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Theorem pjhthlem2 24795
Description: Lemma for pjhth 24796. (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 24371 . . . 4  |-  ~H  =  ( BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
2 eqid 2443 . . . . 5  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
32hhvs 24572 . . . 4  |-  -h  =  ( -v `  <. <.  +h  ,  .h  >. ,  normh >. )
42hhnm 24573 . . . 4  |-  normh  =  (
normCV
`  <. <.  +h  ,  .h  >. ,  normh >. )
5 eqid 2443 . . . . 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 24630 . . . . 5  |-  H  e.  SH
85, 7hhssba 24672 . . . 4  |-  H  =  ( BaseSet `  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) )
>. ,  ( normh  |`  H ) >. )
92hhph 24580 . . . . 5  |-  <. <.  +h  ,  .h  >. ,  normh >.  e.  CPreHil OLD
109a1i 11 . . . 4  |-  ( ph  -> 
<. <.  +h  ,  .h  >. ,  normh >.  e.  CPreHil OLD )
112, 5hhsst 24667 . . . . . . 7  |-  ( H  e.  SH  ->  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) )
>. ,  ( normh  |`  H ) >.  e.  (
SubSp `  <. <.  +h  ,  .h  >. ,  normh >. ) )
127, 11ax-mp 5 . . . . . 6  |-  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) )
>. ,  ( normh  |`  H ) >.  e.  (
SubSp `  <. <.  +h  ,  .h  >. ,  normh >. )
135, 6hhssbn 24681 . . . . . 6  |-  <. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) )
>. ,  ( normh  |`  H ) >.  e.  CBan
14 elin 3539 . . . . . 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 911 . . . . 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 24285 . . 3  |-  ( ph  ->  E! x  e.  H  A. z  e.  H  ( normh `  ( A  -h  x ) )  <_ 
( normh `  ( A  -h  z ) ) )
19 reurex 2937 . . 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 465 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  H  /\  A. z  e.  H  ( normh `  ( A  -h  x
) )  <_  ( normh `  ( A  -h  z ) ) ) )  ->  A  e.  ~H )
226cheli 24635 . . . . . . . 8  |-  ( x  e.  H  ->  x  e.  ~H )
2322ad2antrl 727 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  H  /\  A. z  e.  H  ( normh `  ( A  -h  x
) )  <_  ( normh `  ( A  -h  z ) ) ) )  ->  x  e.  ~H )
24 hvsubcl 24419 . . . . . . 7  |-  ( ( A  e.  ~H  /\  x  e.  ~H )  ->  ( A  -h  x
)  e.  ~H )
2521, 23, 24syl2anc 661 . . . . . 6  |-  ( (
ph  /\  ( x  e.  H  /\  A. z  e.  H  ( normh `  ( A  -h  x
) )  <_  ( normh `  ( A  -h  z ) ) ) )  ->  ( A  -h  x )  e.  ~H )
2621adantr 465 . . . . . . . 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 759 . . . . . . . 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 461 . . . . . . . 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 760 . . . . . . . 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 2443 . . . . . . . 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 24794 . . . . . . 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 2799 . . . . . 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 24685 . . . . . . 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 5 . . . . . 6  |-  ( ( A  -h  x )  e.  ( _|_ `  H
)  <->  ( ( A  -h  x )  e. 
~H  /\  A. y  e.  H  ( ( A  -h  x )  .ih  y )  =  0 ) )
3525, 32, 34sylanbrc 664 . . . . 5  |-  ( (
ph  /\  ( x  e.  H  /\  A. z  e.  H  ( normh `  ( A  -h  x
) )  <_  ( normh `  ( A  -h  z ) ) ) )  ->  ( A  -h  x )  e.  ( _|_ `  H ) )
36 hvpncan3 24444 . . . . . . 7  |-  ( ( x  e.  ~H  /\  A  e.  ~H )  ->  ( x  +h  ( A  -h  x ) )  =  A )
3723, 21, 36syl2anc 661 . . . . . 6  |-  ( (
ph  /\  ( x  e.  H  /\  A. z  e.  H  ( normh `  ( A  -h  x
) )  <_  ( normh `  ( A  -h  z ) ) ) )  ->  ( x  +h  ( A  -h  x
) )  =  A )
3837eqcomd 2448 . . . . 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 6099 . . . . . . 7  |-  ( y  =  ( A  -h  x )  ->  (
x  +h  y )  =  ( x  +h  ( A  -h  x
) ) )
4039eqeq2d 2454 . . . . . 6  |-  ( y  =  ( A  -h  x )  ->  ( A  =  ( x  +h  y )  <->  A  =  ( x  +h  ( A  -h  x ) ) ) )
4140rspcev 3073 . . . . 5  |-  ( ( ( A  -h  x
)  e.  ( _|_ `  H )  /\  A  =  ( x  +h  ( A  -h  x
) ) )  ->  E. y  e.  ( _|_ `  H ) A  =  ( x  +h  y ) )
4235, 38, 41syl2anc 661 . . . 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 615 . . 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 2828 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   E.wrex 2716   E!wreu 2717    i^i cin 3327   <.cop 3883   class class class wbr 4292    X. cxp 4838    |` cres 4842   ` cfv 5418  (class class class)co 6091   CCcc 9280   0cc0 9282   1c1 9283    + caddc 9285    <_ cle 9419    / cdiv 9993   SubSpcss 24119   CPreHil OLDccphlo 24212   CBanccbn 24263   ~Hchil 24321    +h cva 24322    .h csm 24323    .ih csp 24324   normhcno 24325    -h cmv 24327   SHcsh 24330   CHcch 24331   _|_cort 24332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cc 8604  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361  ax-mulf 9362  ax-hilex 24401  ax-hfvadd 24402  ax-hvcom 24403  ax-hvass 24404  ax-hv0cl 24405  ax-hvaddid 24406  ax-hfvmul 24407  ax-hvmulid 24408  ax-hvmulass 24409  ax-hvdistr1 24410  ax-hvdistr2 24411  ax-hvmul0 24412  ax-hfi 24481  ax-his1 24484  ax-his2 24485  ax-his3 24486  ax-his4 24487  ax-hcompl 24604
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-omul 6925  df-er 7101  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fi 7661  df-sup 7691  df-oi 7724  df-card 8109  df-acn 8112  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-n0 10580  df-z 10647  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ico 11306  df-icc 11307  df-fz 11438  df-fl 11642  df-seq 11807  df-exp 11866  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-clim 12966  df-rlim 12967  df-rest 14361  df-topgen 14382  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-fbas 17814  df-fg 17815  df-top 18503  df-bases 18505  df-topon 18506  df-cld 18623  df-ntr 18624  df-cls 18625  df-nei 18702  df-lm 18833  df-haus 18919  df-fil 19419  df-fm 19511  df-flim 19512  df-flf 19513  df-cfil 20766  df-cau 20767  df-cmet 20768  df-grpo 23678  df-gid 23679  df-ginv 23680  df-gdiv 23681  df-ablo 23769  df-subgo 23789  df-vc 23924  df-nv 23970  df-va 23973  df-ba 23974  df-sm 23975  df-0v 23976  df-vs 23977  df-nmcv 23978  df-ims 23979  df-ssp 24120  df-ph 24213  df-cbn 24264  df-hnorm 24370  df-hba 24371  df-hvsub 24373  df-hlim 24374  df-hcau 24375  df-sh 24609  df-ch 24624  df-oc 24655  df-ch0 24656
This theorem is referenced by:  pjhth  24796  omlsii  24806
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