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Theorem strlem1 22830
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 3465 . . 3  |-  ( -.  ( A  \  B
)  =  (/)  <->  E. x  x  e.  ( A  \  B ) )
2 ssdif0 3513 . . 3  |-  ( A 
C_  B  <->  ( A  \  B )  =  (/) )
31, 2xchnxbir 300 . 2  |-  ( -.  A  C_  B  <->  E. x  x  e.  ( A  \  B ) )
4 eldifi 3298 . . . . . . . . . . 11  |-  ( x  e.  ( A  \  B )  ->  x  e.  A )
5 strlem1.1 . . . . . . . . . . . 12  |-  A  e. 
CH
65cheli 21812 . . . . . . . . . . 11  |-  ( x  e.  A  ->  x  e.  ~H )
7 normcl 21704 . . . . . . . . . . 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 21808 . . . . . . . . . . . . . . . 16  |-  ( B  e.  CH  ->  0h  e.  B )
119, 10ax-mp 8 . . . . . . . . . . . . . . 15  |-  0h  e.  B
12 eldifn 3299 . . . . . . . . . . . . . . 15  |-  ( 0h  e.  ( A  \  B )  ->  -.  0h  e.  B )
1311, 12mt2 170 . . . . . . . . . . . . . 14  |-  -.  0h  e.  ( A  \  B
)
14 eleq1 2343 . . . . . . . . . . . . . 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 21708 . . . . . . . . . . . . 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 2448 . . . . . . . . . . 11  |-  ( (
normh `  x )  =/=  0  <->  -.  ( normh `  x )  =  0 )
2119, 20sylibr 203 . . . . . . . . . 10  |-  ( x  e.  ( A  \  B )  ->  ( normh `  x )  =/=  0 )
228, 21rereccld 9587 . . . . . . . . 9  |-  ( x  e.  ( A  \  B )  ->  (
1  /  ( normh `  x ) )  e.  RR )
2322recnd 8861 . . . . . . . 8  |-  ( x  e.  ( A  \  B )  ->  (
1  /  ( normh `  x ) )  e.  CC )
245chshii 21807 . . . . . . . . . 10  |-  A  e.  SH
25 shmulcl 21797 . . . . . . . . . 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 8861 . . . . . . . . . 10  |-  ( x  e.  ( A  \  B )  ->  ( normh `  x )  e.  CC )
309chshii 21807 . . . . . . . . . . . 12  |-  B  e.  SH
31 shmulcl 21797 . . . . . . . . . . . 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 9531 . . . . . . . . . . . 12  |-  ( x  e.  ( A  \  B )  ->  (
( normh `  x )  x.  ( 1  /  ( normh `  x ) ) )  =  1 )
3635oveq1d 5873 . . . . . . . . . . 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 21587 . . . . . . . . . . . 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 21586 . . . . . . . . . . . 12  |-  ( x  e.  ~H  ->  (
1  .h  x )  =  x )
414, 6, 403syl 18 . . . . . . . . . . 11  |-  ( x  e.  ( A  \  B )  ->  (
1  .h  x )  =  x )
4236, 39, 413eqtr3d 2323 . . . . . . . . . 10  |-  ( x  e.  ( A  \  B )  ->  (
( normh `  x )  .h  ( ( 1  / 
( normh `  x )
)  .h  x ) )  =  x )
4342eleq1d 2349 . . . . . . . . 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 3162 . . . . . 6  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
48 eldif 3162 . . . . . 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 21719 . . . . . 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 2491 . . . . . . . . 9  |-  ( x  e.  ( A  \  B )  ->  x  =/=  0h )
54 normgt0 21706 . . . . . . . . . 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 8837 . . . . . . . . 9  |-  1  e.  RR
58 0le1 9297 . . . . . . . . 9  |-  0  <_  1
59 divge0 9625 . . . . . . . . 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 11905 . . . . . 6  |-  ( x  e.  ( A  \  B )  ->  ( abs `  ( 1  / 
( normh `  x )
) )  =  ( 1  /  ( normh `  x ) ) )
6362oveq1d 5873 . . . . 5  |-  ( x  e.  ( A  \  B )  ->  (
( abs `  (
1  /  ( normh `  x ) ) )  x.  ( normh `  x
) )  =  ( ( 1  /  ( normh `  x ) )  x.  ( normh `  x
) ) )
6429, 21recid2d 9532 . . . . 5  |-  ( x  e.  ( A  \  B )  ->  (
( 1  /  ( normh `  x ) )  x.  ( normh `  x
) )  =  1 )
6552, 63, 643eqtrd 2319 . . . 4  |-  ( x  e.  ( A  \  B )  ->  ( normh `  ( ( 1  /  ( normh `  x
) )  .h  x
) )  =  1 )
66 fveq2 5525 . . . . . 6  |-  ( u  =  ( ( 1  /  ( normh `  x
) )  .h  x
)  ->  ( normh `  u )  =  (
normh `  ( ( 1  /  ( normh `  x
) )  .h  x
) ) )
6766eqeq1d 2291 . . . . 5  |-  ( u  =  ( ( 1  /  ( normh `  x
) )  .h  x
)  ->  ( ( normh `  u )  =  1  <->  ( normh `  (
( 1  /  ( normh `  x ) )  .h  x ) )  =  1 ) )
6867rspcev 2884 . . . 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 1666 . 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 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    \ cdif 3149    C_ wss 3152   (/)c0 3455   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    < clt 8867    <_ cle 8868    / cdiv 9423   abscabs 11719   ~Hchil 21499    .h csm 21501   normhcno 21503   0hc0v 21504   SHcsh 21508   CHcch 21509
This theorem is referenced by:  stri  22837  hstri  22845
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  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-hilex 21579  ax-hfvadd 21580  ax-hv0cl 21583  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvmulass 21587  ax-hvmul0 21590  ax-hfi 21658  ax-his1 21661  ax-his3 21663  ax-his4 21664
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-iun 3907  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-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-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  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-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-hnorm 21548  df-sh 21786  df-ch 21801
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