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Theorem mulnqprlemru 6672
Description: Lemma for mulnqpr 6675. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
Assertion
Ref Expression
mulnqprlemru  |-  ( ( 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 } >. ) )
Distinct variable groups:    A, l, u    B, l, u

Proof of Theorem mulnqprlemru
Dummy variables  f  g  h  r  s  t  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nqprlu 6645 . . . . . 6  |-  ( A  e.  Q.  ->  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  e.  P. )
2 nqprlu 6645 . . . . . 6  |-  ( B  e.  Q.  ->  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >.  e.  P. )
3 df-imp 6567 . . . . . . 7  |-  .P.  =  ( x  e.  P. ,  y  e.  P.  |->  <. { f  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 1st `  x )  /\  h  e.  ( 1st `  y
)  /\  f  =  ( g  .Q  h
) ) } ,  { f  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 2nd `  x )  /\  h  e.  ( 2nd `  y
)  /\  f  =  ( g  .Q  h
) ) } >. )
4 mulclnq 6474 . . . . . . 7  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  .Q  h
)  e.  Q. )
53, 4genpelvu 6611 . . . . . 6  |-  ( (
<. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  e.  P.  /\ 
<. { l  |  l 
<Q  B } ,  {
u  |  B  <Q  u } >.  e.  P. )  ->  ( r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
)  <->  E. s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. ) E. t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
r  =  ( s  .Q  t ) ) )
61, 2, 5syl2an 273 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
)  <->  E. s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. ) E. t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
r  =  ( s  .Q  t ) ) )
76biimpa 280 . . . 4  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) ) )  ->  E. s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. ) E. t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
r  =  ( s  .Q  t ) )
8 vex 2560 . . . . . . . . . . . . 13  |-  s  e. 
_V
9 breq2 3768 . . . . . . . . . . . . 13  |-  ( u  =  s  ->  ( A  <Q  u  <->  A  <Q  s ) )
10 ltnqex 6647 . . . . . . . . . . . . . 14  |-  { l  |  l  <Q  A }  e.  _V
11 gtnqex 6648 . . . . . . . . . . . . . 14  |-  { u  |  A  <Q  u }  e.  _V
1210, 11op2nd 5774 . . . . . . . . . . . . 13  |-  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  =  { u  |  A  <Q  u }
138, 9, 12elab2 2690 . . . . . . . . . . . 12  |-  ( s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  <->  A  <Q  s )
1413biimpi 113 . . . . . . . . . . 11  |-  ( s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  ->  A  <Q  s )
1514ad2antrl 459 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  ->  A  <Q  s )
1615adantr 261 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  .Q  t ) )  ->  A  <Q  s )
17 vex 2560 . . . . . . . . . . . . 13  |-  t  e. 
_V
18 breq2 3768 . . . . . . . . . . . . 13  |-  ( u  =  t  ->  ( B  <Q  u  <->  B  <Q  t ) )
19 ltnqex 6647 . . . . . . . . . . . . . 14  |-  { l  |  l  <Q  B }  e.  _V
20 gtnqex 6648 . . . . . . . . . . . . . 14  |-  { u  |  B  <Q  u }  e.  _V
2119, 20op2nd 5774 . . . . . . . . . . . . 13  |-  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )  =  { u  |  B  <Q  u }
2217, 18, 21elab2 2690 . . . . . . . . . . . 12  |-  ( t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )  <->  B  <Q  t )
2322biimpi 113 . . . . . . . . . . 11  |-  ( t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )  ->  B  <Q  t )
2423ad2antll 460 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  ->  B  <Q  t )
2524adantr 261 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  .Q  t ) )  ->  B  <Q  t )
26 ltrelnq 6463 . . . . . . . . . . . 12  |-  <Q  C_  ( Q.  X.  Q. )
2726brel 4392 . . . . . . . . . . 11  |-  ( A 
<Q  s  ->  ( A  e.  Q.  /\  s  e.  Q. ) )
2816, 27syl 14 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  .Q  t ) )  -> 
( A  e.  Q.  /\  s  e.  Q. )
)
2926brel 4392 . . . . . . . . . . 11  |-  ( B 
<Q  t  ->  ( B  e.  Q.  /\  t  e.  Q. ) )
3025, 29syl 14 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  .Q  t ) )  -> 
( B  e.  Q.  /\  t  e.  Q. )
)
31 lt2mulnq 6503 . . . . . . . . . 10  |-  ( ( ( A  e.  Q.  /\  s  e.  Q. )  /\  ( B  e.  Q.  /\  t  e.  Q. )
)  ->  ( ( A  <Q  s  /\  B  <Q  t )  ->  ( A  .Q  B )  <Q 
( s  .Q  t
) ) )
3228, 30, 31syl2anc 391 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  .Q  t ) )  -> 
( ( A  <Q  s  /\  B  <Q  t
)  ->  ( A  .Q  B )  <Q  (
s  .Q  t ) ) )
3316, 25, 32mp2and 409 . . . . . . . 8  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  .Q  t ) )  -> 
( A  .Q  B
)  <Q  ( s  .Q  t ) )
34 breq2 3768 . . . . . . . . 9  |-  ( r  =  ( s  .Q  t )  ->  (
( A  .Q  B
)  <Q  r  <->  ( A  .Q  B )  <Q  (
s  .Q  t ) ) )
3534adantl 262 . . . . . . . 8  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  .Q  t ) )  -> 
( ( A  .Q  B )  <Q  r  <->  ( A  .Q  B ) 
<Q  ( s  .Q  t
) ) )
3633, 35mpbird 156 . . . . . . 7  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  .Q  t ) )  -> 
( A  .Q  B
)  <Q  r )
37 vex 2560 . . . . . . . 8  |-  r  e. 
_V
38 breq2 3768 . . . . . . . 8  |-  ( u  =  r  ->  (
( A  .Q  B
)  <Q  u  <->  ( A  .Q  B )  <Q  r
) )
39 ltnqex 6647 . . . . . . . . 9  |-  { l  |  l  <Q  ( A  .Q  B ) }  e.  _V
40 gtnqex 6648 . . . . . . . . 9  |-  { u  |  ( A  .Q  B )  <Q  u }  e.  _V
4139, 40op2nd 5774 . . . . . . . 8  |-  ( 2nd `  <. { l  |  l  <Q  ( A  .Q  B ) } ,  { u  |  ( A  .Q  B )  <Q  u } >. )  =  {
u  |  ( A  .Q  B )  <Q  u }
4237, 38, 41elab2 2690 . . . . . . 7  |-  ( r  e.  ( 2nd `  <. { l  |  l  <Q 
( A  .Q  B
) } ,  {
u  |  ( A  .Q  B )  <Q  u } >. )  <->  ( A  .Q  B )  <Q  r
)
4336, 42sylibr 137 . . . . . 6  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  .Q  t ) )  -> 
r  e.  ( 2nd `  <. { l  |  l  <Q  ( A  .Q  B ) } ,  { u  |  ( A  .Q  B )  <Q  u } >. ) )
4443ex 108 . . . . 5  |-  ( ( ( ( A  e. 
Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  ->  (
r  =  ( s  .Q  t )  -> 
r  e.  ( 2nd `  <. { l  |  l  <Q  ( A  .Q  B ) } ,  { u  |  ( A  .Q  B )  <Q  u } >. ) ) )
4544rexlimdvva 2440 . . . 4  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) ) )  -> 
( E. s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. ) E. t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) r  =  ( s  .Q  t )  ->  r  e.  ( 2nd `  <. { l  |  l  <Q  ( A  .Q  B ) } ,  { u  |  ( A  .Q  B
)  <Q  u } >. ) ) )
467, 45mpd 13 . . 3  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) ) )  -> 
r  e.  ( 2nd `  <. { l  |  l  <Q  ( A  .Q  B ) } ,  { u  |  ( A  .Q  B )  <Q  u } >. ) )
4746ex 108 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
)  ->  r  e.  ( 2nd `  <. { l  |  l  <Q  ( A  .Q  B ) } ,  { u  |  ( A  .Q  B
)  <Q  u } >. ) ) )
4847ssrdv 2951 1  |-  ( ( 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 } >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243    e. wcel 1393   {cab 2026   E.wrex 2307    C_ wss 2917   <.cop 3378   class class class wbr 3764   ` cfv 4902  (class class class)co 5512   2ndc2nd 5766   Q.cnq 6378    .Q cmq 6381    <Q cltq 6383   P.cnp 6389    .P. cmp 6392
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-inp 6564  df-imp 6567
This theorem is referenced by:  mulnqprlemfl  6673  mulnqpr  6675
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