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Theorem strlem1 23753
Description: Lemma for strong state theorem: if closed subspace  A is not contained in  B, there is a unit vector  u in their difference. (Contributed by NM, 25-Oct-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
strlem1.1  |-  A  e. 
CH
strlem1.2  |-  B  e. 
CH
Assertion
Ref Expression
strlem1  |-  ( -.  A  C_  B  ->  E. u  e.  ( A 
\  B ) (
normh `  u )  =  1 )
Distinct variable groups:    u, A    u, B

Proof of Theorem strlem1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 neq0 3638 . . 3  |-  ( -.  ( A  \  B
)  =  (/)  <->  E. x  x  e.  ( A  \  B ) )
2 ssdif0 3686 . . 3  |-  ( A 
C_  B  <->  ( A  \  B )  =  (/) )
31, 2xchnxbir 301 . 2  |-  ( -.  A  C_  B  <->  E. x  x  e.  ( A  \  B ) )
4 eldifi 3469 . . . . . . . . . . 11  |-  ( x  e.  ( A  \  B )  ->  x  e.  A )
5 strlem1.1 . . . . . . . . . . . 12  |-  A  e. 
CH
65cheli 22735 . . . . . . . . . . 11  |-  ( x  e.  A  ->  x  e.  ~H )
7 normcl 22627 . . . . . . . . . . 11  |-  ( x  e.  ~H  ->  ( normh `  x )  e.  RR )
84, 6, 73syl 19 . . . . . . . . . 10  |-  ( x  e.  ( A  \  B )  ->  ( normh `  x )  e.  RR )
9 strlem1.2 . . . . . . . . . . . . . . . 16  |-  B  e. 
CH
10 ch0 22731 . . . . . . . . . . . . . . . 16  |-  ( B  e.  CH  ->  0h  e.  B )
119, 10ax-mp 8 . . . . . . . . . . . . . . 15  |-  0h  e.  B
12 eldifn 3470 . . . . . . . . . . . . . . 15  |-  ( 0h  e.  ( A  \  B )  ->  -.  0h  e.  B )
1311, 12mt2 172 . . . . . . . . . . . . . 14  |-  -.  0h  e.  ( A  \  B
)
14 eleq1 2496 . . . . . . . . . . . . . 14  |-  ( x  =  0h  ->  (
x  e.  ( A 
\  B )  <->  0h  e.  ( A  \  B ) ) )
1513, 14mtbiri 295 . . . . . . . . . . . . 13  |-  ( x  =  0h  ->  -.  x  e.  ( A  \  B ) )
1615con2i 114 . . . . . . . . . . . 12  |-  ( x  e.  ( A  \  B )  ->  -.  x  =  0h )
17 norm-i 22631 . . . . . . . . . . . . 13  |-  ( x  e.  ~H  ->  (
( normh `  x )  =  0  <->  x  =  0h ) )
184, 6, 173syl 19 . . . . . . . . . . . 12  |-  ( x  e.  ( A  \  B )  ->  (
( normh `  x )  =  0  <->  x  =  0h ) )
1916, 18mtbird 293 . . . . . . . . . . 11  |-  ( x  e.  ( A  \  B )  ->  -.  ( normh `  x )  =  0 )
2019neneqad 2674 . . . . . . . . . 10  |-  ( x  e.  ( A  \  B )  ->  ( normh `  x )  =/=  0 )
218, 20rereccld 9841 . . . . . . . . 9  |-  ( x  e.  ( A  \  B )  ->  (
1  /  ( normh `  x ) )  e.  RR )
2221recnd 9114 . . . . . . . 8  |-  ( x  e.  ( A  \  B )  ->  (
1  /  ( normh `  x ) )  e.  CC )
235chshii 22730 . . . . . . . . . 10  |-  A  e.  SH
24 shmulcl 22720 . . . . . . . . . 10  |-  ( ( A  e.  SH  /\  ( 1  /  ( normh `  x ) )  e.  CC  /\  x  e.  A )  ->  (
( 1  /  ( normh `  x ) )  .h  x )  e.  A )
2523, 24mp3an1 1266 . . . . . . . . 9  |-  ( ( ( 1  /  ( normh `  x ) )  e.  CC  /\  x  e.  A )  ->  (
( 1  /  ( normh `  x ) )  .h  x )  e.  A )
2625ex 424 . . . . . . . 8  |-  ( ( 1  /  ( normh `  x ) )  e.  CC  ->  ( x  e.  A  ->  ( ( 1  /  ( normh `  x ) )  .h  x )  e.  A
) )
2722, 26syl 16 . . . . . . 7  |-  ( x  e.  ( A  \  B )  ->  (
x  e.  A  -> 
( ( 1  / 
( normh `  x )
)  .h  x )  e.  A ) )
288recnd 9114 . . . . . . . . . 10  |-  ( x  e.  ( A  \  B )  ->  ( normh `  x )  e.  CC )
299chshii 22730 . . . . . . . . . . . 12  |-  B  e.  SH
30 shmulcl 22720 . . . . . . . . . . . 12  |-  ( ( B  e.  SH  /\  ( normh `  x )  e.  CC  /\  ( ( 1  /  ( normh `  x ) )  .h  x )  e.  B
)  ->  ( ( normh `  x )  .h  ( ( 1  / 
( normh `  x )
)  .h  x ) )  e.  B )
3129, 30mp3an1 1266 . . . . . . . . . . 11  |-  ( ( ( normh `  x )  e.  CC  /\  ( ( 1  /  ( normh `  x ) )  .h  x )  e.  B
)  ->  ( ( normh `  x )  .h  ( ( 1  / 
( normh `  x )
)  .h  x ) )  e.  B )
3231ex 424 . . . . . . . . . 10  |-  ( (
normh `  x )  e.  CC  ->  ( (
( 1  /  ( normh `  x ) )  .h  x )  e.  B  ->  ( ( normh `  x )  .h  ( ( 1  / 
( normh `  x )
)  .h  x ) )  e.  B ) )
3328, 32syl 16 . . . . . . . . 9  |-  ( x  e.  ( A  \  B )  ->  (
( ( 1  / 
( normh `  x )
)  .h  x )  e.  B  ->  (
( normh `  x )  .h  ( ( 1  / 
( normh `  x )
)  .h  x ) )  e.  B ) )
3428, 20recidd 9785 . . . . . . . . . . . 12  |-  ( x  e.  ( A  \  B )  ->  (
( normh `  x )  x.  ( 1  /  ( normh `  x ) ) )  =  1 )
3534oveq1d 6096 . . . . . . . . . . 11  |-  ( x  e.  ( A  \  B )  ->  (
( ( normh `  x
)  x.  ( 1  /  ( normh `  x
) ) )  .h  x )  =  ( 1  .h  x ) )
364, 6syl 16 . . . . . . . . . . . 12  |-  ( x  e.  ( A  \  B )  ->  x  e.  ~H )
37 ax-hvmulass 22510 . . . . . . . . . . . 12  |-  ( ( ( normh `  x )  e.  CC  /\  ( 1  /  ( normh `  x
) )  e.  CC  /\  x  e.  ~H )  ->  ( ( ( normh `  x )  x.  (
1  /  ( normh `  x ) ) )  .h  x )  =  ( ( normh `  x
)  .h  ( ( 1  /  ( normh `  x ) )  .h  x ) ) )
3828, 22, 36, 37syl3anc 1184 . . . . . . . . . . 11  |-  ( x  e.  ( A  \  B )  ->  (
( ( normh `  x
)  x.  ( 1  /  ( normh `  x
) ) )  .h  x )  =  ( ( normh `  x )  .h  ( ( 1  / 
( normh `  x )
)  .h  x ) ) )
39 ax-hvmulid 22509 . . . . . . . . . . . 12  |-  ( x  e.  ~H  ->  (
1  .h  x )  =  x )
404, 6, 393syl 19 . . . . . . . . . . 11  |-  ( x  e.  ( A  \  B )  ->  (
1  .h  x )  =  x )
4135, 38, 403eqtr3d 2476 . . . . . . . . . 10  |-  ( x  e.  ( A  \  B )  ->  (
( normh `  x )  .h  ( ( 1  / 
( normh `  x )
)  .h  x ) )  =  x )
4241eleq1d 2502 . . . . . . . . 9  |-  ( x  e.  ( A  \  B )  ->  (
( ( normh `  x
)  .h  ( ( 1  /  ( normh `  x ) )  .h  x ) )  e.  B  <->  x  e.  B
) )
4333, 42sylibd 206 . . . . . . . 8  |-  ( x  e.  ( A  \  B )  ->  (
( ( 1  / 
( normh `  x )
)  .h  x )  e.  B  ->  x  e.  B ) )
4443con3d 127 . . . . . . 7  |-  ( x  e.  ( A  \  B )  ->  ( -.  x  e.  B  ->  -.  ( ( 1  /  ( normh `  x
) )  .h  x
)  e.  B ) )
4527, 44anim12d 547 . . . . . 6  |-  ( x  e.  ( A  \  B )  ->  (
( x  e.  A  /\  -.  x  e.  B
)  ->  ( (
( 1  /  ( normh `  x ) )  .h  x )  e.  A  /\  -.  (
( 1  /  ( normh `  x ) )  .h  x )  e.  B ) ) )
46 eldif 3330 . . . . . 6  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
47 eldif 3330 . . . . . 6  |-  ( ( ( 1  /  ( normh `  x ) )  .h  x )  e.  ( A  \  B
)  <->  ( ( ( 1  /  ( normh `  x ) )  .h  x )  e.  A  /\  -.  ( ( 1  /  ( normh `  x
) )  .h  x
)  e.  B ) )
4845, 46, 473imtr4g 262 . . . . 5  |-  ( x  e.  ( A  \  B )  ->  (
x  e.  ( A 
\  B )  -> 
( ( 1  / 
( normh `  x )
)  .h  x )  e.  ( A  \  B ) ) )
4948pm2.43i 45 . . . 4  |-  ( x  e.  ( A  \  B )  ->  (
( 1  /  ( normh `  x ) )  .h  x )  e.  ( A  \  B
) )
50 norm-iii 22642 . . . . . 6  |-  ( ( ( 1  /  ( normh `  x ) )  e.  CC  /\  x  e.  ~H )  ->  ( normh `  ( ( 1  /  ( normh `  x
) )  .h  x
) )  =  ( ( abs `  (
1  /  ( normh `  x ) ) )  x.  ( normh `  x
) ) )
5122, 36, 50syl2anc 643 . . . . 5  |-  ( x  e.  ( A  \  B )  ->  ( normh `  ( ( 1  /  ( normh `  x
) )  .h  x
) )  =  ( ( abs `  (
1  /  ( normh `  x ) ) )  x.  ( normh `  x
) ) )
5215necon2ai 2649 . . . . . . . . 9  |-  ( x  e.  ( A  \  B )  ->  x  =/=  0h )
53 normgt0 22629 . . . . . . . . . 10  |-  ( x  e.  ~H  ->  (
x  =/=  0h  <->  0  <  (
normh `  x ) ) )
544, 6, 533syl 19 . . . . . . . . 9  |-  ( x  e.  ( A  \  B )  ->  (
x  =/=  0h  <->  0  <  (
normh `  x ) ) )
5552, 54mpbid 202 . . . . . . . 8  |-  ( x  e.  ( A  \  B )  ->  0  <  ( normh `  x )
)
56 1re 9090 . . . . . . . . 9  |-  1  e.  RR
57 0le1 9551 . . . . . . . . 9  |-  0  <_  1
58 divge0 9879 . . . . . . . . 9  |-  ( ( ( 1  e.  RR  /\  0  <_  1 )  /\  ( ( normh `  x )  e.  RR  /\  0  <  ( normh `  x ) ) )  ->  0  <_  (
1  /  ( normh `  x ) ) )
5956, 57, 58mpanl12 664 . . . . . . . 8  |-  ( ( ( normh `  x )  e.  RR  /\  0  < 
( normh `  x )
)  ->  0  <_  ( 1  /  ( normh `  x ) ) )
608, 55, 59syl2anc 643 . . . . . . 7  |-  ( x  e.  ( A  \  B )  ->  0  <_  ( 1  /  ( normh `  x ) ) )
6121, 60absidd 12225 . . . . . 6  |-  ( x  e.  ( A  \  B )  ->  ( abs `  ( 1  / 
( normh `  x )
) )  =  ( 1  /  ( normh `  x ) ) )
6261oveq1d 6096 . . . . 5  |-  ( x  e.  ( A  \  B )  ->  (
( abs `  (
1  /  ( normh `  x ) ) )  x.  ( normh `  x
) )  =  ( ( 1  /  ( normh `  x ) )  x.  ( normh `  x
) ) )
6328, 20recid2d 9786 . . . . 5  |-  ( x  e.  ( A  \  B )  ->  (
( 1  /  ( normh `  x ) )  x.  ( normh `  x
) )  =  1 )
6451, 62, 633eqtrd 2472 . . . 4  |-  ( x  e.  ( A  \  B )  ->  ( normh `  ( ( 1  /  ( normh `  x
) )  .h  x
) )  =  1 )
65 fveq2 5728 . . . . . 6  |-  ( u  =  ( ( 1  /  ( normh `  x
) )  .h  x
)  ->  ( normh `  u )  =  (
normh `  ( ( 1  /  ( normh `  x
) )  .h  x
) ) )
6665eqeq1d 2444 . . . . 5  |-  ( u  =  ( ( 1  /  ( normh `  x
) )  .h  x
)  ->  ( ( normh `  u )  =  1  <->  ( normh `  (
( 1  /  ( normh `  x ) )  .h  x ) )  =  1 ) )
6766rspcev 3052 . . . 4  |-  ( ( ( ( 1  / 
( normh `  x )
)  .h  x )  e.  ( A  \  B )  /\  ( normh `  ( ( 1  /  ( normh `  x
) )  .h  x
) )  =  1 )  ->  E. u  e.  ( A  \  B
) ( normh `  u
)  =  1 )
6849, 64, 67syl2anc 643 . . 3  |-  ( x  e.  ( A  \  B )  ->  E. u  e.  ( A  \  B
) ( normh `  u
)  =  1 )
6968exlimiv 1644 . 2  |-  ( E. x  x  e.  ( A  \  B )  ->  E. u  e.  ( A  \  B ) ( normh `  u )  =  1 )
703, 69sylbi 188 1  |-  ( -.  A  C_  B  ->  E. u  e.  ( A 
\  B ) (
normh `  u )  =  1 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706    \ cdif 3317    C_ wss 3320   (/)c0 3628   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    x. cmul 8995    < clt 9120    <_ cle 9121    / cdiv 9677   abscabs 12039   ~Hchil 22422    .h csm 22424   normhcno 22426   0hc0v 22427   SHcsh 22431   CHcch 22432
This theorem is referenced by:  stri  23760  hstri  23768
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-hilex 22502  ax-hfvadd 22503  ax-hv0cl 22506  ax-hfvmul 22508  ax-hvmulid 22509  ax-hvmulass 22510  ax-hvmul0 22513  ax-hfi 22581  ax-his1 22584  ax-his3 22586  ax-his4 22587
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-hnorm 22471  df-sh 22709  df-ch 22724
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