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Theorem pjhthlem2 22743
Description: Lemma for pjhth 22744. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
pjhth.1  |-  H  e. 
CH
pjhth.2  |-  ( ph  ->  A  e.  ~H )
Assertion
Ref Expression
pjhthlem2  |-  ( ph  ->  E. x  e.  H  E. y  e.  ( _|_ `  H ) A  =  ( x  +h  y ) )
Distinct variable groups:    x, y, A    x, H, y    ph, x, y

Proof of Theorem pjhthlem2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-hba 22321 . . . 4  |-  ~H  =  ( BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
2 eqid 2388 . . . . 5  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
32hhvs 22521 . . . 4  |-  -h  =  ( -v `  <. <.  +h  ,  .h  >. ,  normh >. )
42hhnm 22522 . . . 4  |-  normh  =  (
normCV
`  <. <.  +h  ,  .h  >. ,  normh >. )
5 eqid 2388 . . . . 5  |-  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) )
>. ,  ( normh  |`  H ) >.  =  <. <.
(  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) )
>. ,  ( normh  |`  H ) >.
6 pjhth.1 . . . . . 6  |-  H  e. 
CH
76chshii 22579 . . . . 5  |-  H  e.  SH
85, 7hhssba 22620 . . . 4  |-  H  =  ( BaseSet `  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) )
>. ,  ( normh  |`  H ) >. )
92hhph 22529 . . . . 5  |-  <. <.  +h  ,  .h  >. ,  normh >.  e.  CPreHil OLD
109a1i 11 . . . 4  |-  ( ph  -> 
<. <.  +h  ,  .h  >. ,  normh >.  e.  CPreHil OLD )
112, 5hhsst 22615 . . . . . . 7  |-  ( H  e.  SH  ->  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) )
>. ,  ( normh  |`  H ) >.  e.  (
SubSp `  <. <.  +h  ,  .h  >. ,  normh >. ) )
127, 11ax-mp 8 . . . . . 6  |-  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) )
>. ,  ( normh  |`  H ) >.  e.  (
SubSp `  <. <.  +h  ,  .h  >. ,  normh >. )
135, 6hhssbn 22629 . . . . . 6  |-  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) )
>. ,  ( normh  |`  H ) >.  e.  CBan
14 elin 3474 . . . . . 6  |-  ( <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) )
>. ,  ( normh  |`  H ) >.  e.  ( ( SubSp `  <. <.  +h  ,  .h  >. ,  normh >. )  i^i  CBan )  <->  ( <. <.
(  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) )
>. ,  ( normh  |`  H ) >.  e.  (
SubSp `  <. <.  +h  ,  .h  >. ,  normh >. )  /\  <. <.
(  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) )
>. ,  ( normh  |`  H ) >.  e.  CBan ) )
1512, 13, 14mpbir2an 887 . . . . 5  |-  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) )
>. ,  ( normh  |`  H ) >.  e.  ( ( SubSp `  <. <.  +h  ,  .h  >. ,  normh >. )  i^i  CBan )
1615a1i 11 . . . 4  |-  ( ph  -> 
<. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) )
>. ,  ( normh  |`  H ) >.  e.  ( ( SubSp `  <. <.  +h  ,  .h  >. ,  normh >. )  i^i  CBan ) )
17 pjhth.2 . . . 4  |-  ( ph  ->  A  e.  ~H )
181, 3, 4, 8, 10, 16, 17minveco 22235 . . 3  |-  ( ph  ->  E! x  e.  H  A. z  e.  H  ( normh `  ( A  -h  x ) )  <_ 
( normh `  ( A  -h  z ) ) )
19 reurex 2866 . . 3  |-  ( E! x  e.  H  A. z  e.  H  ( normh `  ( A  -h  x ) )  <_ 
( normh `  ( A  -h  z ) )  ->  E. x  e.  H  A. z  e.  H  ( normh `  ( A  -h  x ) )  <_ 
( normh `  ( A  -h  z ) ) )
2018, 19syl 16 . 2  |-  ( ph  ->  E. x  e.  H  A. z  e.  H  ( normh `  ( A  -h  x ) )  <_ 
( normh `  ( A  -h  z ) ) )
2117adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  H  /\  A. z  e.  H  ( normh `  ( A  -h  x
) )  <_  ( normh `  ( A  -h  z ) ) ) )  ->  A  e.  ~H )
226cheli 22584 . . . . . . . 8  |-  ( x  e.  H  ->  x  e.  ~H )
2322ad2antrl 709 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  H  /\  A. z  e.  H  ( normh `  ( A  -h  x
) )  <_  ( normh `  ( A  -h  z ) ) ) )  ->  x  e.  ~H )
24 hvsubcl 22369 . . . . . . 7  |-  ( ( A  e.  ~H  /\  x  e.  ~H )  ->  ( A  -h  x
)  e.  ~H )
2521, 23, 24syl2anc 643 . . . . . 6  |-  ( (
ph  /\  ( x  e.  H  /\  A. z  e.  H  ( normh `  ( A  -h  x
) )  <_  ( normh `  ( A  -h  z ) ) ) )  ->  ( A  -h  x )  e.  ~H )
2621adantr 452 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  H  /\  A. z  e.  H  (
normh `  ( A  -h  x ) )  <_ 
( normh `  ( A  -h  z ) ) ) )  /\  y  e.  H )  ->  A  e.  ~H )
27 simplrl 737 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  H  /\  A. z  e.  H  (
normh `  ( A  -h  x ) )  <_ 
( normh `  ( A  -h  z ) ) ) )  /\  y  e.  H )  ->  x  e.  H )
28 simpr 448 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  H  /\  A. z  e.  H  (
normh `  ( A  -h  x ) )  <_ 
( normh `  ( A  -h  z ) ) ) )  /\  y  e.  H )  ->  y  e.  H )
29 simplrr 738 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  H  /\  A. z  e.  H  (
normh `  ( A  -h  x ) )  <_ 
( normh `  ( A  -h  z ) ) ) )  /\  y  e.  H )  ->  A. z  e.  H  ( normh `  ( A  -h  x
) )  <_  ( normh `  ( A  -h  z ) ) )
30 eqid 2388 . . . . . . . 8  |-  ( ( ( A  -h  x
)  .ih  y )  /  ( ( y 
.ih  y )  +  1 ) )  =  ( ( ( A  -h  x )  .ih  y )  /  (
( y  .ih  y
)  +  1 ) )
316, 26, 27, 28, 29, 30pjhthlem1 22742 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  H  /\  A. z  e.  H  (
normh `  ( A  -h  x ) )  <_ 
( normh `  ( A  -h  z ) ) ) )  /\  y  e.  H )  ->  (
( A  -h  x
)  .ih  y )  =  0 )
3231ralrimiva 2733 . . . . . 6  |-  ( (
ph  /\  ( x  e.  H  /\  A. z  e.  H  ( normh `  ( A  -h  x
) )  <_  ( normh `  ( A  -h  z ) ) ) )  ->  A. y  e.  H  ( ( A  -h  x )  .ih  y )  =  0 )
33 shocel 22633 . . . . . . 7  |-  ( H  e.  SH  ->  (
( A  -h  x
)  e.  ( _|_ `  H )  <->  ( ( A  -h  x )  e. 
~H  /\  A. y  e.  H  ( ( A  -h  x )  .ih  y )  =  0 ) ) )
347, 33ax-mp 8 . . . . . 6  |-  ( ( A  -h  x )  e.  ( _|_ `  H
)  <->  ( ( A  -h  x )  e. 
~H  /\  A. y  e.  H  ( ( A  -h  x )  .ih  y )  =  0 ) )
3525, 32, 34sylanbrc 646 . . . . 5  |-  ( (
ph  /\  ( x  e.  H  /\  A. z  e.  H  ( normh `  ( A  -h  x
) )  <_  ( normh `  ( A  -h  z ) ) ) )  ->  ( A  -h  x )  e.  ( _|_ `  H ) )
36 hvpncan3 22393 . . . . . . 7  |-  ( ( x  e.  ~H  /\  A  e.  ~H )  ->  ( x  +h  ( A  -h  x ) )  =  A )
3723, 21, 36syl2anc 643 . . . . . 6  |-  ( (
ph  /\  ( x  e.  H  /\  A. z  e.  H  ( normh `  ( A  -h  x
) )  <_  ( normh `  ( A  -h  z ) ) ) )  ->  ( x  +h  ( A  -h  x
) )  =  A )
3837eqcomd 2393 . . . . 5  |-  ( (
ph  /\  ( x  e.  H  /\  A. z  e.  H  ( normh `  ( A  -h  x
) )  <_  ( normh `  ( A  -h  z ) ) ) )  ->  A  =  ( x  +h  ( A  -h  x ) ) )
39 oveq2 6029 . . . . . . 7  |-  ( y  =  ( A  -h  x )  ->  (
x  +h  y )  =  ( x  +h  ( A  -h  x
) ) )
4039eqeq2d 2399 . . . . . 6  |-  ( y  =  ( A  -h  x )  ->  ( A  =  ( x  +h  y )  <->  A  =  ( x  +h  ( A  -h  x ) ) ) )
4140rspcev 2996 . . . . 5  |-  ( ( ( A  -h  x
)  e.  ( _|_ `  H )  /\  A  =  ( x  +h  ( A  -h  x
) ) )  ->  E. y  e.  ( _|_ `  H ) A  =  ( x  +h  y ) )
4235, 38, 41syl2anc 643 . . . 4  |-  ( (
ph  /\  ( x  e.  H  /\  A. z  e.  H  ( normh `  ( A  -h  x
) )  <_  ( normh `  ( A  -h  z ) ) ) )  ->  E. y  e.  ( _|_ `  H
) A  =  ( x  +h  y ) )
4342expr 599 . . 3  |-  ( (
ph  /\  x  e.  H )  ->  ( A. z  e.  H  ( normh `  ( A  -h  x ) )  <_ 
( normh `  ( A  -h  z ) )  ->  E. y  e.  ( _|_ `  H ) A  =  ( x  +h  y ) ) )
4443reximdva 2762 . 2  |-  ( ph  ->  ( E. x  e.  H  A. z  e.  H  ( normh `  ( A  -h  x ) )  <_  ( normh `  ( A  -h  z ) )  ->  E. x  e.  H  E. y  e.  ( _|_ `  H ) A  =  ( x  +h  y ) ) )
4520, 44mpd 15 1  |-  ( ph  ->  E. x  e.  H  E. y  e.  ( _|_ `  H ) A  =  ( x  +h  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2650   E.wrex 2651   E!wreu 2652    i^i cin 3263   <.cop 3761   class class class wbr 4154    X. cxp 4817    |` cres 4821   ` cfv 5395  (class class class)co 6021   CCcc 8922   0cc0 8924   1c1 8925    + caddc 8927    <_ cle 9055    / cdiv 9610   SubSpcss 22069   CPreHil OLDccphlo 22162   CBanccbn 22213   ~Hchil 22271    +h cva 22272    .h csm 22273    .ih csp 22274   normhcno 22275    -h cmv 22277   SHcsh 22280   CHcch 22281   _|_cort 22282
This theorem is referenced by:  pjhth  22744  omlsii  22754
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cc 8249  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002  ax-addf 9003  ax-mulf 9004  ax-hilex 22351  ax-hfvadd 22352  ax-hvcom 22353  ax-hvass 22354  ax-hv0cl 22355  ax-hvaddid 22356  ax-hfvmul 22357  ax-hvmulid 22358  ax-hvmulass 22359  ax-hvdistr1 22360  ax-hvdistr2 22361  ax-hvmul0 22362  ax-hfi 22430  ax-his1 22433  ax-his2 22434  ax-his3 22435  ax-his4 22436  ax-hcompl 22553
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-omul 6666  df-er 6842  df-map 6957  df-pm 6958  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-fi 7352  df-sup 7382  df-oi 7413  df-card 7760  df-acn 7763  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-n0 10155  df-z 10216  df-uz 10422  df-q 10508  df-rp 10546  df-xneg 10643  df-xadd 10644  df-xmul 10645  df-ico 10855  df-icc 10856  df-fz 10977  df-fl 11130  df-seq 11252  df-exp 11311  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-clim 12210  df-rlim 12211  df-rest 13578  df-topgen 13595  df-xmet 16620  df-met 16621  df-bl 16622  df-mopn 16623  df-fbas 16624  df-fg 16625  df-top 16887  df-bases 16889  df-topon 16890  df-cld 17007  df-ntr 17008  df-cls 17009  df-nei 17086  df-lm 17216  df-haus 17302  df-fil 17800  df-fm 17892  df-flim 17893  df-flf 17894  df-cfil 19080  df-cau 19081  df-cmet 19082  df-grpo 21628  df-gid 21629  df-ginv 21630  df-gdiv 21631  df-ablo 21719  df-subgo 21739  df-vc 21874  df-nv 21920  df-va 21923  df-ba 21924  df-sm 21925  df-0v 21926  df-vs 21927  df-nmcv 21928  df-ims 21929  df-ssp 22070  df-ph 22163  df-cbn 22214  df-hnorm 22320  df-hba 22321  df-hvsub 22323  df-hlim 22324  df-hcau 22325  df-sh 22558  df-ch 22573  df-oc 22603  df-ch0 22604
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