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

Theorem chscllem3 22652
Description: Lemma for chscl 22654. (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 5632 . . . . . . . 8  |-  ( n  =  N  ->  ( H `  n )  =  ( H `  N ) )
32fveq2d 5636 . . . . . . 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 5646 . . . . . . 7  |-  ( (
proj  h `  A ) `
 ( H `  N ) )  e. 
_V
63, 4, 5fvmpt 5709 . . . . . 6  |-  ( N  e.  NN  ->  ( F `  N )  =  ( ( proj 
h `  A ) `  ( H `  N
) ) )
71, 6syl 15 . . . . 5  |-  ( ph  ->  ( F `  N
)  =  ( (
proj  h `  A ) `
 ( H `  N ) ) )
87eqcomd 2371 . . . 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 22238 . . . . . . . . 9  |-  ( B  e.  CH  ->  B  e.  SH )
1210, 11syl 15 . . . . . . . 8  |-  ( ph  ->  B  e.  SH )
13 chsh 22238 . . . . . . . . . 10  |-  ( A  e.  CH  ->  A  e.  SH )
149, 13syl 15 . . . . . . . . 9  |-  ( ph  ->  A  e.  SH )
15 shocsh 22297 . . . . . . . . 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 22372 . . . . . . . 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 1186 . . . . . . 7  |-  ( ph  ->  ( B  +H  A
)  C_  ( ( _|_ `  A )  +H  A ) )
20 shscom 22332 . . . . . . . 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 22332 . . . . . . . 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 3307 . . . . . 6  |-  ( ph  ->  ( A  +H  B
)  C_  ( A  +H  ( _|_ `  A
) ) )
25 chscl.4 . . . . . . 7  |-  ( ph  ->  H : NN --> ( A  +H  B ) )
26 ffvelrn 5770 . . . . . . 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 3267 . . . . 5  |-  ( ph  ->  ( H `  N
)  e.  ( A  +H  ( _|_ `  A
) ) )
29 pjpreeq 22411 . . . . 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 22309 . . . . . . . 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 3267 . . . . . 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 22650 . . . . . . . 8  |-  ( ph  ->  F : NN --> A )
46 ffvelrn 5770 . . . . . . . 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 2400 . . . . . 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 22313 . . . . 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 2751 . 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 1647    e. wcel 1715   E.wrex 2629    i^i cin 3237    C_ wss 3238   class class class wbr 4125    e. cmpt 4179   -->wf 5354   ` cfv 5358  (class class class)co 5981   NNcn 9893    +h cva 21934    ~~>v chli 21941   SHcsh 21942   CHcch 21943   _|_cort 21944    +H cph 21945   0Hc0h 21949   proj 
hcpjh 21951
This theorem is referenced by:  chscllem4  22653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-hilex 22013  ax-hfvadd 22014  ax-hvcom 22015  ax-hvass 22016  ax-hv0cl 22017  ax-hvaddid 22018  ax-hfvmul 22019  ax-hvmulid 22020  ax-hvmulass 22021  ax-hvdistr1 22022  ax-hvdistr2 22023  ax-hvmul0 22024  ax-hfi 22092  ax-his2 22096  ax-his3 22097  ax-his4 22098
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-po 4417  df-so 4418  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-riota 6446  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-grpo 21290  df-ablo 21381  df-hvsub 21985  df-sh 22220  df-ch 22235  df-oc 22265  df-ch0 22266  df-shs 22321  df-pjh 22408
  Copyright terms: Public domain W3C validator