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Theorem strlem1 27845
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 3715 . . 3  |-  ( -.  ( A  \  B
)  =  (/)  <->  E. x  x  e.  ( A  \  B ) )
2 ssdif0 3796 . . 3  |-  ( A 
C_  B  <->  ( A  \  B )  =  (/) )
31, 2xchnxbir 310 . 2  |-  ( -.  A  C_  B  <->  E. x  x  e.  ( A  \  B ) )
4 eldifi 3530 . . . . . . . . . . 11  |-  ( x  e.  ( A  \  B )  ->  x  e.  A )
5 strlem1.1 . . . . . . . . . . . 12  |-  A  e. 
CH
65cheli 26827 . . . . . . . . . . 11  |-  ( x  e.  A  ->  x  e.  ~H )
7 normcl 26720 . . . . . . . . . . 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 26823 . . . . . . . . . . . . . . . 16  |-  ( B  e.  CH  ->  0h  e.  B )
119, 10ax-mp 5 . . . . . . . . . . . . . . 15  |-  0h  e.  B
12 eldifn 3531 . . . . . . . . . . . . . . 15  |-  ( 0h  e.  ( A  \  B )  ->  -.  0h  e.  B )
1311, 12mt2 182 . . . . . . . . . . . . . 14  |-  -.  0h  e.  ( A  \  B
)
14 eleq1 2494 . . . . . . . . . . . . . 14  |-  ( x  =  0h  ->  (
x  e.  ( A 
\  B )  <->  0h  e.  ( A  \  B ) ) )
1513, 14mtbiri 304 . . . . . . . . . . . . 13  |-  ( x  =  0h  ->  -.  x  e.  ( A  \  B ) )
1615con2i 123 . . . . . . . . . . . 12  |-  ( x  e.  ( A  \  B )  ->  -.  x  =  0h )
17 norm-i 26724 . . . . . . . . . . . . 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 302 . . . . . . . . . . 11  |-  ( x  e.  ( A  \  B )  ->  -.  ( normh `  x )  =  0 )
2019neqned 2608 . . . . . . . . . 10  |-  ( x  e.  ( A  \  B )  ->  ( normh `  x )  =/=  0 )
218, 20rereccld 10385 . . . . . . . . 9  |-  ( x  e.  ( A  \  B )  ->  (
1  /  ( normh `  x ) )  e.  RR )
2221recnd 9620 . . . . . . . 8  |-  ( x  e.  ( A  \  B )  ->  (
1  /  ( normh `  x ) )  e.  CC )
235chshii 26822 . . . . . . . . . 10  |-  A  e.  SH
24 shmulcl 26813 . . . . . . . . . 10  |-  ( ( A  e.  SH  /\  ( 1  /  ( normh `  x ) )  e.  CC  /\  x  e.  A )  ->  (
( 1  /  ( normh `  x ) )  .h  x )  e.  A )
2523, 24mp3an1 1347 . . . . . . . . 9  |-  ( ( ( 1  /  ( normh `  x ) )  e.  CC  /\  x  e.  A )  ->  (
( 1  /  ( normh `  x ) )  .h  x )  e.  A )
2625ex 435 . . . . . . . 8  |-  ( ( 1  /  ( normh `  x ) )  e.  CC  ->  ( x  e.  A  ->  ( ( 1  /  ( normh `  x ) )  .h  x )  e.  A
) )
2722, 26syl 17 . . . . . . 7  |-  ( x  e.  ( A  \  B )  ->  (
x  e.  A  -> 
( ( 1  / 
( normh `  x )
)  .h  x )  e.  A ) )
288recnd 9620 . . . . . . . . . 10  |-  ( x  e.  ( A  \  B )  ->  ( normh `  x )  e.  CC )
299chshii 26822 . . . . . . . . . . . 12  |-  B  e.  SH
30 shmulcl 26813 . . . . . . . . . . . 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 1347 . . . . . . . . . . 11  |-  ( ( ( normh `  x )  e.  CC  /\  ( ( 1  /  ( normh `  x ) )  .h  x )  e.  B
)  ->  ( ( normh `  x )  .h  ( ( 1  / 
( normh `  x )
)  .h  x ) )  e.  B )
3231ex 435 . . . . . . . . . 10  |-  ( (
normh `  x )  e.  CC  ->  ( (
( 1  /  ( normh `  x ) )  .h  x )  e.  B  ->  ( ( normh `  x )  .h  ( ( 1  / 
( normh `  x )
)  .h  x ) )  e.  B ) )
3328, 32syl 17 . . . . . . . . 9  |-  ( x  e.  ( A  \  B )  ->  (
( ( 1  / 
( normh `  x )
)  .h  x )  e.  B  ->  (
( normh `  x )  .h  ( ( 1  / 
( normh `  x )
)  .h  x ) )  e.  B ) )
3428, 20recidd 10329 . . . . . . . . . . . 12  |-  ( x  e.  ( A  \  B )  ->  (
( normh `  x )  x.  ( 1  /  ( normh `  x ) ) )  =  1 )
3534oveq1d 6264 . . . . . . . . . . 11  |-  ( x  e.  ( A  \  B )  ->  (
( ( normh `  x
)  x.  ( 1  /  ( normh `  x
) ) )  .h  x )  =  ( 1  .h  x ) )
364, 6syl 17 . . . . . . . . . . . 12  |-  ( x  e.  ( A  \  B )  ->  x  e.  ~H )
37 ax-hvmulass 26602 . . . . . . . . . . . 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 1264 . . . . . . . . . . 11  |-  ( x  e.  ( A  \  B )  ->  (
( ( normh `  x
)  x.  ( 1  /  ( normh `  x
) ) )  .h  x )  =  ( ( normh `  x )  .h  ( ( 1  / 
( normh `  x )
)  .h  x ) ) )
39 ax-hvmulid 26601 . . . . . . . . . . . 12  |-  ( x  e.  ~H  ->  (
1  .h  x )  =  x )
404, 6, 393syl 18 . . . . . . . . . . 11  |-  ( x  e.  ( A  \  B )  ->  (
1  .h  x )  =  x )
4135, 38, 403eqtr3d 2470 . . . . . . . . . 10  |-  ( x  e.  ( A  \  B )  ->  (
( normh `  x )  .h  ( ( 1  / 
( normh `  x )
)  .h  x ) )  =  x )
4241eleq1d 2490 . . . . . . . . 9  |-  ( x  e.  ( A  \  B )  ->  (
( ( normh `  x
)  .h  ( ( 1  /  ( normh `  x ) )  .h  x ) )  e.  B  <->  x  e.  B
) )
4333, 42sylibd 217 . . . . . . . 8  |-  ( x  e.  ( A  \  B )  ->  (
( ( 1  / 
( normh `  x )
)  .h  x )  e.  B  ->  x  e.  B ) )
4443con3d 138 . . . . . . 7  |-  ( x  e.  ( A  \  B )  ->  ( -.  x  e.  B  ->  -.  ( ( 1  /  ( normh `  x
) )  .h  x
)  e.  B ) )
4527, 44anim12d 565 . . . . . 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 3389 . . . . . 6  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
47 eldif 3389 . . . . . 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 273 . . . . 5  |-  ( x  e.  ( A  \  B )  ->  (
x  e.  ( A 
\  B )  -> 
( ( 1  / 
( normh `  x )
)  .h  x )  e.  ( A  \  B ) ) )
4948pm2.43i 49 . . . 4  |-  ( x  e.  ( A  \  B )  ->  (
( 1  /  ( normh `  x ) )  .h  x )  e.  ( A  \  B
) )
50 norm-iii 26735 . . . . . 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 665 . . . . 5  |-  ( x  e.  ( A  \  B )  ->  ( normh `  ( ( 1  /  ( normh `  x
) )  .h  x
) )  =  ( ( abs `  (
1  /  ( normh `  x ) ) )  x.  ( normh `  x
) ) )
5215necon2ai 2630 . . . . . . . . 9  |-  ( x  e.  ( A  \  B )  ->  x  =/=  0h )
53 normgt0 26722 . . . . . . . . . 10  |-  ( x  e.  ~H  ->  (
x  =/=  0h  <->  0  <  (
normh `  x ) ) )
544, 6, 533syl 18 . . . . . . . . 9  |-  ( x  e.  ( A  \  B )  ->  (
x  =/=  0h  <->  0  <  (
normh `  x ) ) )
5552, 54mpbid 213 . . . . . . . 8  |-  ( x  e.  ( A  \  B )  ->  0  <  ( normh `  x )
)
56 1re 9593 . . . . . . . . 9  |-  1  e.  RR
57 0le1 10088 . . . . . . . . 9  |-  0  <_  1
58 divge0 10425 . . . . . . . . 9  |-  ( ( ( 1  e.  RR  /\  0  <_  1 )  /\  ( ( normh `  x )  e.  RR  /\  0  <  ( normh `  x ) ) )  ->  0  <_  (
1  /  ( normh `  x ) ) )
5956, 57, 58mpanl12 686 . . . . . . . 8  |-  ( ( ( normh `  x )  e.  RR  /\  0  < 
( normh `  x )
)  ->  0  <_  ( 1  /  ( normh `  x ) ) )
608, 55, 59syl2anc 665 . . . . . . 7  |-  ( x  e.  ( A  \  B )  ->  0  <_  ( 1  /  ( normh `  x ) ) )
6121, 60absidd 13428 . . . . . 6  |-  ( x  e.  ( A  \  B )  ->  ( abs `  ( 1  / 
( normh `  x )
) )  =  ( 1  /  ( normh `  x ) ) )
6261oveq1d 6264 . . . . 5  |-  ( x  e.  ( A  \  B )  ->  (
( abs `  (
1  /  ( normh `  x ) ) )  x.  ( normh `  x
) )  =  ( ( 1  /  ( normh `  x ) )  x.  ( normh `  x
) ) )
6328, 20recid2d 10330 . . . . 5  |-  ( x  e.  ( A  \  B )  ->  (
( 1  /  ( normh `  x ) )  x.  ( normh `  x
) )  =  1 )
6451, 62, 633eqtrd 2466 . . . 4  |-  ( x  e.  ( A  \  B )  ->  ( normh `  ( ( 1  /  ( normh `  x
) )  .h  x
) )  =  1 )
65 fveq2 5825 . . . . . 6  |-  ( u  =  ( ( 1  /  ( normh `  x
) )  .h  x
)  ->  ( normh `  u )  =  (
normh `  ( ( 1  /  ( normh `  x
) )  .h  x
) ) )
6665eqeq1d 2430 . . . . 5  |-  ( u  =  ( ( 1  /  ( normh `  x
) )  .h  x
)  ->  ( ( normh `  u )  =  1  <->  ( normh `  (
( 1  /  ( normh `  x ) )  .h  x ) )  =  1 ) )
6766rspcev 3125 . . . 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 665 . . 3  |-  ( x  e.  ( A  \  B )  ->  E. u  e.  ( A  \  B
) ( normh `  u
)  =  1 )
6968exlimiv 1770 . 2  |-  ( E. x  x  e.  ( A  \  B )  ->  E. u  e.  ( A  \  B ) ( normh `  u )  =  1 )
703, 69sylbi 198 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 187    /\ wa 370    = wceq 1437   E.wex 1657    e. wcel 1872    =/= wne 2599   E.wrex 2715    \ cdif 3376    C_ wss 3379   (/)c0 3704   class class class wbr 4366   ` cfv 5544  (class class class)co 6249   CCcc 9488   RRcr 9489   0cc0 9490   1c1 9491    x. cmul 9495    < clt 9626    <_ cle 9627    / cdiv 10220   abscabs 13241   ~Hchil 26514    .h csm 26516   normhcno 26518   0hc0v 26519   SHcsh 26523   CHcch 26524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568  ax-hilex 26594  ax-hfvadd 26595  ax-hv0cl 26598  ax-hfvmul 26600  ax-hvmulid 26601  ax-hvmulass 26602  ax-hvmul0 26605  ax-hfi 26674  ax-his1 26677  ax-his3 26679  ax-his4 26680
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-2nd 6752  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-sup 7909  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-div 10221  df-nn 10561  df-2 10619  df-3 10620  df-n0 10821  df-z 10889  df-uz 11111  df-rp 11254  df-seq 12164  df-exp 12223  df-cj 13106  df-re 13107  df-im 13108  df-sqrt 13242  df-abs 13243  df-hnorm 26563  df-sh 26802  df-ch 26816
This theorem is referenced by:  stri  27852  hstri  27860
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