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Theorem addnqprlemfu 7561
Description: Lemma for addnqpr 7562. 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 7558 . . . . . 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 7399 . . . . . . . . 9  |-  <Q  Or  Q.
3 addclnq 7376 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  +Q  B
)  e.  Q. )
4 sonr 4319 . . . . . . . . 9  |-  ( ( 
<Q  Or  Q.  /\  ( A  +Q  B )  e. 
Q. )  ->  -.  ( A  +Q  B
)  <Q  ( A  +Q  B ) )
52, 3, 4sylancr 414 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  -.  ( A  +Q  B )  <Q  ( A  +Q  B ) )
6 ltrelnq 7366 . . . . . . . . . . . 12  |-  <Q  C_  ( Q.  X.  Q. )
76brel 4680 . . . . . . . . . . 11  |-  ( ( A  +Q  B ) 
<Q  ( A  +Q  B
)  ->  ( ( A  +Q  B )  e. 
Q.  /\  ( A  +Q  B )  e.  Q. ) )
87simpld 112 . . . . . . . . . 10  |-  ( ( A  +Q  B ) 
<Q  ( A  +Q  B
)  ->  ( A  +Q  B )  e.  Q. )
9 elex 2750 . . . . . . . . . 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 4008 . . . . . . . . 9  |-  ( l  =  ( A  +Q  B )  ->  (
l  <Q  ( A  +Q  B )  <->  ( A  +Q  B )  <Q  ( A  +Q  B ) ) )
1210, 11elab3 2891 . . . . . . . 8  |-  ( ( A  +Q  B )  e.  { l  |  l  <Q  ( A  +Q  B ) }  <->  ( A  +Q  B )  <Q  ( A  +Q  B ) )
135, 12sylnibr 677 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  -.  ( A  +Q  B )  e.  {
l  |  l  <Q 
( A  +Q  B
) } )
14 ltnqex 7550 . . . . . . . . 9  |-  { l  |  l  <Q  ( A  +Q  B ) }  e.  _V
15 gtnqex 7551 . . . . . . . . 9  |-  { u  |  ( A  +Q  B )  <Q  u }  e.  _V
1614, 15op1st 6149 . . . . . . . 8  |-  ( 1st `  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B )  <Q  u } >. )  =  {
l  |  l  <Q 
( A  +Q  B
) }
1716eleq2i 2244 . . . . . . 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 677 . . . . . 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 3160 . . . . 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 276 . . . 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 7548 . . . . . . . 8  |-  ( A  e.  Q.  ->  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  e.  P. )
22 nqprlu 7548 . . . . . . . 8  |-  ( B  e.  Q.  ->  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >.  e.  P. )
23 addclpr 7538 . . . . . . . 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 289 . . . . . . 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 7476 . . . . . . 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 2742 . . . . . . . 8  |-  r  e. 
_V
28 breq2 4009 . . . . . . . 8  |-  ( u  =  r  ->  (
( A  +Q  B
)  <Q  u  <->  ( A  +Q  B )  <Q  r
) )
2914, 15op2nd 6150 . . . . . . . 8  |-  ( 2nd `  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B )  <Q  u } >. )  =  {
u  |  ( A  +Q  B )  <Q  u }
3027, 28, 29elab2 2887 . . . . . . 7  |-  ( r  e.  ( 2nd `  <. { l  |  l  <Q 
( A  +Q  B
) } ,  {
u  |  ( A  +Q  B )  <Q  u } >. )  <->  ( A  +Q  B )  <Q  r
)
3130biimpi 120 . . . . . 6  |-  ( r  e.  ( 2nd `  <. { l  |  l  <Q 
( A  +Q  B
) } ,  {
u  |  ( A  +Q  B )  <Q  u } >. )  ->  ( A  +Q  B )  <Q 
r )
32 prloc 7492 . . . . . 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 289 . . . . 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 729 . . . 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 1349 . . 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 115 . 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 3163 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 104    \/ wo 708    e. wcel 2148   {cab 2163   _Vcvv 2739    C_ wss 3131   <.cop 3597   class class class wbr 4005    Or wor 4297   ` cfv 5218  (class class class)co 5877   1stc1st 6141   2ndc2nd 6142   Q.cnq 7281    +Q cplq 7283    <Q cltq 7286   P.cnp 7292    +P. cpp 7294
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
This theorem is referenced by:  addnqpr  7562
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