl 16 . . . . . . . 8  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( normh `  ( S `  ( _|_ `  A
) ) )  =  0  <->  ( S `  ( _|_ `  A ) )  =  0h )
)
3229, 31bitrd 253 . . . . . . 7  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( ( normh `  ( S `  ( _|_ `  A ) ) ) ^ 2 )  =  0  <->  ( S `  ( _|_ `  A ) )  =  0h )
)
3326, 32sylibd 214 . . . . . 6  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( normh `  ( S `  A ) )  =  1  ->  ( S `  ( _|_ `  A
) )  =  0h ) )
3433imp 429 . . . . 5  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  ( S `  ( _|_ `  A ) )  =  0h )
3534oveq2d 6096 . . . 4  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  (
( S `  A
)  +h  ( S `
 ( _|_ `  A
) ) )  =  ( ( S `  A )  +h  0h ) )
36 hstoc 25449 . . . . 5  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( S `  A
)  +h  ( S `
 ( _|_ `  A
) ) )  =  ( S `  ~H ) )
3736adantr 462 . . . 4  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  (
( S `  A
)  +h  ( S `
 ( _|_ `  A
) ) )  =  ( S `  ~H ) )
3835, 37eqtr3d 2467 . . 3  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  (
( S `  A
)  +h  0h )  =  ( S `  ~H ) )
394, 38eqtr3d 2467 . 2  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  ( S `  A )  =  ( S `  ~H ) )
40 fveq2 5679 . . 3  |-  ( ( S `  A )  =  ( S `  ~H )  ->  ( normh `  ( S `  A
) )  =  (
normh `  ( S `  ~H ) ) )
41 hst1a 25445 . . . 4  |-  ( S  e.  CHStates  ->  ( normh `  ( S `  ~H )
)  =  1 )
4241adantr 462 . . 3  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  ( normh `  ( S `  ~H ) )  =  1 )
4340, 42sylan9eqr 2487 . 2  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( S `  A
)  =  ( S `
 ~H ) )  ->  ( normh `  ( S `  A )
)  =  1 )
4439, 43impbida 821 1  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( normh `  ( S `  A ) )  =  1  <->  ( S `  A )  =  ( S `  ~H )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1362    e. wcel 1755   ` cfv 5406  (class class class)co 6080   CCcc 9268   RRcr 9269   0cc0 9270   1c1 9271    + caddc 9273    - cmin 9583   2c2 10359   ^cexp 11849   ~Hchil 24144    +h cva 24145   normhcno 24148   0hc0v 24149   CHcch 24154   _|_cort 24155   CHStateschst 24188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349  ax-mulf 9350  ax-hilex 24224  ax-hfvadd 24225  ax-hvcom 24226  ax-hvass 24227  ax-hv0cl 24228  ax-hvaddid 24229  ax-hfvmul 24230  ax-hvmulid 24231  ax-hvmulass 24232  ax-hvdistr1 24233  ax-hvdistr2 24234  ax-hvmul0 24235  ax-hfi 24304  ax-his1 24307  ax-his2 24308  ax-his3 24309  ax-his4 24310  ax-hcompl 24427
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-ioo 11292  df-icc 11295  df-fz 11425  df-fzo 11533  df-seq 11791  df-exp 11850  df-hash 12088  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-clim 12950  df-sum 13148  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-starv 14236  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-unif 14244  df-hom 14245  df-cco 14246  df-rest 14344  df-topn 14345  df-0g 14363  df-gsum 14364  df-topgen 14365  df-pt 14366  df-prds 14369  df-xrs 14423  df-qtop 14428  df-imas 14429  df-xps 14431  df-mre 14507  df-mrc 14508  df-acs 14510  df-mnd 15398  df-submnd 15448  df-mulg 15528  df-cntz 15815  df-cmn 16259  df-psmet 17653  df-xmet 17654  df-met 17655  df-bl 17656  df-mopn 17657  df-cnfld 17663  df-top 18345  df-bases 18347  df-topon 18348  df-topsp 18349  df-cn 18673  df-cnp 18674  df-lm 18675  df-haus 18761  df-tx 18977  df-hmeo 19170  df-xms 19737  df-ms 19738  df-tms 19739  df-cau 20609  df-grpo 23501  df-gid 23502  df-ginv 23503  df-gdiv 23504  df-ablo 23592  df-vc 23747  df-nv 23793  df-va 23796  df-ba 23797  df-sm 23798  df-0v 23799  df-vs 23800  df-nmcv 23801  df-ims 23802  df-dip 23919  df-hnorm 24193  df-hvsub 24196  df-hlim 24197  df-hcau 24198  df-sh 24432  df-ch 24447  df-oc 24478  df-ch0 24479  df-chj 24536  df-hst 25439
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator