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Theorem caucvgprprlemloccalc 7680
Description: Lemma for caucvgprpr 7708. Rearranging some expressions for caucvgprprlemloc 7699. (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 7361 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
32brel 4677 . . . . . 6  |-  ( S 
<Q  T  ->  ( S  e.  Q.  /\  T  e.  Q. ) )
41, 3syl 14 . . . . 5  |-  ( ph  ->  ( S  e.  Q.  /\  T  e.  Q. )
)
54simpld 112 . . . 4  |-  ( ph  ->  S  e.  Q. )
6 caucvgprprlemloccalc.m . . . . 5  |-  ( ph  ->  M  e.  N. )
7 nnnq 7418 . . . . 5  |-  ( M  e.  N.  ->  [ <. M ,  1o >. ]  ~Q  e.  Q. )
8 recclnq 7388 . . . . 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 7371 . . . 4  |-  ( ( S  e.  Q.  /\  ( *Q `  [ <. M ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( S  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  )
)  e.  Q. )
115, 9, 10syl2anc 411 . . 3  |-  ( ph  ->  ( S  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  )
)  e.  Q. )
12 addnqpr 7557 . . 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 411 . 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 7378 . . . . 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 1238 . . . 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 7400 . . . . . . . . 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 1239 . . . . . . . 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 433 . . . . . . 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 7394 . . . . . . . 8  |-  <Q  Or  Q.
2322, 2sotri 5023 . . . . . . 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 411 . . . . . 6  |-  ( ph  ->  ( ( *Q `  [ <. M ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  ) )  <Q  Y )
25 ltanqi 7398 . . . . . 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 411 . . . . 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 4028 . . . 4  |-  ( ph  ->  ( S  +Q  (
( *Q `  [ <. M ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  )
) )  <Q  T )
2915, 28eqbrtrd 4024 . . 3  |-  ( ph  ->  ( ( S  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  )
)  <Q  T )
30 ltnqpri 7590 . . 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 4026 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 104    = wceq 1353    e. wcel 2148   {cab 2163   <.cop 3595   class class class wbr 4002   ` cfv 5215  (class class class)co 5872   1oc1o 6407   [cec 6530   N.cnpi 7268    ~Q ceq 7275   Q.cnq 7276    +Q cplq 7278   *Qcrq 7280    <Q cltq 7281    +P. cpp 7289    <P cltp 7291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-iinf 4586
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-tr 4101  df-eprel 4288  df-id 4292  df-po 4295  df-iso 4296  df-iord 4365  df-on 4367  df-suc 4370  df-iom 4589  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-ov 5875  df-oprab 5876  df-mpo 5877  df-1st 6138  df-2nd 6139  df-recs 6303  df-irdg 6368  df-1o 6414  df-2o 6415  df-oadd 6418  df-omul 6419  df-er 6532  df-ec 6534  df-qs 6538  df-ni 7300  df-pli 7301  df-mi 7302  df-lti 7303  df-plpq 7340  df-mpq 7341  df-enq 7343  df-nqqs 7344  df-plqqs 7345  df-mqqs 7346  df-1nqqs 7347  df-rq 7348  df-ltnqqs 7349  df-enq0 7420  df-nq0 7421  df-0nq0 7422  df-plq0 7423  df-mq0 7424  df-inp 7462  df-iplp 7464  df-iltp 7466
This theorem is referenced by:  caucvgprprlemloc  7699
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