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Theorem mulnqprlemfu 7135
Description: Lemma for mulnqpr 7136. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
Assertion
Ref Expression
mulnqprlemfu  |-  ( ( 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 mulnqprlemfu
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 mulnqprlemrl 7132 . . . . . 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 6957 . . . . . . . . 9  |-  <Q  Or  Q.
3 mulclnq 6935 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  .Q  B
)  e.  Q. )
4 sonr 4144 . . . . . . . . 9  |-  ( ( 
<Q  Or  Q.  /\  ( A  .Q  B )  e. 
Q. )  ->  -.  ( A  .Q  B
)  <Q  ( A  .Q  B ) )
52, 3, 4sylancr 405 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  -.  ( A  .Q  B )  <Q  ( A  .Q  B ) )
6 ltrelnq 6924 . . . . . . . . . . . 12  |-  <Q  C_  ( Q.  X.  Q. )
76brel 4490 . . . . . . . . . . 11  |-  ( ( A  .Q  B ) 
<Q  ( A  .Q  B
)  ->  ( ( A  .Q  B )  e. 
Q.  /\  ( A  .Q  B )  e.  Q. ) )
87simpld 110 . . . . . . . . . 10  |-  ( ( A  .Q  B ) 
<Q  ( A  .Q  B
)  ->  ( A  .Q  B )  e.  Q. )
9 elex 2630 . . . . . . . . . 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 3848 . . . . . . . . 9  |-  ( l  =  ( A  .Q  B )  ->  (
l  <Q  ( A  .Q  B )  <->  ( A  .Q  B )  <Q  ( A  .Q  B ) ) )
1210, 11elab3 2767 . . . . . . . 8  |-  ( ( A  .Q  B )  e.  { l  |  l  <Q  ( A  .Q  B ) }  <->  ( A  .Q  B )  <Q  ( A  .Q  B ) )
135, 12sylnibr 637 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  -.  ( A  .Q  B )  e.  {
l  |  l  <Q 
( A  .Q  B
) } )
14 ltnqex 7108 . . . . . . . . 9  |-  { l  |  l  <Q  ( A  .Q  B ) }  e.  _V
15 gtnqex 7109 . . . . . . . . 9  |-  { u  |  ( A  .Q  B )  <Q  u }  e.  _V
1614, 15op1st 5917 . . . . . . . 8  |-  ( 1st `  <. { l  |  l  <Q  ( A  .Q  B ) } ,  { u  |  ( A  .Q  B )  <Q  u } >. )  =  {
l  |  l  <Q 
( A  .Q  B
) }
1716eleq2i 2154 . . . . . . 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 637 . . . . . 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 3028 . . . . 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 270 . . . 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 7106 . . . . . . . 8  |-  ( A  e.  Q.  ->  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  e.  P. )
22 nqprlu 7106 . . . . . . . 8  |-  ( B  e.  Q.  ->  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >.  e.  P. )
23 mulclpr 7131 . . . . . . . 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 283 . . . . . . 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 7034 . . . . . . 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 2622 . . . . . . . 8  |-  r  e. 
_V
28 breq2 3849 . . . . . . . 8  |-  ( u  =  r  ->  (
( A  .Q  B
)  <Q  u  <->  ( A  .Q  B )  <Q  r
) )
2914, 15op2nd 5918 . . . . . . . 8  |-  ( 2nd `  <. { l  |  l  <Q  ( A  .Q  B ) } ,  { u  |  ( A  .Q  B )  <Q  u } >. )  =  {
u  |  ( A  .Q  B )  <Q  u }
3027, 28, 29elab2 2763 . . . . . . 7  |-  ( r  e.  ( 2nd `  <. { l  |  l  <Q 
( A  .Q  B
) } ,  {
u  |  ( A  .Q  B )  <Q  u } >. )  <->  ( A  .Q  B )  <Q  r
)
3130biimpi 118 . . . . . 6  |-  ( r  e.  ( 2nd `  <. { l  |  l  <Q 
( A  .Q  B
) } ,  {
u  |  ( A  .Q  B )  <Q  u } >. )  ->  ( A  .Q  B )  <Q 
r )
32 prloc 7050 . . . . . 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 283 . . . . 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 683 . . . 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 1285 . . 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 113 . 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 3031 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 102    \/ wo 664    e. wcel 1438   {cab 2074   _Vcvv 2619    C_ wss 2999   <.cop 3449   class class class wbr 3845    Or wor 4122   ` cfv 5015  (class class class)co 5652   1stc1st 5909   2ndc2nd 5910   Q.cnq 6839    .Q cmq 6842    <Q cltq 6844   P.cnp 6850    .P. cmp 6853
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-1o 6181  df-2o 6182  df-oadd 6185  df-omul 6186  df-er 6292  df-ec 6294  df-qs 6298  df-ni 6863  df-pli 6864  df-mi 6865  df-lti 6866  df-plpq 6903  df-mpq 6904  df-enq 6906  df-nqqs 6907  df-plqqs 6908  df-mqqs 6909  df-1nqqs 6910  df-rq 6911  df-ltnqqs 6912  df-enq0 6983  df-nq0 6984  df-0nq0 6985  df-plq0 6986  df-mq0 6987  df-inp 7025  df-imp 7028
This theorem is referenced by:  mulnqpr  7136
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