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Theorem caucvgprprlemloccalc 6936
Description: Lemma for caucvgprpr 6964. Rearranging some expressions for caucvgprprlemloc 6955. (Contributed by Jim Kingdon, 8-Feb-2021.)
Hypotheses
Ref Expression
caucvgprprlemloccalc.st  |-  ( ph  ->  S  <Q  T )
caucvgprprlemloccalc.y  |-  ( ph  ->  Y  e.  Q. )
caucvgprprlemloccalc.syt  |-  ( ph  ->  ( S  +Q  Y
)  =  T )
caucvgprprlemloccalc.x  |-  ( ph  ->  X  e.  Q. )
caucvgprprlemloccalc.xxy  |-  ( ph  ->  ( X  +Q  X
)  <Q  Y )
caucvgprprlemloccalc.m  |-  ( ph  ->  M  e.  N. )
caucvgprprlemloccalc.mx  |-  ( ph  ->  ( *Q `  [ <. M ,  1o >. ]  ~Q  )  <Q  X )
Assertion
Ref Expression
caucvgprprlemloccalc  |-  ( ph  ->  ( <. { l  |  l  <Q  ( S  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  ) ) } ,  { u  |  ( S  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  )
)  <Q  u } >.  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. M ,  1o >. ]  ~Q  )  <Q  u } >. )  <P 
<. { l  |  l 
<Q  T } ,  {
u  |  T  <Q  u } >. )
Distinct variable groups:    M, l, u    S, l, u    T, l, u
Allowed substitution hints:    ph( u, l)    X( u, l)    Y( u, l)

Proof of Theorem caucvgprprlemloccalc
StepHypRef Expression
1 caucvgprprlemloccalc.st . . . . . 6  |-  ( ph  ->  S  <Q  T )
2 ltrelnq 6617 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
32brel 4418 . . . . . 6  |-  ( S 
<Q  T  ->  ( S  e.  Q.  /\  T  e.  Q. ) )
41, 3syl 14 . . . . 5  |-  ( ph  ->  ( S  e.  Q.  /\  T  e.  Q. )
)
54simpld 110 . . . 4  |-  ( ph  ->  S  e.  Q. )
6 caucvgprprlemloccalc.m . . . . 5  |-  ( ph  ->  M  e.  N. )
7 nnnq 6674 . . . . 5  |-  ( M  e.  N.  ->  [ <. M ,  1o >. ]  ~Q  e.  Q. )
8 recclnq 6644 . . . . 5  |-  ( [
<. M ,  1o >. ]  ~Q  e.  Q.  ->  ( *Q `  [ <. M ,  1o >. ]  ~Q  )  e.  Q. )
96, 7, 83syl 17 . . . 4  |-  ( ph  ->  ( *Q `  [ <. M ,  1o >. ]  ~Q  )  e.  Q. )
10 addclnq 6627 . . . 4  |-  ( ( S  e.  Q.  /\  ( *Q `  [ <. M ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( S  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  )
)  e.  Q. )
115, 9, 10syl2anc 403 . . 3  |-  ( ph  ->  ( S  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  )
)  e.  Q. )
12 addnqpr 6813 . . 3  |-  ( ( ( S  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  )
)  e.  Q.  /\  ( *Q `  [ <. M ,  1o >. ]  ~Q  )  e.  Q. )  -> 
<. { l  |  l 
<Q  ( ( S  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  )
) } ,  {
u  |  ( ( S  +Q  ( *Q
`  [ <. M ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. M ,  1o >. ]  ~Q  )
)  <Q  u } >.  =  ( <. { l  |  l  <Q  ( S  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  ) ) } ,  { u  |  ( S  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  )
)  <Q  u } >.  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. M ,  1o >. ]  ~Q  )  <Q  u } >. )
)
1311, 9, 12syl2anc 403 . 2  |-  ( ph  -> 
<. { l  |  l 
<Q  ( ( S  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  )
) } ,  {
u  |  ( ( S  +Q  ( *Q
`  [ <. M ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. M ,  1o >. ]  ~Q  )
)  <Q  u } >.  =  ( <. { l  |  l  <Q  ( S  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  ) ) } ,  { u  |  ( S  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  )
)  <Q  u } >.  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. M ,  1o >. ]  ~Q  )  <Q  u } >. )
)
14 addassnqg 6634 . . . . 5  |-  ( ( S  e.  Q.  /\  ( *Q `  [ <. M ,  1o >. ]  ~Q  )  e.  Q.  /\  ( *Q `  [ <. M ,  1o >. ]  ~Q  )  e.  Q. )  ->  (
( S  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. M ,  1o >. ]  ~Q  )
)  =  ( S  +Q  ( ( *Q
`  [ <. M ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  ) ) ) )
155, 9, 9, 14syl3anc 1170 . . . 4  |-  ( ph  ->  ( ( S  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  )
)  =  ( S  +Q  ( ( *Q
`  [ <. M ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  ) ) ) )
16 caucvgprprlemloccalc.mx . . . . . . . 8  |-  ( ph  ->  ( *Q `  [ <. M ,  1o >. ]  ~Q  )  <Q  X )
17 caucvgprprlemloccalc.x . . . . . . . . 9  |-  ( ph  ->  X  e.  Q. )
18 lt2addnq 6656 . . . . . . . . 9  |-  ( ( ( ( *Q `  [ <. M ,  1o >. ]  ~Q  )  e. 
Q.  /\  X  e.  Q. )  /\  (
( *Q `  [ <. M ,  1o >. ]  ~Q  )  e.  Q.  /\  X  e.  Q. )
)  ->  ( (
( *Q `  [ <. M ,  1o >. ]  ~Q  )  <Q  X  /\  ( *Q `  [ <. M ,  1o >. ]  ~Q  )  <Q  X )  -> 
( ( *Q `  [ <. M ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  ) )  <Q 
( X  +Q  X
) ) )
199, 17, 9, 17, 18syl22anc 1171 . . . . . . . 8  |-  ( ph  ->  ( ( ( *Q
`  [ <. M ,  1o >. ]  ~Q  )  <Q  X  /\  ( *Q
`  [ <. M ,  1o >. ]  ~Q  )  <Q  X )  ->  (
( *Q `  [ <. M ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  )
)  <Q  ( X  +Q  X ) ) )
2016, 16, 19mp2and 424 . . . . . . 7  |-  ( ph  ->  ( ( *Q `  [ <. M ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  ) )  <Q 
( X  +Q  X
) )
21 caucvgprprlemloccalc.xxy . . . . . . 7  |-  ( ph  ->  ( X  +Q  X
)  <Q  Y )
22 ltsonq 6650 . . . . . . . 8  |-  <Q  Or  Q.
2322, 2sotri 4750 . . . . . . 7  |-  ( ( ( ( *Q `  [ <. M ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  ) )  <Q 
( X  +Q  X
)  /\  ( X  +Q  X )  <Q  Y )  ->  ( ( *Q
`  [ <. M ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  ) )  <Q  Y )
2420, 21, 23syl2anc 403 . . . . . 6  |-  ( ph  ->  ( ( *Q `  [ <. M ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  ) )  <Q  Y )
25 ltanqi 6654 . . . . . 6  |-  ( ( ( ( *Q `  [ <. M ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  ) )  <Q  Y  /\  S  e.  Q. )  ->  ( S  +Q  ( ( *Q `  [ <. M ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  ) ) ) 
<Q  ( S  +Q  Y
) )
2624, 5, 25syl2anc 403 . . . . 5  |-  ( ph  ->  ( S  +Q  (
( *Q `  [ <. M ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  )
) )  <Q  ( S  +Q  Y ) )
27 caucvgprprlemloccalc.syt . . . . 5  |-  ( ph  ->  ( S  +Q  Y
)  =  T )
2826, 27breqtrd 3817 . . . 4  |-  ( ph  ->  ( S  +Q  (
( *Q `  [ <. M ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  )
) )  <Q  T )
2915, 28eqbrtrd 3813 . . 3  |-  ( ph  ->  ( ( S  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  )
)  <Q  T )
30 ltnqpri 6846 . . 3  |-  ( ( ( S  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. M ,  1o >. ]  ~Q  )
)  <Q  T  ->  <. { l  |  l  <Q  (
( S  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. M ,  1o >. ]  ~Q  )
) } ,  {
u  |  ( ( S  +Q  ( *Q
`  [ <. M ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. M ,  1o >. ]  ~Q  )
)  <Q  u } >.  <P  <. { l  |  l 
<Q  T } ,  {
u  |  T  <Q  u } >. )
3129, 30syl 14 . 2  |-  ( ph  -> 
<. { l  |  l 
<Q  ( ( S  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  )
) } ,  {
u  |  ( ( S  +Q  ( *Q
`  [ <. M ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. M ,  1o >. ]  ~Q  )
)  <Q  u } >.  <P  <. { l  |  l 
<Q  T } ,  {
u  |  T  <Q  u } >. )
3213, 31eqbrtrrd 3815 1  |-  ( ph  ->  ( <. { l  |  l  <Q  ( S  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  ) ) } ,  { u  |  ( S  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  )
)  <Q  u } >.  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. M ,  1o >. ]  ~Q  )  <Q  u } >. )  <P 
<. { l  |  l 
<Q  T } ,  {
u  |  T  <Q  u } >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   {cab 2068   <.cop 3409   class class class wbr 3793   ` cfv 4932  (class class class)co 5543   1oc1o 6058   [cec 6170   N.cnpi 6524    ~Q ceq 6531   Q.cnq 6532    +Q cplq 6534   *Qcrq 6536    <Q cltq 6537    +P. cpp 6545    <P cltp 6547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-2o 6066  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-enq0 6676  df-nq0 6677  df-0nq0 6678  df-plq0 6679  df-mq0 6680  df-inp 6718  df-iplp 6720  df-iltp 6722
This theorem is referenced by:  caucvgprprlemloc  6955
  Copyright terms: Public domain W3C validator