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Theorem pjhthlem2 22847
Description: Lemma for pjhth 22848. (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 22425 . . . 4  |-  ~H  =  ( BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
2 eqid 2404 . . . . 5  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
32hhvs 22625 . . . 4  |-  -h  =  ( -v `  <. <.  +h  ,  .h  >. ,  normh >. )
42hhnm 22626 . . . 4  |-  normh  =  (
normCV
`  <. <.  +h  ,  .h  >. ,  normh >. )
5 eqid 2404 . . . . 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 22683 . . . . 5  |-  H  e.  SH
85, 7hhssba 22724 . . . 4  |-  H  =  ( BaseSet `  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) )
>. ,  ( normh  |`  H ) >. )
92hhph 22633 . . . . 5  |-  <. <.  +h  ,  .h  >. ,  normh >.  e.  CPreHil OLD
109a1i 11 . . . 4  |-  ( ph  -> 
<. <.  +h  ,  .h  >. ,  normh >.  e.  CPreHil OLD )
112, 5hhsst 22719 . . . . . . 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 22733 . . . . . 6  |-  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) )
>. ,  ( normh  |`  H ) >.  e.  CBan
14 elin 3490 . . . . . 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 22339 . . 3  |-  ( ph  ->  E! x  e.  H  A. z  e.  H  ( normh `  ( A  -h  x ) )  <_ 
( normh `  ( A  -h  z ) ) )
19 reurex 2882 . . 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 22688 . . . . . . . 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 22473 . . . . . . 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 2404 . . . . . . . 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 22846 . . . . . . 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 2749 . . . . . 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 22737 . . . . . . 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 22497 . . . . . . 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 2409 . . . . 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 6048 . . . . . . 7  |-  ( y  =  ( A  -h  x )  ->  (
x  +h  y )  =  ( x  +h  ( A  -h  x
) ) )
4039eqeq2d 2415 . . . . . 6  |-  ( y  =  ( A  -h  x )  ->  ( A  =  ( x  +h  y )  <->  A  =  ( x  +h  ( A  -h  x ) ) ) )
4140rspcev 3012 . . . . 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 2778 . 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 1721   A.wral 2666   E.wrex 2667   E!wreu 2668    i^i cin 3279   <.cop 3777   class class class wbr 4172    X. cxp 4835    |` cres 4839   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949    <_ cle 9077    / cdiv 9633   SubSpcss 22173   CPreHil OLDccphlo 22266   CBanccbn 22317   ~Hchil 22375    +h cva 22376    .h csm 22377    .ih csp 22378   normhcno 22379    -h cmv 22381   SHcsh 22384   CHcch 22385   _|_cort 22386
This theorem is referenced by:  pjhth  22848  omlsii  22858
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cc 8271  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026  ax-hilex 22455  ax-hfvadd 22456  ax-hvcom 22457  ax-hvass 22458  ax-hv0cl 22459  ax-hvaddid 22460  ax-hfvmul 22461  ax-hvmulid 22462  ax-hvmulass 22463  ax-hvdistr1 22464  ax-hvdistr2 22465  ax-hvmul0 22466  ax-hfi 22534  ax-his1 22537  ax-his2 22538  ax-his3 22539  ax-his4 22540  ax-hcompl 22657
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-omul 6688  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-acn 7785  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ico 10878  df-icc 10879  df-fz 11000  df-fl 11157  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-rest 13605  df-topgen 13622  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-top 16918  df-bases 16920  df-topon 16921  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lm 17247  df-haus 17333  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-cfil 19161  df-cau 19162  df-cmet 19163  df-grpo 21732  df-gid 21733  df-ginv 21734  df-gdiv 21735  df-ablo 21823  df-subgo 21843  df-vc 21978  df-nv 22024  df-va 22027  df-ba 22028  df-sm 22029  df-0v 22030  df-vs 22031  df-nmcv 22032  df-ims 22033  df-ssp 22174  df-ph 22267  df-cbn 22318  df-hnorm 22424  df-hba 22425  df-hvsub 22427  df-hlim 22428  df-hcau 22429  df-sh 22662  df-ch 22677  df-oc 22707  df-ch0 22708
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