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Theorem mulnqprlemfl 6730
Description: Lemma for mulnqpr 6732. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
Assertion
Ref Expression
mulnqprlemfl  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( 1st `  <. { l  |  l  <Q 
( A  .Q  B
) } ,  {
u  |  ( A  .Q  B )  <Q  u } >. )  C_  ( 1st `  ( <. { 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 mulnqprlemfl
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 mulnqprlemru 6729 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
)  C_  ( 2nd ` 
<. { l  |  l 
<Q  ( A  .Q  B
) } ,  {
u  |  ( A  .Q  B )  <Q  u } >. ) )
2 ltsonq 6553 . . . . . . . . 9  |-  <Q  Or  Q.
3 mulclnq 6531 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  .Q  B
)  e.  Q. )
4 sonr 4081 . . . . . . . . 9  |-  ( ( 
<Q  Or  Q.  /\  ( A  .Q  B )  e. 
Q. )  ->  -.  ( A  .Q  B
)  <Q  ( A  .Q  B ) )
52, 3, 4sylancr 399 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  -.  ( A  .Q  B )  <Q  ( A  .Q  B ) )
6 ltrelnq 6520 . . . . . . . . . . . 12  |-  <Q  C_  ( Q.  X.  Q. )
76brel 4419 . . . . . . . . . . 11  |-  ( ( A  .Q  B ) 
<Q  ( A  .Q  B
)  ->  ( ( A  .Q  B )  e. 
Q.  /\  ( A  .Q  B )  e.  Q. ) )
87simpld 109 . . . . . . . . . 10  |-  ( ( A  .Q  B ) 
<Q  ( A  .Q  B
)  ->  ( A  .Q  B )  e.  Q. )
9 elex 2583 . . . . . . . . . 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 breq2 3795 . . . . . . . . 9  |-  ( u  =  ( A  .Q  B )  ->  (
( A  .Q  B
)  <Q  u  <->  ( A  .Q  B )  <Q  ( A  .Q  B ) ) )
1210, 11elab3 2716 . . . . . . . 8  |-  ( ( A  .Q  B )  e.  { u  |  ( A  .Q  B
)  <Q  u }  <->  ( A  .Q  B )  <Q  ( A  .Q  B ) )
135, 12sylnibr 612 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  -.  ( A  .Q  B )  e.  {
u  |  ( A  .Q  B )  <Q  u } )
14 ltnqex 6704 . . . . . . . . 9  |-  { l  |  l  <Q  ( A  .Q  B ) }  e.  _V
15 gtnqex 6705 . . . . . . . . 9  |-  { u  |  ( A  .Q  B )  <Q  u }  e.  _V
1614, 15op2nd 5801 . . . . . . . 8  |-  ( 2nd `  <. { l  |  l  <Q  ( A  .Q  B ) } ,  { u  |  ( A  .Q  B )  <Q  u } >. )  =  {
u  |  ( A  .Q  B )  <Q  u }
1716eleq2i 2120 . . . . . . 7  |-  ( ( A  .Q  B )  e.  ( 2nd `  <. { l  |  l  <Q 
( A  .Q  B
) } ,  {
u  |  ( A  .Q  B )  <Q  u } >. )  <->  ( A  .Q  B )  e.  {
u  |  ( A  .Q  B )  <Q  u } )
1813, 17sylnibr 612 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  -.  ( A  .Q  B )  e.  ( 2nd `  <. { l  |  l  <Q  ( A  .Q  B ) } ,  { u  |  ( A  .Q  B
)  <Q  u } >. ) )
191, 18ssneldd 2975 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  -.  ( A  .Q  B )  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )
2019adantr 265 . . . 4  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  <. { l  |  l  <Q  ( A  .Q  B ) } ,  { u  |  ( A  .Q  B )  <Q  u } >. ) )  ->  -.  ( A  .Q  B
)  e.  ( 2nd `  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) ) )
21 nqprlu 6702 . . . . . . 7  |-  ( A  e.  Q.  ->  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  e.  P. )
22 nqprlu 6702 . . . . . . 7  |-  ( B  e.  Q.  ->  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >.  e.  P. )
23 mulclpr 6727 . . . . . . 7  |-  ( (
<. { 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 277 . . . . . 6  |-  ( ( 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 6630 . . . . . 6  |-  ( (
<. { 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 . . . . 5  |-  ( ( 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 2577 . . . . . . 7  |-  r  e. 
_V
28 breq1 3794 . . . . . . 7  |-  ( l  =  r  ->  (
l  <Q  ( A  .Q  B )  <->  r  <Q  ( A  .Q  B ) ) )
2914, 15op1st 5800 . . . . . . 7  |-  ( 1st `  <. { l  |  l  <Q  ( A  .Q  B ) } ,  { u  |  ( A  .Q  B )  <Q  u } >. )  =  {
l  |  l  <Q 
( A  .Q  B
) }
3027, 28, 29elab2 2712 . . . . . 6  |-  ( r  e.  ( 1st `  <. { l  |  l  <Q 
( A  .Q  B
) } ,  {
u  |  ( A  .Q  B )  <Q  u } >. )  <->  r  <Q  ( A  .Q  B ) )
3130biimpi 117 . . . . 5  |-  ( r  e.  ( 1st `  <. { l  |  l  <Q 
( A  .Q  B
) } ,  {
u  |  ( A  .Q  B )  <Q  u } >. )  ->  r  <Q  ( A  .Q  B
) )
32 prloc 6646 . . . . 5  |-  ( (
<. ( 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.  /\  r  <Q  ( A  .Q  B ) )  -> 
( r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
)  \/  ( A  .Q  B )  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) ) )
3326, 31, 32syl2an 277 . . . 4  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  <. { l  |  l  <Q  ( A  .Q  B ) } ,  { u  |  ( A  .Q  B )  <Q  u } >. ) )  -> 
( r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
)  \/  ( A  .Q  B )  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) ) )
3420, 33ecased 1255 . . 3  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  <. { l  |  l  <Q  ( A  .Q  B ) } ,  { u  |  ( A  .Q  B )  <Q  u } >. ) )  -> 
r  e.  ( 1st `  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) ) )
3534ex 112 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( r  e.  ( 1st `  <. { l  |  l  <Q  ( A  .Q  B ) } ,  { u  |  ( A  .Q  B
)  <Q  u } >. )  ->  r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) ) )
3635ssrdv 2978 1  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( 1st `  <. { l  |  l  <Q 
( A  .Q  B
) } ,  {
u  |  ( A  .Q  B )  <Q  u } >. )  C_  ( 1st `  ( <. { 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 101    \/ wo 639    e. wcel 1409   {cab 2042   _Vcvv 2574    C_ wss 2944   <.cop 3405   class class class wbr 3791    Or wor 4059   ` cfv 4929  (class class class)co 5539   1stc1st 5792   2ndc2nd 5793   Q.cnq 6435    .Q cmq 6438    <Q cltq 6440   P.cnp 6446    .P. cmp 6449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-2o 6032  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-enq0 6579  df-nq0 6580  df-0nq0 6581  df-plq0 6582  df-mq0 6583  df-inp 6621  df-imp 6624
This theorem is referenced by:  mulnqpr  6732
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