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Theorem caucvgprprlemloccalc 7685
Description: Lemma for caucvgprpr 7713. Rearranging some expressions for caucvgprprlemloc 7704. (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 7366 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
32brel 4680 . . . . . 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 7423 . . . . 5  |-  ( M  e.  N.  ->  [ <. M ,  1o >. ]  ~Q  e.  Q. )
8 recclnq 7393 . . . . 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 7376 . . . 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 7562 . . 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 7383 . . . . 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 7405 . . . . . . . . 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 7399 . . . . . . . 8  |-  <Q  Or  Q.
2322, 2sotri 5026 . . . . . . 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 7403 . . . . . 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 4031 . . . 4  |-  ( ph  ->  ( S  +Q  (
( *Q `  [ <. M ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  )
) )  <Q  T )
2915, 28eqbrtrd 4027 . . 3  |-  ( ph  ->  ( ( S  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  )
)  <Q  T )
30 ltnqpri 7595 . . 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 4029 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 3597   class class class wbr 4005   ` cfv 5218  (class class class)co 5877   1oc1o 6412   [cec 6535   N.cnpi 7273    ~Q ceq 7280   Q.cnq 7281    +Q cplq 7283   *Qcrq 7285    <Q cltq 7286    +P. cpp 7294    <P cltp 7296
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 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589
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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-2o 6420  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-enq0 7425  df-nq0 7426  df-0nq0 7427  df-plq0 7428  df-mq0 7429  df-inp 7467  df-iplp 7469  df-iltp 7471
This theorem is referenced by:  caucvgprprlemloc  7704
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