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Theorem hst1h 22807
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 22797 . . . . 5  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  ( S `  A )  e.  ~H )
2 ax-hvaddid 21584 . . . . 5  |-  ( ( S `  A )  e.  ~H  ->  (
( S `  A
)  +h  0h )  =  ( S `  A ) )
31, 2syl 15 . . . 4  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( S `  A
)  +h  0h )  =  ( S `  A ) )
43adantr 451 . . 3  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  (
( S `  A
)  +h  0h )  =  ( S `  A ) )
5 ax-1cn 8795 . . . . . . . . . . . 12  |-  1  e.  CC
6 choccl 21885 . . . . . . . . . . . . . . . 16  |-  ( A  e.  CH  ->  ( _|_ `  A )  e. 
CH )
7 hstcl 22797 . . . . . . . . . . . . . . . 16  |-  ( ( S  e.  CHStates  /\  ( _|_ `  A )  e. 
CH )  ->  ( S `  ( _|_ `  A ) )  e. 
~H )
86, 7sylan2 460 . . . . . . . . . . . . . . 15  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  ( S `  ( _|_ `  A ) )  e. 
~H )
9 normcl 21704 . . . . . . . . . . . . . . 15  |-  ( ( S `  ( _|_ `  A ) )  e. 
~H  ->  ( normh `  ( S `  ( _|_ `  A ) ) )  e.  RR )
108, 9syl 15 . . . . . . . . . . . . . 14  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  ( normh `  ( S `  ( _|_ `  A ) ) )  e.  RR )
1110resqcld 11271 . . . . . . . . . . . . 13  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 )  e.  RR )
1211recnd 8861 . . . . . . . . . . . 12  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 )  e.  CC )
13 pncan2 9058 . . . . . . . . . . . 12  |-  ( ( 1  e.  CC  /\  ( ( normh `  ( S `  ( _|_ `  A ) ) ) ^ 2 )  e.  CC )  ->  (
( 1  +  ( ( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 ) )  - 
1 )  =  ( ( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 ) )
145, 12, 13sylancr 644 . . . . . . . . . . 11  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( 1  +  ( ( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 ) )  - 
1 )  =  ( ( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 ) )
1514adantr 451 . . . . . . . . . 10  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  (
( 1  +  ( ( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 ) )  - 
1 )  =  ( ( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 ) )
16 oveq1 5865 . . . . . . . . . . . . . 14  |-  ( (
normh `  ( S `  A ) )  =  1  ->  ( ( normh `  ( S `  A ) ) ^
2 )  =  ( 1 ^ 2 ) )
17 sq1 11198 . . . . . . . . . . . . . 14  |-  ( 1 ^ 2 )  =  1
1816, 17syl6req 2332 . . . . . . . . . . . . 13  |-  ( (
normh `  ( S `  A ) )  =  1  ->  1  =  ( ( normh `  ( S `  A )
) ^ 2 ) )
1918oveq1d 5873 . . . . . . . . . . . 12  |-  ( (
normh `  ( S `  A ) )  =  1  ->  ( 1  +  ( ( normh `  ( S `  ( _|_ `  A ) ) ) ^ 2 ) )  =  ( ( ( normh `  ( S `  A ) ) ^
2 )  +  ( ( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 ) ) )
20 hstnmoc 22803 . . . . . . . . . . . 12  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( ( normh `  ( S `  A )
) ^ 2 )  +  ( ( normh `  ( S `  ( _|_ `  A ) ) ) ^ 2 ) )  =  1 )
2119, 20sylan9eqr 2337 . . . . . . . . . . 11  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  (
1  +  ( (
normh `  ( S `  ( _|_ `  A ) ) ) ^ 2 ) )  =  1 )
2221oveq1d 5873 . . . . . . . . . 10  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  (
( 1  +  ( ( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 ) )  - 
1 )  =  ( 1  -  1 ) )
2315, 22eqtr3d 2317 . . . . . . . . 9  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  (
( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 )  =  ( 1  -  1 ) )
24 1m1e0 9814 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
2523, 24syl6eq 2331 . . . . . . . 8  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  (
( normh `  ( S `  ( _|_ `  A
) ) ) ^
2 )  =  0 )
2625ex 423 . . . . . . 7  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( normh `  ( S `  A ) )  =  1  ->  ( ( normh `  ( S `  ( _|_ `  A ) ) ) ^ 2 )  =  0 ) )
2710recnd 8861 . . . . . . . . 9  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  ( normh `  ( S `  ( _|_ `  A ) ) )  e.  CC )
28 sqeq0 11168 . . . . . . . . 9  |-  ( (
normh `  ( S `  ( _|_ `  A ) ) )  e.  CC  ->  ( ( ( normh `  ( S `  ( _|_ `  A ) ) ) ^ 2 )  =  0  <->  ( normh `  ( S `  ( _|_ `  A ) ) )  =  0 ) )
2927, 28syl 15 . . . . . . . 8  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( ( normh `  ( S `  ( _|_ `  A ) ) ) ^ 2 )  =  0  <->  ( normh `  ( S `  ( _|_ `  A ) ) )  =  0 ) )
30 norm-i 21708 . . . . . . . . 9  |-  ( ( S `  ( _|_ `  A ) )  e. 
~H  ->  ( ( normh `  ( S `  ( _|_ `  A ) ) )  =  0  <->  ( S `  ( _|_ `  A ) )  =  0h ) )
318, 30syl 15 . . . . . . . 8  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( normh `  ( S `  ( _|_ `  A
) ) )  =  0  <->  ( S `  ( _|_ `  A ) )  =  0h )
)
3229, 31bitrd 244 . . . . . . 7  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( ( normh `  ( S `  ( _|_ `  A ) ) ) ^ 2 )  =  0  <->  ( S `  ( _|_ `  A ) )  =  0h )
)
3326, 32sylibd 205 . . . . . 6  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( normh `  ( S `  A ) )  =  1  ->  ( S `  ( _|_ `  A
) )  =  0h ) )
3433imp 418 . . . . 5  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  ( S `  ( _|_ `  A ) )  =  0h )
3534oveq2d 5874 . . . 4  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  (
( S `  A
)  +h  ( S `
 ( _|_ `  A
) ) )  =  ( ( S `  A )  +h  0h ) )
36 hstoc 22802 . . . . 5  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( S `  A
)  +h  ( S `
 ( _|_ `  A
) ) )  =  ( S `  ~H ) )
3736adantr 451 . . . 4  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  (
( S `  A
)  +h  ( S `
 ( _|_ `  A
) ) )  =  ( S `  ~H ) )
3835, 37eqtr3d 2317 . . 3  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  (
( S `  A
)  +h  0h )  =  ( S `  ~H ) )
394, 38eqtr3d 2317 . 2  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( normh `  ( S `  A ) )  =  1 )  ->  ( S `  A )  =  ( S `  ~H ) )
40 fveq2 5525 . . 3  |-  ( ( S `  A )  =  ( S `  ~H )  ->  ( normh `  ( S `  A
) )  =  (
normh `  ( S `  ~H ) ) )
41 hst1a 22798 . . . 4  |-  ( S  e.  CHStates  ->  ( normh `  ( S `  ~H )
)  =  1 )
4241adantr 451 . . 3  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  ( normh `  ( S `  ~H ) )  =  1 )
4340, 42sylan9eqr 2337 . 2  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( S `  A
)  =  ( S `
 ~H ) )  ->  ( normh `  ( S `  A )
)  =  1 )
4439, 43impbida 805 1  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  (
( normh `  ( S `  A ) )  =  1  <->  ( S `  A )  =  ( S `  ~H )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    - cmin 9037   2c2 9795   ^cexp 11104   ~Hchil 21499    +h cva 21500   normhcno 21503   0hc0v 21504   CHcch 21509   _|_cort 21510   CHStateschst 21543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817  ax-hilex 21579  ax-hfvadd 21580  ax-hvcom 21581  ax-hvass 21582  ax-hv0cl 21583  ax-hvaddid 21584  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvmulass 21587  ax-hvdistr1 21588  ax-hvdistr2 21589  ax-hvmul0 21590  ax-hfi 21658  ax-his1 21661  ax-his2 21662  ax-his3 21663  ax-his4 21664  ax-hcompl 21781
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cn 16957  df-cnp 16958  df-lm 16959  df-haus 17043  df-tx 17257  df-hmeo 17446  df-xms 17885  df-ms 17886  df-tms 17887  df-cau 18682  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-vs 21155  df-nmcv 21156  df-ims 21157  df-dip 21274  df-hnorm 21548  df-hvsub 21551  df-hlim 21552  df-hcau 21553  df-sh 21786  df-ch 21801  df-oc 21831  df-ch0 21832  df-chj 21889  df-hst 22792
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