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Theorem strlem1 22846
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 3478 . . 3  |-  ( -.  ( A  \  B
)  =  (/)  <->  E. x  x  e.  ( A  \  B ) )
2 ssdif0 3526 . . 3  |-  ( A 
C_  B  <->  ( A  \  B )  =  (/) )
31, 2xchnxbir 300 . 2  |-  ( -.  A  C_  B  <->  E. x  x  e.  ( A  \  B ) )
4 eldifi 3311 . . . . . . . . . . 11  |-  ( x  e.  ( A  \  B )  ->  x  e.  A )
5 strlem1.1 . . . . . . . . . . . 12  |-  A  e. 
CH
65cheli 21828 . . . . . . . . . . 11  |-  ( x  e.  A  ->  x  e.  ~H )
7 normcl 21720 . . . . . . . . . . 11  |-  ( x  e.  ~H  ->  ( normh `  x )  e.  RR )
84, 6, 73syl 18 . . . . . . . . . 10  |-  ( x  e.  ( A  \  B )  ->  ( normh `  x )  e.  RR )
9 strlem1.2 . . . . . . . . . . . . . . . 16  |-  B  e. 
CH
10 ch0 21824 . . . . . . . . . . . . . . . 16  |-  ( B  e.  CH  ->  0h  e.  B )
119, 10ax-mp 8 . . . . . . . . . . . . . . 15  |-  0h  e.  B
12 eldifn 3312 . . . . . . . . . . . . . . 15  |-  ( 0h  e.  ( A  \  B )  ->  -.  0h  e.  B )
1311, 12mt2 170 . . . . . . . . . . . . . 14  |-  -.  0h  e.  ( A  \  B
)
14 eleq1 2356 . . . . . . . . . . . . . 14  |-  ( x  =  0h  ->  (
x  e.  ( A 
\  B )  <->  0h  e.  ( A  \  B ) ) )
1513, 14mtbiri 294 . . . . . . . . . . . . 13  |-  ( x  =  0h  ->  -.  x  e.  ( A  \  B ) )
1615con2i 112 . . . . . . . . . . . 12  |-  ( x  e.  ( A  \  B )  ->  -.  x  =  0h )
17 norm-i 21724 . . . . . . . . . . . . 13  |-  ( x  e.  ~H  ->  (
( normh `  x )  =  0  <->  x  =  0h ) )
184, 6, 173syl 18 . . . . . . . . . . . 12  |-  ( x  e.  ( A  \  B )  ->  (
( normh `  x )  =  0  <->  x  =  0h ) )
1916, 18mtbird 292 . . . . . . . . . . 11  |-  ( x  e.  ( A  \  B )  ->  -.  ( normh `  x )  =  0 )
20 df-ne 2461 . . . . . . . . . . 11  |-  ( (
normh `  x )  =/=  0  <->  -.  ( normh `  x )  =  0 )
2119, 20sylibr 203 . . . . . . . . . 10  |-  ( x  e.  ( A  \  B )  ->  ( normh `  x )  =/=  0 )
228, 21rereccld 9603 . . . . . . . . 9  |-  ( x  e.  ( A  \  B )  ->  (
1  /  ( normh `  x ) )  e.  RR )
2322recnd 8877 . . . . . . . 8  |-  ( x  e.  ( A  \  B )  ->  (
1  /  ( normh `  x ) )  e.  CC )
245chshii 21823 . . . . . . . . . 10  |-  A  e.  SH
25 shmulcl 21813 . . . . . . . . . 10  |-  ( ( A  e.  SH  /\  ( 1  /  ( normh `  x ) )  e.  CC  /\  x  e.  A )  ->  (
( 1  /  ( normh `  x ) )  .h  x )  e.  A )
2624, 25mp3an1 1264 . . . . . . . . 9  |-  ( ( ( 1  /  ( normh `  x ) )  e.  CC  /\  x  e.  A )  ->  (
( 1  /  ( normh `  x ) )  .h  x )  e.  A )
2726ex 423 . . . . . . . 8  |-  ( ( 1  /  ( normh `  x ) )  e.  CC  ->  ( x  e.  A  ->  ( ( 1  /  ( normh `  x ) )  .h  x )  e.  A
) )
2823, 27syl 15 . . . . . . 7  |-  ( x  e.  ( A  \  B )  ->  (
x  e.  A  -> 
( ( 1  / 
( normh `  x )
)  .h  x )  e.  A ) )
298recnd 8877 . . . . . . . . . 10  |-  ( x  e.  ( A  \  B )  ->  ( normh `  x )  e.  CC )
309chshii 21823 . . . . . . . . . . . 12  |-  B  e.  SH
31 shmulcl 21813 . . . . . . . . . . . 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 )
3230, 31mp3an1 1264 . . . . . . . . . . 11  |-  ( ( ( normh `  x )  e.  CC  /\  ( ( 1  /  ( normh `  x ) )  .h  x )  e.  B
)  ->  ( ( normh `  x )  .h  ( ( 1  / 
( normh `  x )
)  .h  x ) )  e.  B )
3332ex 423 . . . . . . . . . 10  |-  ( (
normh `  x )  e.  CC  ->  ( (
( 1  /  ( normh `  x ) )  .h  x )  e.  B  ->  ( ( normh `  x )  .h  ( ( 1  / 
( normh `  x )
)  .h  x ) )  e.  B ) )
3429, 33syl 15 . . . . . . . . 9  |-  ( x  e.  ( A  \  B )  ->  (
( ( 1  / 
( normh `  x )
)  .h  x )  e.  B  ->  (
( normh `  x )  .h  ( ( 1  / 
( normh `  x )
)  .h  x ) )  e.  B ) )
3529, 21recidd 9547 . . . . . . . . . . . 12  |-  ( x  e.  ( A  \  B )  ->  (
( normh `  x )  x.  ( 1  /  ( normh `  x ) ) )  =  1 )
3635oveq1d 5889 . . . . . . . . . . 11  |-  ( x  e.  ( A  \  B )  ->  (
( ( normh `  x
)  x.  ( 1  /  ( normh `  x
) ) )  .h  x )  =  ( 1  .h  x ) )
374, 6syl 15 . . . . . . . . . . . 12  |-  ( x  e.  ( A  \  B )  ->  x  e.  ~H )
38 ax-hvmulass 21603 . . . . . . . . . . . 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 ) ) )
3929, 23, 37, 38syl3anc 1182 . . . . . . . . . . 11  |-  ( x  e.  ( A  \  B )  ->  (
( ( normh `  x
)  x.  ( 1  /  ( normh `  x
) ) )  .h  x )  =  ( ( normh `  x )  .h  ( ( 1  / 
( normh `  x )
)  .h  x ) ) )
40 ax-hvmulid 21602 . . . . . . . . . . . 12  |-  ( x  e.  ~H  ->  (
1  .h  x )  =  x )
414, 6, 403syl 18 . . . . . . . . . . 11  |-  ( x  e.  ( A  \  B )  ->  (
1  .h  x )  =  x )
4236, 39, 413eqtr3d 2336 . . . . . . . . . 10  |-  ( x  e.  ( A  \  B )  ->  (
( normh `  x )  .h  ( ( 1  / 
( normh `  x )
)  .h  x ) )  =  x )
4342eleq1d 2362 . . . . . . . . 9  |-  ( x  e.  ( A  \  B )  ->  (
( ( normh `  x
)  .h  ( ( 1  /  ( normh `  x ) )  .h  x ) )  e.  B  <->  x  e.  B
) )
4434, 43sylibd 205 . . . . . . . 8  |-  ( x  e.  ( A  \  B )  ->  (
( ( 1  / 
( normh `  x )
)  .h  x )  e.  B  ->  x  e.  B ) )
4544con3d 125 . . . . . . 7  |-  ( x  e.  ( A  \  B )  ->  ( -.  x  e.  B  ->  -.  ( ( 1  /  ( normh `  x
) )  .h  x
)  e.  B ) )
4628, 45anim12d 546 . . . . . 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 ) ) )
47 eldif 3175 . . . . . 6  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
48 eldif 3175 . . . . . 6  |-  ( ( ( 1  /  ( normh `  x ) )  .h  x )  e.  ( A  \  B
)  <->  ( ( ( 1  /  ( normh `  x ) )  .h  x )  e.  A  /\  -.  ( ( 1  /  ( normh `  x
) )  .h  x
)  e.  B ) )
4946, 47, 483imtr4g 261 . . . . 5  |-  ( x  e.  ( A  \  B )  ->  (
x  e.  ( A 
\  B )  -> 
( ( 1  / 
( normh `  x )
)  .h  x )  e.  ( A  \  B ) ) )
5049pm2.43i 43 . . . 4  |-  ( x  e.  ( A  \  B )  ->  (
( 1  /  ( normh `  x ) )  .h  x )  e.  ( A  \  B
) )
51 norm-iii 21735 . . . . . 6  |-  ( ( ( 1  /  ( normh `  x ) )  e.  CC  /\  x  e.  ~H )  ->  ( normh `  ( ( 1  /  ( normh `  x
) )  .h  x
) )  =  ( ( abs `  (
1  /  ( normh `  x ) ) )  x.  ( normh `  x
) ) )
5223, 37, 51syl2anc 642 . . . . 5  |-  ( x  e.  ( A  \  B )  ->  ( normh `  ( ( 1  /  ( normh `  x
) )  .h  x
) )  =  ( ( abs `  (
1  /  ( normh `  x ) ) )  x.  ( normh `  x
) ) )
5315necon2ai 2504 . . . . . . . . 9  |-  ( x  e.  ( A  \  B )  ->  x  =/=  0h )
54 normgt0 21722 . . . . . . . . . 10  |-  ( x  e.  ~H  ->  (
x  =/=  0h  <->  0  <  (
normh `  x ) ) )
554, 6, 543syl 18 . . . . . . . . 9  |-  ( x  e.  ( A  \  B )  ->  (
x  =/=  0h  <->  0  <  (
normh `  x ) ) )
5653, 55mpbid 201 . . . . . . . 8  |-  ( x  e.  ( A  \  B )  ->  0  <  ( normh `  x )
)
57 1re 8853 . . . . . . . . 9  |-  1  e.  RR
58 0le1 9313 . . . . . . . . 9  |-  0  <_  1
59 divge0 9641 . . . . . . . . 9  |-  ( ( ( 1  e.  RR  /\  0  <_  1 )  /\  ( ( normh `  x )  e.  RR  /\  0  <  ( normh `  x ) ) )  ->  0  <_  (
1  /  ( normh `  x ) ) )
6057, 58, 59mpanl12 663 . . . . . . . 8  |-  ( ( ( normh `  x )  e.  RR  /\  0  < 
( normh `  x )
)  ->  0  <_  ( 1  /  ( normh `  x ) ) )
618, 56, 60syl2anc 642 . . . . . . 7  |-  ( x  e.  ( A  \  B )  ->  0  <_  ( 1  /  ( normh `  x ) ) )
6222, 61absidd 11921 . . . . . 6  |-  ( x  e.  ( A  \  B )  ->  ( abs `  ( 1  / 
( normh `  x )
) )  =  ( 1  /  ( normh `  x ) ) )
6362oveq1d 5889 . . . . 5  |-  ( x  e.  ( A  \  B )  ->  (
( abs `  (
1  /  ( normh `  x ) ) )  x.  ( normh `  x
) )  =  ( ( 1  /  ( normh `  x ) )  x.  ( normh `  x
) ) )
6429, 21recid2d 9548 . . . . 5  |-  ( x  e.  ( A  \  B )  ->  (
( 1  /  ( normh `  x ) )  x.  ( normh `  x
) )  =  1 )
6552, 63, 643eqtrd 2332 . . . 4  |-  ( x  e.  ( A  \  B )  ->  ( normh `  ( ( 1  /  ( normh `  x
) )  .h  x
) )  =  1 )
66 fveq2 5541 . . . . . 6  |-  ( u  =  ( ( 1  /  ( normh `  x
) )  .h  x
)  ->  ( normh `  u )  =  (
normh `  ( ( 1  /  ( normh `  x
) )  .h  x
) ) )
6766eqeq1d 2304 . . . . 5  |-  ( u  =  ( ( 1  /  ( normh `  x
) )  .h  x
)  ->  ( ( normh `  u )  =  1  <->  ( normh `  (
( 1  /  ( normh `  x ) )  .h  x ) )  =  1 ) )
6867rspcev 2897 . . . 4  |-  ( ( ( ( 1  / 
( normh `  x )
)  .h  x )  e.  ( A  \  B )  /\  ( normh `  ( ( 1  /  ( normh `  x
) )  .h  x
) )  =  1 )  ->  E. u  e.  ( A  \  B
) ( normh `  u
)  =  1 )
6950, 65, 68syl2anc 642 . . 3  |-  ( x  e.  ( A  \  B )  ->  E. u  e.  ( A  \  B
) ( normh `  u
)  =  1 )
7069exlimiv 1624 . 2  |-  ( E. x  x  e.  ( A  \  B )  ->  E. u  e.  ( A  \  B ) ( normh `  u )  =  1 )
713, 70sylbi 187 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 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557    \ cdif 3162    C_ wss 3165   (/)c0 3468   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758    < clt 8883    <_ cle 8884    / cdiv 9439   abscabs 11735   ~Hchil 21515    .h csm 21517   normhcno 21519   0hc0v 21520   SHcsh 21524   CHcch 21525
This theorem is referenced by:  stri  22853  hstri  22861
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-hilex 21595  ax-hfvadd 21596  ax-hv0cl 21599  ax-hfvmul 21601  ax-hvmulid 21602  ax-hvmulass 21603  ax-hvmul0 21606  ax-hfi 21674  ax-his1 21677  ax-his3 21679  ax-his4 21680
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-hnorm 21564  df-sh 21802  df-ch 21817
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