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Theorem norm1exi 21829
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 2454 . . 3  |-  ( x  =  z  ->  (
x  =/=  0h  <->  z  =/=  0h ) )
21cbvrexv 2765 . 2  |-  ( E. x  e.  H  x  =/=  0h  <->  E. z  e.  H  z  =/=  0h )
3 norm1ex.1 . . . . . . . . . . 11  |-  H  e.  SH
43sheli 21793 . . . . . . . . . 10  |-  ( z  e.  H  ->  z  e.  ~H )
5 normcl 21704 . . . . . . . . . 10  |-  ( z  e.  ~H  ->  ( normh `  z )  e.  RR )
64, 5syl 15 . . . . . . . . 9  |-  ( z  e.  H  ->  ( normh `  z )  e.  RR )
76adantr 451 . . . . . . . 8  |-  ( ( z  e.  H  /\  z  =/=  0h )  -> 
( normh `  z )  e.  RR )
8 normne0 21709 . . . . . . . . . 10  |-  ( z  e.  ~H  ->  (
( normh `  z )  =/=  0  <->  z  =/=  0h ) )
94, 8syl 15 . . . . . . . . 9  |-  ( z  e.  H  ->  (
( normh `  z )  =/=  0  <->  z  =/=  0h ) )
109biimpar 471 . . . . . . . 8  |-  ( ( z  e.  H  /\  z  =/=  0h )  -> 
( normh `  z )  =/=  0 )
117, 10rereccld 9587 . . . . . . 7  |-  ( ( z  e.  H  /\  z  =/=  0h )  -> 
( 1  /  ( normh `  z ) )  e.  RR )
1211recnd 8861 . . . . . 6  |-  ( ( z  e.  H  /\  z  =/=  0h )  -> 
( 1  /  ( normh `  z ) )  e.  CC )
13 simpl 443 . . . . . 6  |-  ( ( z  e.  H  /\  z  =/=  0h )  -> 
z  e.  H )
14 shmulcl 21797 . . . . . . 7  |-  ( ( H  e.  SH  /\  ( 1  /  ( normh `  z ) )  e.  CC  /\  z  e.  H )  ->  (
( 1  /  ( normh `  z ) )  .h  z )  e.  H )
153, 14mp3an1 1264 . . . . . 6  |-  ( ( ( 1  /  ( normh `  z ) )  e.  CC  /\  z  e.  H )  ->  (
( 1  /  ( normh `  z ) )  .h  z )  e.  H )
1612, 13, 15syl2anc 642 . . . . 5  |-  ( ( z  e.  H  /\  z  =/=  0h )  -> 
( ( 1  / 
( normh `  z )
)  .h  z )  e.  H )
17 norm1 21828 . . . . . 6  |-  ( ( z  e.  ~H  /\  z  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  z ) )  .h  z ) )  =  1 )
184, 17sylan 457 . . . . 5  |-  ( ( z  e.  H  /\  z  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  z ) )  .h  z ) )  =  1 )
19 fveq2 5525 . . . . . . 7  |-  ( y  =  ( ( 1  /  ( normh `  z
) )  .h  z
)  ->  ( normh `  y )  =  (
normh `  ( ( 1  /  ( normh `  z
) )  .h  z
) ) )
2019eqeq1d 2291 . . . . . 6  |-  ( y  =  ( ( 1  /  ( normh `  z
) )  .h  z
)  ->  ( ( normh `  y )  =  1  <->  ( normh `  (
( 1  /  ( normh `  z ) )  .h  z ) )  =  1 ) )
2120rspcev 2884 . . . . 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 642 . . . 4  |-  ( ( z  e.  H  /\  z  =/=  0h )  ->  E. y  e.  H  ( normh `  y )  =  1 )
2322rexlimiva 2662 . . 3  |-  ( E. z  e.  H  z  =/=  0h  ->  E. y  e.  H  ( normh `  y )  =  1 )
24 ax-1ne0 8806 . . . . . . . 8  |-  1  =/=  0
25 df-ne 2448 . . . . . . . 8  |-  ( 1  =/=  0  <->  -.  1  =  0 )
2624, 25mpbi 199 . . . . . . 7  |-  -.  1  =  0
27 eqeq1 2289 . . . . . . 7  |-  ( (
normh `  y )  =  1  ->  ( ( normh `  y )  =  0  <->  1  =  0 ) )
2826, 27mtbiri 294 . . . . . 6  |-  ( (
normh `  y )  =  1  ->  -.  ( normh `  y )  =  0 )
293sheli 21793 . . . . . . . 8  |-  ( y  e.  H  ->  y  e.  ~H )
30 norm-i 21708 . . . . . . . 8  |-  ( y  e.  ~H  ->  (
( normh `  y )  =  0  <->  y  =  0h ) )
3129, 30syl 15 . . . . . . 7  |-  ( y  e.  H  ->  (
( normh `  y )  =  0  <->  y  =  0h ) )
3231necon3bbid 2480 . . . . . 6  |-  ( y  e.  H  ->  ( -.  ( normh `  y )  =  0  <->  y  =/=  0h ) )
3328, 32syl5ib 210 . . . . 5  |-  ( y  e.  H  ->  (
( normh `  y )  =  1  ->  y  =/=  0h ) )
3433reximia 2648 . . . 4  |-  ( E. y  e.  H  (
normh `  y )  =  1  ->  E. y  e.  H  y  =/=  0h )
35 neeq1 2454 . . . . 5  |-  ( y  =  z  ->  (
y  =/=  0h  <->  z  =/=  0h ) )
3635cbvrexv 2765 . . . 4  |-  ( E. y  e.  H  y  =/=  0h  <->  E. z  e.  H  z  =/=  0h )
3734, 36sylib 188 . . 3  |-  ( E. y  e.  H  (
normh `  y )  =  1  ->  E. z  e.  H  z  =/=  0h )
3823, 37impbii 180 . 2  |-  ( E. z  e.  H  z  =/=  0h  <->  E. y  e.  H  ( normh `  y )  =  1 )
392, 38bitri 240 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 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    / cdiv 9423   ~Hchil 21499    .h csm 21501   normhcno 21503   0hc0v 21504   SHcsh 21508
This theorem is referenced by:  norm1hex  21830  pjnmopi  22728
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-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
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