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Theorem strlem1 27285
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 3722 . . 3  |-  ( -.  ( A  \  B
)  =  (/)  <->  E. x  x  e.  ( A  \  B ) )
2 ssdif0 3801 . . 3  |-  ( A 
C_  B  <->  ( A  \  B )  =  (/) )
31, 2xchnxbir 307 . 2  |-  ( -.  A  C_  B  <->  E. x  x  e.  ( A  \  B ) )
4 eldifi 3540 . . . . . . . . . . 11  |-  ( x  e.  ( A  \  B )  ->  x  e.  A )
5 strlem1.1 . . . . . . . . . . . 12  |-  A  e. 
CH
65cheli 26267 . . . . . . . . . . 11  |-  ( x  e.  A  ->  x  e.  ~H )
7 normcl 26159 . . . . . . . . . . 11  |-  ( x  e.  ~H  ->  ( normh `  x )  e.  RR )
84, 6, 73syl 20 . . . . . . . . . 10  |-  ( x  e.  ( A  \  B )  ->  ( normh `  x )  e.  RR )
9 strlem1.2 . . . . . . . . . . . . . . . 16  |-  B  e. 
CH
10 ch0 26263 . . . . . . . . . . . . . . . 16  |-  ( B  e.  CH  ->  0h  e.  B )
119, 10ax-mp 5 . . . . . . . . . . . . . . 15  |-  0h  e.  B
12 eldifn 3541 . . . . . . . . . . . . . . 15  |-  ( 0h  e.  ( A  \  B )  ->  -.  0h  e.  B )
1311, 12mt2 179 . . . . . . . . . . . . . 14  |-  -.  0h  e.  ( A  \  B
)
14 eleq1 2454 . . . . . . . . . . . . . 14  |-  ( x  =  0h  ->  (
x  e.  ( A 
\  B )  <->  0h  e.  ( A  \  B ) ) )
1513, 14mtbiri 301 . . . . . . . . . . . . 13  |-  ( x  =  0h  ->  -.  x  e.  ( A  \  B ) )
1615con2i 120 . . . . . . . . . . . 12  |-  ( x  e.  ( A  \  B )  ->  -.  x  =  0h )
17 norm-i 26163 . . . . . . . . . . . . 13  |-  ( x  e.  ~H  ->  (
( normh `  x )  =  0  <->  x  =  0h ) )
184, 6, 173syl 20 . . . . . . . . . . . 12  |-  ( x  e.  ( A  \  B )  ->  (
( normh `  x )  =  0  <->  x  =  0h ) )
1916, 18mtbird 299 . . . . . . . . . . 11  |-  ( x  e.  ( A  \  B )  ->  -.  ( normh `  x )  =  0 )
2019neqned 2585 . . . . . . . . . 10  |-  ( x  e.  ( A  \  B )  ->  ( normh `  x )  =/=  0 )
218, 20rereccld 10288 . . . . . . . . 9  |-  ( x  e.  ( A  \  B )  ->  (
1  /  ( normh `  x ) )  e.  RR )
2221recnd 9533 . . . . . . . 8  |-  ( x  e.  ( A  \  B )  ->  (
1  /  ( normh `  x ) )  e.  CC )
235chshii 26262 . . . . . . . . . 10  |-  A  e.  SH
24 shmulcl 26252 . . . . . . . . . 10  |-  ( ( A  e.  SH  /\  ( 1  /  ( normh `  x ) )  e.  CC  /\  x  e.  A )  ->  (
( 1  /  ( normh `  x ) )  .h  x )  e.  A )
2523, 24mp3an1 1309 . . . . . . . . 9  |-  ( ( ( 1  /  ( normh `  x ) )  e.  CC  /\  x  e.  A )  ->  (
( 1  /  ( normh `  x ) )  .h  x )  e.  A )
2625ex 432 . . . . . . . 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 9533 . . . . . . . . . 10  |-  ( x  e.  ( A  \  B )  ->  ( normh `  x )  e.  CC )
299chshii 26262 . . . . . . . . . . . 12  |-  B  e.  SH
30 shmulcl 26252 . . . . . . . . . . . 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 1309 . . . . . . . . . . 11  |-  ( ( ( normh `  x )  e.  CC  /\  ( ( 1  /  ( normh `  x ) )  .h  x )  e.  B
)  ->  ( ( normh `  x )  .h  ( ( 1  / 
( normh `  x )
)  .h  x ) )  e.  B )
3231ex 432 . . . . . . . . . 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 10232 . . . . . . . . . . . 12  |-  ( x  e.  ( A  \  B )  ->  (
( normh `  x )  x.  ( 1  /  ( normh `  x ) ) )  =  1 )
3534oveq1d 6211 . . . . . . . . . . 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 26041 . . . . . . . . . . . 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 1226 . . . . . . . . . . 11  |-  ( x  e.  ( A  \  B )  ->  (
( ( normh `  x
)  x.  ( 1  /  ( normh `  x
) ) )  .h  x )  =  ( ( normh `  x )  .h  ( ( 1  / 
( normh `  x )
)  .h  x ) ) )
39 ax-hvmulid 26040 . . . . . . . . . . . 12  |-  ( x  e.  ~H  ->  (
1  .h  x )  =  x )
404, 6, 393syl 20 . . . . . . . . . . 11  |-  ( x  e.  ( A  \  B )  ->  (
1  .h  x )  =  x )
4135, 38, 403eqtr3d 2431 . . . . . . . . . 10  |-  ( x  e.  ( A  \  B )  ->  (
( normh `  x )  .h  ( ( 1  / 
( normh `  x )
)  .h  x ) )  =  x )
4241eleq1d 2451 . . . . . . . . 9  |-  ( x  e.  ( A  \  B )  ->  (
( ( normh `  x
)  .h  ( ( 1  /  ( normh `  x ) )  .h  x ) )  e.  B  <->  x  e.  B
) )
4333, 42sylibd 214 . . . . . . . 8  |-  ( x  e.  ( A  \  B )  ->  (
( ( 1  / 
( normh `  x )
)  .h  x )  e.  B  ->  x  e.  B ) )
4443con3d 133 . . . . . . 7  |-  ( x  e.  ( A  \  B )  ->  ( -.  x  e.  B  ->  -.  ( ( 1  /  ( normh `  x
) )  .h  x
)  e.  B ) )
4527, 44anim12d 561 . . . . . 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 3399 . . . . . 6  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
47 eldif 3399 . . . . . 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 270 . . . . 5  |-  ( x  e.  ( A  \  B )  ->  (
x  e.  ( A 
\  B )  -> 
( ( 1  / 
( normh `  x )
)  .h  x )  e.  ( A  \  B ) ) )
4948pm2.43i 47 . . . 4  |-  ( x  e.  ( A  \  B )  ->  (
( 1  /  ( normh `  x ) )  .h  x )  e.  ( A  \  B
) )
50 norm-iii 26174 . . . . . 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 659 . . . . 5  |-  ( x  e.  ( A  \  B )  ->  ( normh `  ( ( 1  /  ( normh `  x
) )  .h  x
) )  =  ( ( abs `  (
1  /  ( normh `  x ) ) )  x.  ( normh `  x
) ) )
5215necon2ai 2617 . . . . . . . . 9  |-  ( x  e.  ( A  \  B )  ->  x  =/=  0h )
53 normgt0 26161 . . . . . . . . . 10  |-  ( x  e.  ~H  ->  (
x  =/=  0h  <->  0  <  (
normh `  x ) ) )
544, 6, 533syl 20 . . . . . . . . 9  |-  ( x  e.  ( A  \  B )  ->  (
x  =/=  0h  <->  0  <  (
normh `  x ) ) )
5552, 54mpbid 210 . . . . . . . 8  |-  ( x  e.  ( A  \  B )  ->  0  <  ( normh `  x )
)
56 1re 9506 . . . . . . . . 9  |-  1  e.  RR
57 0le1 9993 . . . . . . . . 9  |-  0  <_  1
58 divge0 10328 . . . . . . . . 9  |-  ( ( ( 1  e.  RR  /\  0  <_  1 )  /\  ( ( normh `  x )  e.  RR  /\  0  <  ( normh `  x ) ) )  ->  0  <_  (
1  /  ( normh `  x ) ) )
5956, 57, 58mpanl12 680 . . . . . . . 8  |-  ( ( ( normh `  x )  e.  RR  /\  0  < 
( normh `  x )
)  ->  0  <_  ( 1  /  ( normh `  x ) ) )
608, 55, 59syl2anc 659 . . . . . . 7  |-  ( x  e.  ( A  \  B )  ->  0  <_  ( 1  /  ( normh `  x ) ) )
6121, 60absidd 13256 . . . . . 6  |-  ( x  e.  ( A  \  B )  ->  ( abs `  ( 1  / 
( normh `  x )
) )  =  ( 1  /  ( normh `  x ) ) )
6261oveq1d 6211 . . . . 5  |-  ( x  e.  ( A  \  B )  ->  (
( abs `  (
1  /  ( normh `  x ) ) )  x.  ( normh `  x
) )  =  ( ( 1  /  ( normh `  x ) )  x.  ( normh `  x
) ) )
6328, 20recid2d 10233 . . . . 5  |-  ( x  e.  ( A  \  B )  ->  (
( 1  /  ( normh `  x ) )  x.  ( normh `  x
) )  =  1 )
6451, 62, 633eqtrd 2427 . . . 4  |-  ( x  e.  ( A  \  B )  ->  ( normh `  ( ( 1  /  ( normh `  x
) )  .h  x
) )  =  1 )
65 fveq2 5774 . . . . . 6  |-  ( u  =  ( ( 1  /  ( normh `  x
) )  .h  x
)  ->  ( normh `  u )  =  (
normh `  ( ( 1  /  ( normh `  x
) )  .h  x
) ) )
6665eqeq1d 2384 . . . . 5  |-  ( u  =  ( ( 1  /  ( normh `  x
) )  .h  x
)  ->  ( ( normh `  u )  =  1  <->  ( normh `  (
( 1  /  ( normh `  x ) )  .h  x ) )  =  1 ) )
6766rspcev 3135 . . . 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 659 . . 3  |-  ( x  e.  ( A  \  B )  ->  E. u  e.  ( A  \  B
) ( normh `  u
)  =  1 )
6968exlimiv 1730 . 2  |-  ( E. x  x  e.  ( A  \  B )  ->  E. u  e.  ( A  \  B ) ( normh `  u )  =  1 )
703, 69sylbi 195 1  |-  ( -.  A  C_  B  ->  E. u  e.  ( A 
\  B ) (
normh `  u )  =  1 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399   E.wex 1620    e. wcel 1826    =/= wne 2577   E.wrex 2733    \ cdif 3386    C_ wss 3389   (/)c0 3711   class class class wbr 4367   ` cfv 5496  (class class class)co 6196   CCcc 9401   RRcr 9402   0cc0 9403   1c1 9404    x. cmul 9408    < clt 9539    <_ cle 9540    / cdiv 10123   abscabs 13069   ~Hchil 25953    .h csm 25955   normhcno 25957   0hc0v 25958   SHcsh 25962   CHcch 25963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481  ax-hilex 26033  ax-hfvadd 26034  ax-hv0cl 26037  ax-hfvmul 26039  ax-hvmulid 26040  ax-hvmulass 26041  ax-hvmul0 26044  ax-hfi 26113  ax-his1 26116  ax-his3 26118  ax-his4 26119
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-sup 7816  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-n0 10713  df-z 10782  df-uz 11002  df-rp 11140  df-seq 12011  df-exp 12070  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-hnorm 26002  df-sh 26241  df-ch 26256
This theorem is referenced by:  stri  27292  hstri  27300
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