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Theorem addnqprlemfu 7336
Description: Lemma for addnqpr 7337. 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 7333 . . . . . 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 7174 . . . . . . . . 9  |-  <Q  Or  Q.
3 addclnq 7151 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  +Q  B
)  e.  Q. )
4 sonr 4209 . . . . . . . . 9  |-  ( ( 
<Q  Or  Q.  /\  ( A  +Q  B )  e. 
Q. )  ->  -.  ( A  +Q  B
)  <Q  ( A  +Q  B ) )
52, 3, 4sylancr 410 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  -.  ( A  +Q  B )  <Q  ( A  +Q  B ) )
6 ltrelnq 7141 . . . . . . . . . . . 12  |-  <Q  C_  ( Q.  X.  Q. )
76brel 4561 . . . . . . . . . . 11  |-  ( ( A  +Q  B ) 
<Q  ( A  +Q  B
)  ->  ( ( A  +Q  B )  e. 
Q.  /\  ( A  +Q  B )  e.  Q. ) )
87simpld 111 . . . . . . . . . 10  |-  ( ( A  +Q  B ) 
<Q  ( A  +Q  B
)  ->  ( A  +Q  B )  e.  Q. )
9 elex 2671 . . . . . . . . . 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 3902 . . . . . . . . 9  |-  ( l  =  ( A  +Q  B )  ->  (
l  <Q  ( A  +Q  B )  <->  ( A  +Q  B )  <Q  ( A  +Q  B ) ) )
1210, 11elab3 2809 . . . . . . . 8  |-  ( ( A  +Q  B )  e.  { l  |  l  <Q  ( A  +Q  B ) }  <->  ( A  +Q  B )  <Q  ( A  +Q  B ) )
135, 12sylnibr 651 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  -.  ( A  +Q  B )  e.  {
l  |  l  <Q 
( A  +Q  B
) } )
14 ltnqex 7325 . . . . . . . . 9  |-  { l  |  l  <Q  ( A  +Q  B ) }  e.  _V
15 gtnqex 7326 . . . . . . . . 9  |-  { u  |  ( A  +Q  B )  <Q  u }  e.  _V
1614, 15op1st 6012 . . . . . . . 8  |-  ( 1st `  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B )  <Q  u } >. )  =  {
l  |  l  <Q 
( A  +Q  B
) }
1716eleq2i 2184 . . . . . . 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 651 . . . . . 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 3070 . . . . 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 274 . . . 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 7323 . . . . . . . 8  |-  ( A  e.  Q.  ->  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  e.  P. )
22 nqprlu 7323 . . . . . . . 8  |-  ( B  e.  Q.  ->  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >.  e.  P. )
23 addclpr 7313 . . . . . . . 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 287 . . . . . . 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 7251 . . . . . . 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 2663 . . . . . . . 8  |-  r  e. 
_V
28 breq2 3903 . . . . . . . 8  |-  ( u  =  r  ->  (
( A  +Q  B
)  <Q  u  <->  ( A  +Q  B )  <Q  r
) )
2914, 15op2nd 6013 . . . . . . . 8  |-  ( 2nd `  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B )  <Q  u } >. )  =  {
u  |  ( A  +Q  B )  <Q  u }
3027, 28, 29elab2 2805 . . . . . . 7  |-  ( r  e.  ( 2nd `  <. { l  |  l  <Q 
( A  +Q  B
) } ,  {
u  |  ( A  +Q  B )  <Q  u } >. )  <->  ( A  +Q  B )  <Q  r
)
3130biimpi 119 . . . . . 6  |-  ( r  e.  ( 2nd `  <. { l  |  l  <Q 
( A  +Q  B
) } ,  {
u  |  ( A  +Q  B )  <Q  u } >. )  ->  ( A  +Q  B )  <Q 
r )
32 prloc 7267 . . . . . 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 287 . . . . 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 703 . . . 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 1312 . . 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 114 . 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 3073 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 103    \/ wo 682    e. wcel 1465   {cab 2103   _Vcvv 2660    C_ wss 3041   <.cop 3500   class class class wbr 3899    Or wor 4187   ` cfv 5093  (class class class)co 5742   1stc1st 6004   2ndc2nd 6005   Q.cnq 7056    +Q cplq 7058    <Q cltq 7061   P.cnp 7067    +P. cpp 7069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-2o 6282  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-enq0 7200  df-nq0 7201  df-0nq0 7202  df-plq0 7203  df-mq0 7204  df-inp 7242  df-iplp 7244
This theorem is referenced by:  addnqpr  7337
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