HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  chscllem1 Structured version   Unicode version

Theorem chscllem1 23170
Description: Lemma for chscl 23174. (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 )
) )
Assertion
Ref Expression
chscllem1  |-  ( ph  ->  F : NN --> A )
Distinct variable groups:    u, n, A    ph, n    B, n, u    n, H, u
Allowed substitution hints:    ph( u)    F( u, n)

Proof of Theorem chscllem1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2442 . . . 4  |-  ( (
proj  h `  A ) `
 ( H `  n ) )  =  ( ( proj  h `  A ) `  ( H `  n )
)
2 chscl.1 . . . . . 6  |-  ( ph  ->  A  e.  CH )
32adantr 453 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  A  e. 
CH )
4 chscl.4 . . . . . . 7  |-  ( ph  ->  H : NN --> ( A  +H  B ) )
54ffvelrnda 5899 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( H `
 n )  e.  ( A  +H  B
) )
6 chscl.2 . . . . . . . . . 10  |-  ( ph  ->  B  e.  CH )
7 chsh 22758 . . . . . . . . . 10  |-  ( B  e.  CH  ->  B  e.  SH )
86, 7syl 16 . . . . . . . . 9  |-  ( ph  ->  B  e.  SH )
9 chsh 22758 . . . . . . . . . . 11  |-  ( A  e.  CH  ->  A  e.  SH )
102, 9syl 16 . . . . . . . . . 10  |-  ( ph  ->  A  e.  SH )
11 shocsh 22817 . . . . . . . . . 10  |-  ( A  e.  SH  ->  ( _|_ `  A )  e.  SH )
1210, 11syl 16 . . . . . . . . 9  |-  ( ph  ->  ( _|_ `  A
)  e.  SH )
13 chscl.3 . . . . . . . . 9  |-  ( ph  ->  B  C_  ( _|_ `  A ) )
14 shless 22892 . . . . . . . . 9  |-  ( ( ( B  e.  SH  /\  ( _|_ `  A
)  e.  SH  /\  A  e.  SH )  /\  B  C_  ( _|_ `  A ) )  -> 
( B  +H  A
)  C_  ( ( _|_ `  A )  +H  A ) )
158, 12, 10, 13, 14syl31anc 1188 . . . . . . . 8  |-  ( ph  ->  ( B  +H  A
)  C_  ( ( _|_ `  A )  +H  A ) )
16 shscom 22852 . . . . . . . . 9  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( A  +H  B
)  =  ( B  +H  A ) )
1710, 8, 16syl2anc 644 . . . . . . . 8  |-  ( ph  ->  ( A  +H  B
)  =  ( B  +H  A ) )
18 shscom 22852 . . . . . . . . 9  |-  ( ( A  e.  SH  /\  ( _|_ `  A )  e.  SH )  -> 
( A  +H  ( _|_ `  A ) )  =  ( ( _|_ `  A )  +H  A
) )
1910, 12, 18syl2anc 644 . . . . . . . 8  |-  ( ph  ->  ( A  +H  ( _|_ `  A ) )  =  ( ( _|_ `  A )  +H  A
) )
2015, 17, 193sstr4d 3377 . . . . . . 7  |-  ( ph  ->  ( A  +H  B
)  C_  ( A  +H  ( _|_ `  A
) ) )
2120sselda 3334 . . . . . 6  |-  ( (
ph  /\  ( H `  n )  e.  ( A  +H  B ) )  ->  ( H `  n )  e.  ( A  +H  ( _|_ `  A ) ) )
225, 21syldan 458 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  ( H `
 n )  e.  ( A  +H  ( _|_ `  A ) ) )
23 pjpreeq 22931 . . . . 5  |-  ( ( A  e.  CH  /\  ( H `  n )  e.  ( A  +H  ( _|_ `  A ) ) )  ->  (
( ( proj  h `  A ) `  ( H `  n )
)  =  ( (
proj  h `  A ) `
 ( H `  n ) )  <->  ( (
( proj  h `  A
) `  ( H `  n ) )  e.  A  /\  E. x  e.  ( _|_ `  A
) ( H `  n )  =  ( ( ( proj  h `  A ) `  ( H `  n )
)  +h  x ) ) ) )
243, 22, 23syl2anc 644 . . . 4  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( proj  h `  A
) `  ( H `  n ) )  =  ( ( proj  h `  A ) `  ( H `  n )
)  <->  ( ( (
proj  h `  A ) `
 ( H `  n ) )  e.  A  /\  E. x  e.  ( _|_ `  A
) ( H `  n )  =  ( ( ( proj  h `  A ) `  ( H `  n )
)  +h  x ) ) ) )
251, 24mpbii 204 . . 3  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( proj  h `  A
) `  ( H `  n ) )  e.  A  /\  E. x  e.  ( _|_ `  A
) ( H `  n )  =  ( ( ( proj  h `  A ) `  ( H `  n )
)  +h  x ) ) )
2625simpld 447 . 2  |-  ( (
ph  /\  n  e.  NN )  ->  ( (
proj  h `  A ) `
 ( H `  n ) )  e.  A )
27 chscl.6 . 2  |-  F  =  ( n  e.  NN  |->  ( ( proj  h `  A ) `  ( H `  n )
) )
2826, 27fmptd 5922 1  |-  ( ph  ->  F : NN --> A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1727   E.wrex 2712    C_ wss 3306   class class class wbr 4237    e. cmpt 4291   -->wf 5479   ` cfv 5483  (class class class)co 6110   NNcn 10031    +h cva 22454    ~~>v chli 22461   SHcsh 22462   CHcch 22463   _|_cort 22464    +H cph 22465   proj 
hcpjh 22471
This theorem is referenced by:  chscllem2  23171  chscllem3  23172  chscllem4  23173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098  ax-hilex 22533  ax-hfvadd 22534  ax-hvcom 22535  ax-hvass 22536  ax-hv0cl 22537  ax-hvaddid 22538  ax-hfvmul 22539  ax-hvmulid 22540  ax-hvmulass 22541  ax-hvdistr1 22542  ax-hvdistr2 22543  ax-hvmul0 22544  ax-hfi 22612  ax-his2 22616  ax-his3 22617  ax-his4 22618
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-po 4532  df-so 4533  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-riota 6578  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-div 9709  df-grpo 21810  df-ablo 21901  df-hvsub 22505  df-sh 22740  df-ch 22755  df-oc 22785  df-ch0 22786  df-shs 22841  df-pjh 22928
  Copyright terms: Public domain W3C validator