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Theorem norm1exi 22744
Description: A normalized vector exists in a subspace iff the subspace has a nonzero vector. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
norm1ex.1  |-  H  e.  SH
Assertion
Ref Expression
norm1exi  |-  ( E. x  e.  H  x  =/=  0h  <->  E. y  e.  H  ( normh `  y )  =  1 )
Distinct variable groups:    x, H    y, H

Proof of Theorem norm1exi
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 neeq1 2606 . . 3  |-  ( x  =  z  ->  (
x  =/=  0h  <->  z  =/=  0h ) )
21cbvrexv 2925 . 2  |-  ( E. x  e.  H  x  =/=  0h  <->  E. z  e.  H  z  =/=  0h )
3 norm1ex.1 . . . . . . . . . . 11  |-  H  e.  SH
43sheli 22708 . . . . . . . . . 10  |-  ( z  e.  H  ->  z  e.  ~H )
5 normcl 22619 . . . . . . . . . 10  |-  ( z  e.  ~H  ->  ( normh `  z )  e.  RR )
64, 5syl 16 . . . . . . . . 9  |-  ( z  e.  H  ->  ( normh `  z )  e.  RR )
76adantr 452 . . . . . . . 8  |-  ( ( z  e.  H  /\  z  =/=  0h )  -> 
( normh `  z )  e.  RR )
8 normne0 22624 . . . . . . . . . 10  |-  ( z  e.  ~H  ->  (
( normh `  z )  =/=  0  <->  z  =/=  0h ) )
94, 8syl 16 . . . . . . . . 9  |-  ( z  e.  H  ->  (
( normh `  z )  =/=  0  <->  z  =/=  0h ) )
109biimpar 472 . . . . . . . 8  |-  ( ( z  e.  H  /\  z  =/=  0h )  -> 
( normh `  z )  =/=  0 )
117, 10rereccld 9833 . . . . . . 7  |-  ( ( z  e.  H  /\  z  =/=  0h )  -> 
( 1  /  ( normh `  z ) )  e.  RR )
1211recnd 9106 . . . . . 6  |-  ( ( z  e.  H  /\  z  =/=  0h )  -> 
( 1  /  ( normh `  z ) )  e.  CC )
13 simpl 444 . . . . . 6  |-  ( ( z  e.  H  /\  z  =/=  0h )  -> 
z  e.  H )
14 shmulcl 22712 . . . . . . 7  |-  ( ( H  e.  SH  /\  ( 1  /  ( normh `  z ) )  e.  CC  /\  z  e.  H )  ->  (
( 1  /  ( normh `  z ) )  .h  z )  e.  H )
153, 14mp3an1 1266 . . . . . 6  |-  ( ( ( 1  /  ( normh `  z ) )  e.  CC  /\  z  e.  H )  ->  (
( 1  /  ( normh `  z ) )  .h  z )  e.  H )
1612, 13, 15syl2anc 643 . . . . 5  |-  ( ( z  e.  H  /\  z  =/=  0h )  -> 
( ( 1  / 
( normh `  z )
)  .h  z )  e.  H )
17 norm1 22743 . . . . . 6  |-  ( ( z  e.  ~H  /\  z  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  z ) )  .h  z ) )  =  1 )
184, 17sylan 458 . . . . 5  |-  ( ( z  e.  H  /\  z  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  z ) )  .h  z ) )  =  1 )
19 fveq2 5720 . . . . . . 7  |-  ( y  =  ( ( 1  /  ( normh `  z
) )  .h  z
)  ->  ( normh `  y )  =  (
normh `  ( ( 1  /  ( normh `  z
) )  .h  z
) ) )
2019eqeq1d 2443 . . . . . 6  |-  ( y  =  ( ( 1  /  ( normh `  z
) )  .h  z
)  ->  ( ( normh `  y )  =  1  <->  ( normh `  (
( 1  /  ( normh `  z ) )  .h  z ) )  =  1 ) )
2120rspcev 3044 . . . . 5  |-  ( ( ( ( 1  / 
( normh `  z )
)  .h  z )  e.  H  /\  ( normh `  ( ( 1  /  ( normh `  z
) )  .h  z
) )  =  1 )  ->  E. y  e.  H  ( normh `  y )  =  1 )
2216, 18, 21syl2anc 643 . . . 4  |-  ( ( z  e.  H  /\  z  =/=  0h )  ->  E. y  e.  H  ( normh `  y )  =  1 )
2322rexlimiva 2817 . . 3  |-  ( E. z  e.  H  z  =/=  0h  ->  E. y  e.  H  ( normh `  y )  =  1 )
24 ax-1ne0 9051 . . . . . . . 8  |-  1  =/=  0
25 df-ne 2600 . . . . . . . 8  |-  ( 1  =/=  0  <->  -.  1  =  0 )
2624, 25mpbi 200 . . . . . . 7  |-  -.  1  =  0
27 eqeq1 2441 . . . . . . 7  |-  ( (
normh `  y )  =  1  ->  ( ( normh `  y )  =  0  <->  1  =  0 ) )
2826, 27mtbiri 295 . . . . . 6  |-  ( (
normh `  y )  =  1  ->  -.  ( normh `  y )  =  0 )
293sheli 22708 . . . . . . . 8  |-  ( y  e.  H  ->  y  e.  ~H )
30 norm-i 22623 . . . . . . . 8  |-  ( y  e.  ~H  ->  (
( normh `  y )  =  0  <->  y  =  0h ) )
3129, 30syl 16 . . . . . . 7  |-  ( y  e.  H  ->  (
( normh `  y )  =  0  <->  y  =  0h ) )
3231necon3bbid 2632 . . . . . 6  |-  ( y  e.  H  ->  ( -.  ( normh `  y )  =  0  <->  y  =/=  0h ) )
3328, 32syl5ib 211 . . . . 5  |-  ( y  e.  H  ->  (
( normh `  y )  =  1  ->  y  =/=  0h ) )
3433reximia 2803 . . . 4  |-  ( E. y  e.  H  (
normh `  y )  =  1  ->  E. y  e.  H  y  =/=  0h )
35 neeq1 2606 . . . . 5  |-  ( y  =  z  ->  (
y  =/=  0h  <->  z  =/=  0h ) )
3635cbvrexv 2925 . . . 4  |-  ( E. y  e.  H  y  =/=  0h  <->  E. z  e.  H  z  =/=  0h )
3734, 36sylib 189 . . 3  |-  ( E. y  e.  H  (
normh `  y )  =  1  ->  E. z  e.  H  z  =/=  0h )
3823, 37impbii 181 . 2  |-  ( E. z  e.  H  z  =/=  0h  <->  E. y  e.  H  ( normh `  y )  =  1 )
392, 38bitri 241 1  |-  ( E. x  e.  H  x  =/=  0h  <->  E. y  e.  H  ( normh `  y )  =  1 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    / cdiv 9669   ~Hchil 22414    .h csm 22416   normhcno 22418   0hc0v 22419   SHcsh 22423
This theorem is referenced by:  norm1hex  22745  pjnmopi  23643
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-hilex 22494  ax-hfvadd 22495  ax-hv0cl 22498  ax-hfvmul 22500  ax-hvmul0 22505  ax-hfi 22573  ax-his1 22576  ax-his3 22578  ax-his4 22579
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-hnorm 22463  df-sh 22701
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