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Theorem mulnqprlemfl 7594
Description: Lemma for mulnqpr 7596. 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 7593 . . . . . 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 7417 . . . . . . . . 9  |-  <Q  Or  Q.
3 mulclnq 7395 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  .Q  B
)  e.  Q. )
4 sonr 4332 . . . . . . . . 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 7384 . . . . . . . . . . . 12  |-  <Q  C_  ( Q.  X.  Q. )
76brel 4693 . . . . . . . . . . 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 2763 . . . . . . . . . 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 4022 . . . . . . . . 9  |-  ( u  =  ( A  .Q  B )  ->  (
( A  .Q  B
)  <Q  u  <->  ( A  .Q  B )  <Q  ( A  .Q  B ) ) )
1210, 11elab3 2904 . . . . . . . 8  |-  ( ( A  .Q  B )  e.  { u  |  ( A  .Q  B
)  <Q  u }  <->  ( A  .Q  B )  <Q  ( A  .Q  B ) )
135, 12sylnibr 678 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  -.  ( A  .Q  B )  e.  {
u  |  ( A  .Q  B )  <Q  u } )
14 ltnqex 7568 . . . . . . . . 9  |-  { l  |  l  <Q  ( A  .Q  B ) }  e.  _V
15 gtnqex 7569 . . . . . . . . 9  |-  { u  |  ( A  .Q  B )  <Q  u }  e.  _V
1614, 15op2nd 6167 . . . . . . . 8  |-  ( 2nd `  <. { l  |  l  <Q  ( A  .Q  B ) } ,  { u  |  ( A  .Q  B )  <Q  u } >. )  =  {
u  |  ( A  .Q  B )  <Q  u }
1716eleq2i 2256 . . . . . . 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 678 . . . . . 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 3173 . . . . 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 276 . . . 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 7566 . . . . . . 7  |-  ( A  e.  Q.  ->  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  e.  P. )
22 nqprlu 7566 . . . . . . 7  |-  ( B  e.  Q.  ->  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >.  e.  P. )
23 mulclpr 7591 . . . . . . 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 289 . . . . . 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 7494 . . . . . 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 2755 . . . . . . 7  |-  r  e. 
_V
28 breq1 4021 . . . . . . 7  |-  ( l  =  r  ->  (
l  <Q  ( A  .Q  B )  <->  r  <Q  ( A  .Q  B ) ) )
2914, 15op1st 6166 . . . . . . 7  |-  ( 1st `  <. { l  |  l  <Q  ( A  .Q  B ) } ,  { u  |  ( A  .Q  B )  <Q  u } >. )  =  {
l  |  l  <Q 
( A  .Q  B
) }
3027, 28, 29elab2 2900 . . . . . 6  |-  ( r  e.  ( 1st `  <. { l  |  l  <Q 
( A  .Q  B
) } ,  {
u  |  ( A  .Q  B )  <Q  u } >. )  <->  r  <Q  ( A  .Q  B ) )
3130biimpi 120 . . . . 5  |-  ( r  e.  ( 1st `  <. { l  |  l  <Q 
( A  .Q  B
) } ,  {
u  |  ( A  .Q  B )  <Q  u } >. )  ->  r  <Q  ( A  .Q  B
) )
32 prloc 7510 . . . . 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 289 . . . 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 1360 . . 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 115 . 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 3176 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 104    \/ wo 709    e. wcel 2160   {cab 2175   _Vcvv 2752    C_ wss 3144   <.cop 3610   class class class wbr 4018    Or wor 4310   ` cfv 5232  (class class class)co 5892   1stc1st 6158   2ndc2nd 6159   Q.cnq 7299    .Q cmq 7302    <Q cltq 7304   P.cnp 7310    .P. cmp 7313
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4304  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-recs 6325  df-irdg 6390  df-1o 6436  df-2o 6437  df-oadd 6440  df-omul 6441  df-er 6554  df-ec 6556  df-qs 6560  df-ni 7323  df-pli 7324  df-mi 7325  df-lti 7326  df-plpq 7363  df-mpq 7364  df-enq 7366  df-nqqs 7367  df-plqqs 7368  df-mqqs 7369  df-1nqqs 7370  df-rq 7371  df-ltnqqs 7372  df-enq0 7443  df-nq0 7444  df-0nq0 7445  df-plq0 7446  df-mq0 7447  df-inp 7485  df-imp 7488
This theorem is referenced by:  mulnqpr  7596
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