HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  chscllem3 Unicode version

Theorem chscllem3 22218
Description: Lemma for chscl 22220. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
chscl.1  |-  ( ph  ->  A  e.  CH )
chscl.2  |-  ( ph  ->  B  e.  CH )
chscl.3  |-  ( ph  ->  B  C_  ( _|_ `  A ) )
chscl.4  |-  ( ph  ->  H : NN --> ( A  +H  B ) )
chscl.5  |-  ( ph  ->  H  ~~>v  u )
chscl.6  |-  F  =  ( n  e.  NN  |->  ( ( proj  h `  A ) `  ( H `  n )
) )
chscllem3.7  |-  ( ph  ->  N  e.  NN )
chscllem3.8  |-  ( ph  ->  C  e.  A )
chscllem3.9  |-  ( ph  ->  D  e.  B )
chscllem3.10  |-  ( ph  ->  ( H `  N
)  =  ( C  +h  D ) )
Assertion
Ref Expression
chscllem3  |-  ( ph  ->  C  =  ( F `
 N ) )
Distinct variable groups:    u, n, A    ph, n    B, n, u    n, H, u    n, N
Allowed substitution hints:    ph( u)    C( u, n)    D( u, n)    F( u, n)    N( u)

Proof of Theorem chscllem3
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 chscllem3.7 . . . . . 6  |-  ( ph  ->  N  e.  NN )
2 fveq2 5525 . . . . . . . 8  |-  ( n  =  N  ->  ( H `  n )  =  ( H `  N ) )
32fveq2d 5529 . . . . . . 7  |-  ( n  =  N  ->  (
( proj  h `  A
) `  ( H `  n ) )  =  ( ( proj  h `  A ) `  ( H `  N )
) )
4 chscl.6 . . . . . . 7  |-  F  =  ( n  e.  NN  |->  ( ( proj  h `  A ) `  ( H `  n )
) )
5 fvex 5539 . . . . . . 7  |-  ( (
proj  h `  A ) `
 ( H `  N ) )  e. 
_V
63, 4, 5fvmpt 5602 . . . . . 6  |-  ( N  e.  NN  ->  ( F `  N )  =  ( ( proj 
h `  A ) `  ( H `  N
) ) )
71, 6syl 15 . . . . 5  |-  ( ph  ->  ( F `  N
)  =  ( (
proj  h `  A ) `
 ( H `  N ) ) )
87eqcomd 2288 . . . 4  |-  ( ph  ->  ( ( proj  h `  A ) `  ( H `  N )
)  =  ( F `
 N ) )
9 chscl.1 . . . . 5  |-  ( ph  ->  A  e.  CH )
10 chscl.2 . . . . . . . . 9  |-  ( ph  ->  B  e.  CH )
11 chsh 21804 . . . . . . . . 9  |-  ( B  e.  CH  ->  B  e.  SH )
1210, 11syl 15 . . . . . . . 8  |-  ( ph  ->  B  e.  SH )
13 chsh 21804 . . . . . . . . . 10  |-  ( A  e.  CH  ->  A  e.  SH )
149, 13syl 15 . . . . . . . . 9  |-  ( ph  ->  A  e.  SH )
15 shocsh 21863 . . . . . . . . 9  |-  ( A  e.  SH  ->  ( _|_ `  A )  e.  SH )
1614, 15syl 15 . . . . . . . 8  |-  ( ph  ->  ( _|_ `  A
)  e.  SH )
17 chscl.3 . . . . . . . 8  |-  ( ph  ->  B  C_  ( _|_ `  A ) )
18 shless 21938 . . . . . . . 8  |-  ( ( ( B  e.  SH  /\  ( _|_ `  A
)  e.  SH  /\  A  e.  SH )  /\  B  C_  ( _|_ `  A ) )  -> 
( B  +H  A
)  C_  ( ( _|_ `  A )  +H  A ) )
1912, 16, 14, 17, 18syl31anc 1185 . . . . . . 7  |-  ( ph  ->  ( B  +H  A
)  C_  ( ( _|_ `  A )  +H  A ) )
20 shscom 21898 . . . . . . . 8  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( A  +H  B
)  =  ( B  +H  A ) )
2114, 12, 20syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( A  +H  B
)  =  ( B  +H  A ) )
22 shscom 21898 . . . . . . . 8  |-  ( ( A  e.  SH  /\  ( _|_ `  A )  e.  SH )  -> 
( A  +H  ( _|_ `  A ) )  =  ( ( _|_ `  A )  +H  A
) )
2314, 16, 22syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( A  +H  ( _|_ `  A ) )  =  ( ( _|_ `  A )  +H  A
) )
2419, 21, 233sstr4d 3221 . . . . . 6  |-  ( ph  ->  ( A  +H  B
)  C_  ( A  +H  ( _|_ `  A
) ) )
25 chscl.4 . . . . . . 7  |-  ( ph  ->  H : NN --> ( A  +H  B ) )
26 ffvelrn 5663 . . . . . . 7  |-  ( ( H : NN --> ( A  +H  B )  /\  N  e.  NN )  ->  ( H `  N
)  e.  ( A  +H  B ) )
2725, 1, 26syl2anc 642 . . . . . 6  |-  ( ph  ->  ( H `  N
)  e.  ( A  +H  B ) )
2824, 27sseldd 3181 . . . . 5  |-  ( ph  ->  ( H `  N
)  e.  ( A  +H  ( _|_ `  A
) ) )
29 pjpreeq 21977 . . . . 5  |-  ( ( A  e.  CH  /\  ( H `  N )  e.  ( A  +H  ( _|_ `  A ) ) )  ->  (
( ( proj  h `  A ) `  ( H `  N )
)  =  ( F `
 N )  <->  ( ( F `  N )  e.  A  /\  E. z  e.  ( _|_ `  A
) ( H `  N )  =  ( ( F `  N
)  +h  z ) ) ) )
309, 28, 29syl2anc 642 . . . 4  |-  ( ph  ->  ( ( ( proj 
h `  A ) `  ( H `  N
) )  =  ( F `  N )  <-> 
( ( F `  N )  e.  A  /\  E. z  e.  ( _|_ `  A ) ( H `  N
)  =  ( ( F `  N )  +h  z ) ) ) )
318, 30mpbid 201 . . 3  |-  ( ph  ->  ( ( F `  N )  e.  A  /\  E. z  e.  ( _|_ `  A ) ( H `  N
)  =  ( ( F `  N )  +h  z ) ) )
3231simprd 449 . 2  |-  ( ph  ->  E. z  e.  ( _|_ `  A ) ( H `  N
)  =  ( ( F `  N )  +h  z ) )
3314adantr 451 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  A  e.  SH )
3416adantr 451 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  ( _|_ `  A )  e.  SH )
35 ocin 21875 . . . . . . . 8  |-  ( A  e.  SH  ->  ( A  i^i  ( _|_ `  A
) )  =  0H )
3614, 35syl 15 . . . . . . 7  |-  ( ph  ->  ( A  i^i  ( _|_ `  A ) )  =  0H )
3736adantr 451 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  ( A  i^i  ( _|_ `  A
) )  =  0H )
38 chscllem3.8 . . . . . . 7  |-  ( ph  ->  C  e.  A )
3938adantr 451 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  C  e.  A )
4017adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  B  C_  ( _|_ `  A
) )
41 chscllem3.9 . . . . . . . 8  |-  ( ph  ->  D  e.  B )
4241adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  D  e.  B )
4340, 42sseldd 3181 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  D  e.  ( _|_ `  A
) )
44 chscl.5 . . . . . . . . 9  |-  ( ph  ->  H  ~~>v  u )
459, 10, 17, 25, 44, 4chscllem1 22216 . . . . . . . 8  |-  ( ph  ->  F : NN --> A )
46 ffvelrn 5663 . . . . . . . 8  |-  ( ( F : NN --> A  /\  N  e.  NN )  ->  ( F `  N
)  e.  A )
4745, 1, 46syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( F `  N
)  e.  A )
4847adantr 451 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  ( F `  N )  e.  A )
49 simprl 732 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  z  e.  ( _|_ `  A
) )
50 chscllem3.10 . . . . . . . 8  |-  ( ph  ->  ( H `  N
)  =  ( C  +h  D ) )
5150adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  ( H `  N )  =  ( C  +h  D ) )
52 simprr 733 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  ( H `  N )  =  ( ( F `
 N )  +h  z ) )
5351, 52eqtr3d 2317 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  ( C  +h  D )  =  ( ( F `  N )  +h  z
) )
5433, 34, 37, 39, 43, 48, 49, 53shuni 21879 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  ( C  =  ( F `  N )  /\  D  =  z ) )
5554simpld 445 . . . 4  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  C  =  ( F `  N ) )
5655exp32 588 . . 3  |-  ( ph  ->  ( z  e.  ( _|_ `  A )  ->  ( ( H `
 N )  =  ( ( F `  N )  +h  z
)  ->  C  =  ( F `  N ) ) ) )
5756rexlimdv 2666 . 2  |-  ( ph  ->  ( E. z  e.  ( _|_ `  A
) ( H `  N )  =  ( ( F `  N
)  +h  z )  ->  C  =  ( F `  N ) ) )
5832, 57mpd 14 1  |-  ( ph  ->  C  =  ( F `
 N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544    i^i cin 3151    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858   NNcn 9746    +h cva 21500    ~~>v chli 21507   SHcsh 21508   CHcch 21509   _|_cort 21510    +H cph 21511   0Hc0h 21515   proj 
hcpjh 21517
This theorem is referenced by:  chscllem4  22219
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  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-hilex 21579  ax-hfvadd 21580  ax-hvcom 21581  ax-hvass 21582  ax-hv0cl 21583  ax-hvaddid 21584  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvmulass 21587  ax-hvdistr1 21588  ax-hvdistr2 21589  ax-hvmul0 21590  ax-hfi 21658  ax-his2 21662  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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  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-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  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-grpo 20858  df-ablo 20949  df-hvsub 21551  df-sh 21786  df-ch 21801  df-oc 21831  df-ch0 21832  df-shs 21887  df-pjh 21974
  Copyright terms: Public domain W3C validator