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Theorem hst1h 25634
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 25624 . . . . 5  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  ( S `  A )  e.  ~H )
2 ax-hvaddid 24409 . . . . 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 465 . . 3  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  (
( S `  A
)  +h  0h )  =  ( S `  A ) )
5 ax-1cn 9343 . . . . . . . . . . . 12  |-  1  e.  CC
6 choccl 24712 . . . . . . . . . . . . . . . 16  |-  ( A  e.  CH  ->  ( _|_ `  A )  e. 
CH )
7 hstcl 25624 . . . . . . . . . . . . . . . 16  |-  ( ( S  e.  CHStates  /\  ( _|_ `  A )  e. 
CH )  ->  ( S `  ( _|_ `  A ) )  e. 
~H )
86, 7sylan2 474 . . . . . . . . . . . . . . 15  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  ( S `  ( _|_ `  A ) )  e. 
~H )
9 normcl 24530 . . . . . . . . . . . . . . 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 12037 . . . . . . . . . . . . 13  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 )  e.  RR )
1211recnd 9415 . . . . . . . . . . . 12  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 )  e.  CC )
13 pncan2 9620 . . . . . . . . . . . 12  |-  ( ( 1  e.  CC  /\  ( ( normh `  ( S `  ( _|_ `  A ) ) ) ^ 2 )  e.  CC )  ->  (
( 1  +  ( ( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 ) )  - 
1 )  =  ( ( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 ) )
145, 12, 13sylancr 663 . . . . . . . . . . 11  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( 1  +  ( ( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 ) )  - 
1 )  =  ( ( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 ) )
1514adantr 465 . . . . . . . . . 10  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  (
( 1  +  ( ( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 ) )  - 
1 )  =  ( ( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 ) )
16 oveq1 6101 . . . . . . . . . . . . . 14  |-  ( (
normh `  ( S `  A ) )  =  1  ->  ( ( normh `  ( S `  A ) ) ^
2 )  =  ( 1 ^ 2 ) )
17 sq1 11963 . . . . . . . . . . . . . 14  |-  ( 1 ^ 2 )  =  1
1816, 17syl6req 2492 . . . . . . . . . . . . 13  |-  ( (
normh `  ( S `  A ) )  =  1  ->  1  =  ( ( normh `  ( S `  A )
) ^ 2 ) )
1918oveq1d 6109 . . . . . . . . . . . 12  |-  ( (
normh `  ( S `  A ) )  =  1  ->  ( 1  +  ( ( normh `  ( S `  ( _|_ `  A ) ) ) ^ 2 ) )  =  ( ( ( normh `  ( S `  A ) ) ^
2 )  +  ( ( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 ) ) )
20 hstnmoc 25630 . . . . . . . . . . . 12  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( ( normh `  ( S `  A )
) ^ 2 )  +  ( ( normh `  ( S `  ( _|_ `  A ) ) ) ^ 2 ) )  =  1 )
2119, 20sylan9eqr 2497 . . . . . . . . . . 11  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  (
1  +  ( (
normh `  ( S `  ( _|_ `  A ) ) ) ^ 2 ) )  =  1 )
2221oveq1d 6109 . . . . . . . . . 10  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  (
( 1  +  ( ( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 ) )  - 
1 )  =  ( 1  -  1 ) )
2315, 22eqtr3d 2477 . . . . . . . . 9  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  (
( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 )  =  ( 1  -  1 ) )
24 1m1e0 10393 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
2523, 24syl6eq 2491 . . . . . . . 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 9415 . . . . . . . . 9  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  ( normh `  ( S `  ( _|_ `  A ) ) )  e.  CC )
28 sqeq0 11933 . . . . . . . . 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 24534 . . . . . . . . 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 6110 . . . 4  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  (
( S `  A
)  +h  ( S `
 ( _|_ `  A
) ) )  =  ( ( S `  A )  +h  0h ) )
36 hstoc 25629 . . . . 5  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( S `  A
)  +h  ( S `
 ( _|_ `  A
) ) )  =  ( S `  ~H ) )
3736adantr 465 . . . 4  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  (
( S `  A
)  +h  ( S `
 ( _|_ `  A
) ) )  =  ( S `  ~H ) )
3835, 37eqtr3d 2477 . . 3  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  (
( S `  A
)  +h  0h )  =  ( S `  ~H ) )
394, 38eqtr3d 2477 . 2  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  ( S `  A )  =  ( S `  ~H ) )
40 fveq2 5694 . . 3  |-  ( ( S `  A )  =  ( S `  ~H )  ->  ( normh `  ( S `  A
) )  =  (
normh `  ( S `  ~H ) ) )
41 hst1a 25625 . . . 4  |-  ( S  e.  CHStates  ->  ( normh `  ( S `  ~H )
)  =  1 )
4241adantr 465 . . 3  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  ( normh `  ( S `  ~H ) )  =  1 )
4340, 42sylan9eqr 2497 . 2  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( S `  A
)  =  ( S `
 ~H ) )  ->  ( normh `  ( S `  A )
)  =  1 )
4439, 43impbida 828 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 1369    e. wcel 1756   ` cfv 5421  (class class class)co 6094   CCcc 9283   RRcr 9284   0cc0 9285   1c1 9286    + caddc 9288    - cmin 9598   2c2 10374   ^cexp 11868   ~Hchil 24324    +h cva 24325   normhcno 24328   0hc0v 24329   CHcch 24334   _|_cort 24335   CHStateschst 24368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-inf2 7850  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363  ax-addf 9364  ax-mulf 9365  ax-hilex 24404  ax-hfvadd 24405  ax-hvcom 24406  ax-hvass 24407  ax-hv0cl 24408  ax-hvaddid 24409  ax-hfvmul 24410  ax-hvmulid 24411  ax-hvmulass 24412  ax-hvdistr1 24413  ax-hvdistr2 24414  ax-hvmul0 24415  ax-hfi 24484  ax-his1 24487  ax-his2 24488  ax-his3 24489  ax-his4 24490  ax-hcompl 24607
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-iin 4177  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-se 4683  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-of 6323  df-om 6480  df-1st 6580  df-2nd 6581  df-supp 6694  df-recs 6835  df-rdg 6869  df-1o 6923  df-2o 6924  df-oadd 6927  df-er 7104  df-map 7219  df-pm 7220  df-ixp 7267  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-fsupp 7624  df-fi 7664  df-sup 7694  df-oi 7727  df-card 8112  df-cda 8340  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997  df-nn 10326  df-2 10383  df-3 10384  df-4 10385  df-5 10386  df-6 10387  df-7 10388  df-8 10389  df-9 10390  df-10 10391  df-n0 10583  df-z 10650  df-dec 10759  df-uz 10865  df-q 10957  df-rp 10995  df-xneg 11092  df-xadd 11093  df-xmul 11094  df-ioo 11307  df-icc 11310  df-fz 11441  df-fzo 11552  df-seq 11810  df-exp 11869  df-hash 12107  df-cj 12591  df-re 12592  df-im 12593  df-sqr 12727  df-abs 12728  df-clim 12969  df-sum 13167  df-struct 14179  df-ndx 14180  df-slot 14181  df-base 14182  df-sets 14183  df-ress 14184  df-plusg 14254  df-mulr 14255  df-starv 14256  df-sca 14257  df-vsca 14258  df-ip 14259  df-tset 14260  df-ple 14261  df-ds 14263  df-unif 14264  df-hom 14265  df-cco 14266  df-rest 14364  df-topn 14365  df-0g 14383  df-gsum 14384  df-topgen 14385  df-pt 14386  df-prds 14389  df-xrs 14443  df-qtop 14448  df-imas 14449  df-xps 14451  df-mre 14527  df-mrc 14528  df-acs 14530  df-mnd 15418  df-submnd 15468  df-mulg 15551  df-cntz 15838  df-cmn 16282  df-psmet 17812  df-xmet 17813  df-met 17814  df-bl 17815  df-mopn 17816  df-cnfld 17822  df-top 18506  df-bases 18508  df-topon 18509  df-topsp 18510  df-cn 18834  df-cnp 18835  df-lm 18836  df-haus 18922  df-tx 19138  df-hmeo 19331  df-xms 19898  df-ms 19899  df-tms 19900  df-cau 20770  df-grpo 23681  df-gid 23682  df-ginv 23683  df-gdiv 23684  df-ablo 23772  df-vc 23927  df-nv 23973  df-va 23976  df-ba 23977  df-sm 23978  df-0v 23979  df-vs 23980  df-nmcv 23981  df-ims 23982  df-dip 24099  df-hnorm 24373  df-hvsub 24376  df-hlim 24377  df-hcau 24378  df-sh 24612  df-ch 24627  df-oc 24658  df-ch0 24659  df-chj 24716  df-hst 25619
This theorem is referenced by: (None)
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