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Theorem pjhthlem2 27037
Description: Lemma for pjhth 27038. (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 pjhth.2 . . . . . 6  |-  ( ph  ->  A  e.  ~H )
21adantr 467 . . . . 5  |-  ( (
ph  /\  ( x  e.  H  /\  A. z  e.  H  ( normh `  ( A  -h  x
) )  <_  ( normh `  ( A  -h  z ) ) ) )  ->  A  e.  ~H )
3 pjhth.1 . . . . . . 7  |-  H  e. 
CH
43cheli 26877 . . . . . 6  |-  ( x  e.  H  ->  x  e.  ~H )
54ad2antrl 733 . . . . 5  |-  ( (
ph  /\  ( x  e.  H  /\  A. z  e.  H  ( normh `  ( A  -h  x
) )  <_  ( normh `  ( A  -h  z ) ) ) )  ->  x  e.  ~H )
6 hvsubcl 26662 . . . . 5  |-  ( ( A  e.  ~H  /\  x  e.  ~H )  ->  ( A  -h  x
)  e.  ~H )
72, 5, 6syl2anc 666 . . . 4  |-  ( (
ph  /\  ( x  e.  H  /\  A. z  e.  H  ( normh `  ( A  -h  x
) )  <_  ( normh `  ( A  -h  z ) ) ) )  ->  ( A  -h  x )  e.  ~H )
82adantr 467 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  H  /\  A. z  e.  H  (
normh `  ( A  -h  x ) )  <_ 
( normh `  ( A  -h  z ) ) ) )  /\  y  e.  H )  ->  A  e.  ~H )
9 simplrl 769 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  H  /\  A. z  e.  H  (
normh `  ( A  -h  x ) )  <_ 
( normh `  ( A  -h  z ) ) ) )  /\  y  e.  H )  ->  x  e.  H )
10 simpr 463 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  H  /\  A. z  e.  H  (
normh `  ( A  -h  x ) )  <_ 
( normh `  ( A  -h  z ) ) ) )  /\  y  e.  H )  ->  y  e.  H )
11 simplrr 770 . . . . . 6  |-  ( ( ( 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 ) ) )
12 eqid 2423 . . . . . 6  |-  ( ( ( A  -h  x
)  .ih  y )  /  ( ( y 
.ih  y )  +  1 ) )  =  ( ( ( A  -h  x )  .ih  y )  /  (
( y  .ih  y
)  +  1 ) )
133, 8, 9, 10, 11, 12pjhthlem1 27036 . . . . 5  |-  ( ( ( 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 )
1413ralrimiva 2840 . . . 4  |-  ( (
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 )
153chshii 26872 . . . . 5  |-  H  e.  SH
16 shocel 26927 . . . . 5  |-  ( H  e.  SH  ->  (
( A  -h  x
)  e.  ( _|_ `  H )  <->  ( ( A  -h  x )  e. 
~H  /\  A. y  e.  H  ( ( A  -h  x )  .ih  y )  =  0 ) ) )
1715, 16ax-mp 5 . . . 4  |-  ( ( A  -h  x )  e.  ( _|_ `  H
)  <->  ( ( A  -h  x )  e. 
~H  /\  A. y  e.  H  ( ( A  -h  x )  .ih  y )  =  0 ) )
187, 14, 17sylanbrc 669 . . 3  |-  ( (
ph  /\  ( x  e.  H  /\  A. z  e.  H  ( normh `  ( A  -h  x
) )  <_  ( normh `  ( A  -h  z ) ) ) )  ->  ( A  -h  x )  e.  ( _|_ `  H ) )
19 hvpncan3 26687 . . . . 5  |-  ( ( x  e.  ~H  /\  A  e.  ~H )  ->  ( x  +h  ( A  -h  x ) )  =  A )
205, 2, 19syl2anc 666 . . . 4  |-  ( (
ph  /\  ( x  e.  H  /\  A. z  e.  H  ( normh `  ( A  -h  x
) )  <_  ( normh `  ( A  -h  z ) ) ) )  ->  ( x  +h  ( A  -h  x
) )  =  A )
2120eqcomd 2431 . . 3  |-  ( (
ph  /\  ( x  e.  H  /\  A. z  e.  H  ( normh `  ( A  -h  x
) )  <_  ( normh `  ( A  -h  z ) ) ) )  ->  A  =  ( x  +h  ( A  -h  x ) ) )
22 oveq2 6311 . . . . 5  |-  ( y  =  ( A  -h  x )  ->  (
x  +h  y )  =  ( x  +h  ( A  -h  x
) ) )
2322eqeq2d 2437 . . . 4  |-  ( y  =  ( A  -h  x )  ->  ( A  =  ( x  +h  y )  <->  A  =  ( x  +h  ( A  -h  x ) ) ) )
2423rspcev 3183 . . 3  |-  ( ( ( A  -h  x
)  e.  ( _|_ `  H )  /\  A  =  ( x  +h  ( A  -h  x
) ) )  ->  E. y  e.  ( _|_ `  H ) A  =  ( x  +h  y ) )
2518, 21, 24syl2anc 666 . 2  |-  ( (
ph  /\  ( x  e.  H  /\  A. z  e.  H  ( normh `  ( A  -h  x
) )  <_  ( normh `  ( A  -h  z ) ) ) )  ->  E. y  e.  ( _|_ `  H
) A  =  ( x  +h  y ) )
26 df-hba 26614 . . . 4  |-  ~H  =  ( BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
27 eqid 2423 . . . . 5  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
2827hhvs 26815 . . . 4  |-  -h  =  ( -v `  <. <.  +h  ,  .h  >. ,  normh >. )
2927hhnm 26816 . . . 4  |-  normh  =  (
normCV
`  <. <.  +h  ,  .h  >. ,  normh >. )
30 eqid 2423 . . . . 5  |-  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) )
>. ,  ( normh  |`  H ) >.  =  <. <.
(  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) )
>. ,  ( normh  |`  H ) >.
3130, 15hhssba 26914 . . . 4  |-  H  =  ( BaseSet `  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) )
>. ,  ( normh  |`  H ) >. )
3227hhph 26823 . . . . 5  |-  <. <.  +h  ,  .h  >. ,  normh >.  e.  CPreHil OLD
3332a1i 11 . . . 4  |-  ( ph  -> 
<. <.  +h  ,  .h  >. ,  normh >.  e.  CPreHil OLD )
3427, 30hhsst 26909 . . . . . . 7  |-  ( H  e.  SH  ->  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) )
>. ,  ( normh  |`  H ) >.  e.  (
SubSp `  <. <.  +h  ,  .h  >. ,  normh >. ) )
3515, 34ax-mp 5 . . . . . 6  |-  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) )
>. ,  ( normh  |`  H ) >.  e.  (
SubSp `  <. <.  +h  ,  .h  >. ,  normh >. )
3630, 3hhssbn 26923 . . . . . 6  |-  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) )
>. ,  ( normh  |`  H ) >.  e.  CBan
37 elin 3650 . . . . . 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 ) )
3835, 36, 37mpbir2an 929 . . . . 5  |-  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) )
>. ,  ( normh  |`  H ) >.  e.  ( ( SubSp `  <. <.  +h  ,  .h  >. ,  normh >. )  i^i  CBan )
3938a1i 11 . . . 4  |-  ( ph  -> 
<. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) )
>. ,  ( normh  |`  H ) >.  e.  ( ( SubSp `  <. <.  +h  ,  .h  >. ,  normh >. )  i^i  CBan ) )
4026, 28, 29, 31, 33, 39, 1minveco 26518 . . 3  |-  ( ph  ->  E! x  e.  H  A. z  e.  H  ( normh `  ( A  -h  x ) )  <_ 
( normh `  ( A  -h  z ) ) )
41 reurex 3046 . . 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 ) ) )
4240, 41syl 17 . 2  |-  ( ph  ->  E. x  e.  H  A. z  e.  H  ( normh `  ( A  -h  x ) )  <_ 
( normh `  ( A  -h  z ) ) )
4325, 42reximddv 2902 1  |-  ( ph  ->  E. x  e.  H  E. y  e.  ( _|_ `  H ) A  =  ( x  +h  y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1438    e. wcel 1869   A.wral 2776   E.wrex 2777   E!wreu 2778    i^i cin 3436   <.cop 4003   class class class wbr 4421    X. cxp 4849    |` cres 4853   ` cfv 5599  (class class class)co 6303   CCcc 9539   0cc0 9541   1c1 9542    + caddc 9544    <_ cle 9678    / cdiv 10271   SubSpcss 26352   CPreHil OLDccphlo 26445   CBanccbn 26496   ~Hchil 26564    +h cva 26565    .h csm 26566    .ih csp 26567   normhcno 26568    -h cmv 26570   SHcsh 26573   CHcch 26574   _|_cort 26575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-inf2 8150  ax-cc 8867  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-pre-sup 9619  ax-addf 9620  ax-mulf 9621  ax-hilex 26644  ax-hfvadd 26645  ax-hvcom 26646  ax-hvass 26647  ax-hv0cl 26648  ax-hvaddid 26649  ax-hfvmul 26650  ax-hvmulid 26651  ax-hvmulass 26652  ax-hvdistr1 26653  ax-hvdistr2 26654  ax-hvmul0 26655  ax-hfi 26724  ax-his1 26727  ax-his2 26728  ax-his3 26729  ax-his4 26730  ax-hcompl 26847
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-iin 4300  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-se 4811  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-oadd 7192  df-omul 7193  df-er 7369  df-map 7480  df-pm 7481  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-fi 7929  df-sup 7960  df-inf 7961  df-oi 8029  df-card 8376  df-acn 8379  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-2 10670  df-3 10671  df-4 10672  df-n0 10872  df-z 10940  df-uz 11162  df-q 11267  df-rp 11305  df-xneg 11411  df-xadd 11412  df-xmul 11413  df-ico 11643  df-icc 11644  df-fz 11787  df-fl 12029  df-seq 12215  df-exp 12274  df-cj 13156  df-re 13157  df-im 13158  df-sqrt 13292  df-abs 13293  df-clim 13545  df-rlim 13546  df-rest 15314  df-topgen 15335  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-fbas 18960  df-fg 18961  df-top 19913  df-bases 19914  df-topon 19915  df-cld 20026  df-ntr 20027  df-cls 20028  df-nei 20106  df-lm 20237  df-haus 20323  df-fil 20853  df-fm 20945  df-flim 20946  df-flf 20947  df-cfil 22217  df-cau 22218  df-cmet 22219  df-grpo 25911  df-gid 25912  df-ginv 25913  df-gdiv 25914  df-ablo 26002  df-subgo 26022  df-vc 26157  df-nv 26203  df-va 26206  df-ba 26207  df-sm 26208  df-0v 26209  df-vs 26210  df-nmcv 26211  df-ims 26212  df-ssp 26353  df-ph 26446  df-cbn 26497  df-hnorm 26613  df-hba 26614  df-hvsub 26616  df-hlim 26617  df-hcau 26618  df-sh 26852  df-ch 26866  df-oc 26897  df-ch0 26898
This theorem is referenced by:  pjhth  27038  omlsii  27048
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