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Theorem addnqprlemfu 6658
Description: Lemma for addnqpr 6659. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
Assertion
Ref Expression
addnqprlemfu  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( 2nd `  <. { l  |  l  <Q 
( A  +Q  B
) } ,  {
u  |  ( A  +Q  B )  <Q  u } >. )  C_  ( 2nd `  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) ) )
Distinct variable groups:    A, l, u    B, l, u

Proof of Theorem addnqprlemfu
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 addnqprlemrl 6655 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
)  C_  ( 1st ` 
<. { l  |  l 
<Q  ( A  +Q  B
) } ,  {
u  |  ( A  +Q  B )  <Q  u } >. ) )
2 ltsonq 6496 . . . . . . . . 9  |-  <Q  Or  Q.
3 addclnq 6473 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  +Q  B
)  e.  Q. )
4 sonr 4054 . . . . . . . . 9  |-  ( ( 
<Q  Or  Q.  /\  ( A  +Q  B )  e. 
Q. )  ->  -.  ( A  +Q  B
)  <Q  ( A  +Q  B ) )
52, 3, 4sylancr 393 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  -.  ( A  +Q  B )  <Q  ( A  +Q  B ) )
6 ltrelnq 6463 . . . . . . . . . . . 12  |-  <Q  C_  ( Q.  X.  Q. )
76brel 4392 . . . . . . . . . . 11  |-  ( ( A  +Q  B ) 
<Q  ( A  +Q  B
)  ->  ( ( A  +Q  B )  e. 
Q.  /\  ( A  +Q  B )  e.  Q. ) )
87simpld 105 . . . . . . . . . 10  |-  ( ( A  +Q  B ) 
<Q  ( A  +Q  B
)  ->  ( A  +Q  B )  e.  Q. )
9 elex 2566 . . . . . . . . . 10  |-  ( ( A  +Q  B )  e.  Q.  ->  ( A  +Q  B )  e. 
_V )
108, 9syl 14 . . . . . . . . 9  |-  ( ( A  +Q  B ) 
<Q  ( A  +Q  B
)  ->  ( A  +Q  B )  e.  _V )
11 breq1 3767 . . . . . . . . 9  |-  ( l  =  ( A  +Q  B )  ->  (
l  <Q  ( A  +Q  B )  <->  ( A  +Q  B )  <Q  ( A  +Q  B ) ) )
1210, 11elab3 2694 . . . . . . . 8  |-  ( ( A  +Q  B )  e.  { l  |  l  <Q  ( A  +Q  B ) }  <->  ( A  +Q  B )  <Q  ( A  +Q  B ) )
135, 12sylnibr 602 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  -.  ( A  +Q  B )  e.  {
l  |  l  <Q 
( A  +Q  B
) } )
14 ltnqex 6647 . . . . . . . . 9  |-  { l  |  l  <Q  ( A  +Q  B ) }  e.  _V
15 gtnqex 6648 . . . . . . . . 9  |-  { u  |  ( A  +Q  B )  <Q  u }  e.  _V
1614, 15op1st 5773 . . . . . . . 8  |-  ( 1st `  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B )  <Q  u } >. )  =  {
l  |  l  <Q 
( A  +Q  B
) }
1716eleq2i 2104 . . . . . . 7  |-  ( ( A  +Q  B )  e.  ( 1st `  <. { l  |  l  <Q 
( A  +Q  B
) } ,  {
u  |  ( A  +Q  B )  <Q  u } >. )  <->  ( A  +Q  B )  e.  {
l  |  l  <Q 
( A  +Q  B
) } )
1813, 17sylnibr 602 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  -.  ( A  +Q  B )  e.  ( 1st `  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B
)  <Q  u } >. ) )
191, 18ssneldd 2948 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  -.  ( A  +Q  B )  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )
2019adantr 261 . . . 4  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B )  <Q  u } >. ) )  ->  -.  ( A  +Q  B
)  e.  ( 1st `  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) ) )
21 nqprlu 6645 . . . . . . . 8  |-  ( A  e.  Q.  ->  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  e.  P. )
22 nqprlu 6645 . . . . . . . 8  |-  ( B  e.  Q.  ->  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >.  e.  P. )
23 addclpr 6635 . . . . . . . 8  |-  ( (
<. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  e.  P.  /\ 
<. { l  |  l 
<Q  B } ,  {
u  |  B  <Q  u } >.  e.  P. )  ->  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )  e.  P. )
2421, 22, 23syl2an 273 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )  e.  P. )
25 prop 6573 . . . . . . 7  |-  ( (
<. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )  e.  P.  ->  <. ( 1st `  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) ) ,  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) >.  e.  P. )
2624, 25syl 14 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  -> 
<. ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) ,  ( 2nd `  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) ) >.  e.  P. )
27 vex 2560 . . . . . . . 8  |-  r  e. 
_V
28 breq2 3768 . . . . . . . 8  |-  ( u  =  r  ->  (
( A  +Q  B
)  <Q  u  <->  ( A  +Q  B )  <Q  r
) )
2914, 15op2nd 5774 . . . . . . . 8  |-  ( 2nd `  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B )  <Q  u } >. )  =  {
u  |  ( A  +Q  B )  <Q  u }
3027, 28, 29elab2 2690 . . . . . . 7  |-  ( r  e.  ( 2nd `  <. { l  |  l  <Q 
( A  +Q  B
) } ,  {
u  |  ( A  +Q  B )  <Q  u } >. )  <->  ( A  +Q  B )  <Q  r
)
3130biimpi 113 . . . . . 6  |-  ( r  e.  ( 2nd `  <. { l  |  l  <Q 
( A  +Q  B
) } ,  {
u  |  ( A  +Q  B )  <Q  u } >. )  ->  ( A  +Q  B )  <Q 
r )
32 prloc 6589 . . . . . 6  |-  ( (
<. ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) ,  ( 2nd `  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) ) >.  e.  P.  /\  ( A  +Q  B
)  <Q  r )  -> 
( ( A  +Q  B )  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
)  \/  r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) ) )
3326, 31, 32syl2an 273 . . . . 5  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B )  <Q  u } >. ) )  -> 
( ( A  +Q  B )  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
)  \/  r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) ) )
3433orcomd 648 . . . 4  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B )  <Q  u } >. ) )  -> 
( r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
)  \/  ( A  +Q  B )  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) ) )
3520, 34ecased 1239 . . 3  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B )  <Q  u } >. ) )  -> 
r  e.  ( 2nd `  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) ) )
3635ex 108 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( r  e.  ( 2nd `  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B
)  <Q  u } >. )  ->  r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) ) )
3736ssrdv 2951 1  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( 2nd `  <. { l  |  l  <Q 
( A  +Q  B
) } ,  {
u  |  ( A  +Q  B )  <Q  u } >. )  C_  ( 2nd `  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 97    \/ wo 629    e. wcel 1393   {cab 2026   _Vcvv 2557    C_ wss 2917   <.cop 3378   class class class wbr 3764    Or wor 4032   ` cfv 4902  (class class class)co 5512   1stc1st 5765   2ndc2nd 5766   Q.cnq 6378    +Q cplq 6380    <Q cltq 6383   P.cnp 6389    +P. cpp 6391
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-iplp 6566
This theorem is referenced by:  addnqpr  6659
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