ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mulnqprlemrl Unicode version

Theorem mulnqprlemrl 6902
Description: Lemma for mulnqpr 6906. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
Assertion
Ref Expression
mulnqprlemrl  |-  ( ( 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 } >. ) )
Distinct variable groups:    A, l, u    B, l, u

Proof of Theorem mulnqprlemrl
Dummy variables  f  g  h  r  s  t  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nqprlu 6876 . . . . . 6  |-  ( A  e.  Q.  ->  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  e.  P. )
2 nqprlu 6876 . . . . . 6  |-  ( B  e.  Q.  ->  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >.  e.  P. )
3 df-imp 6798 . . . . . . 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 6705 . . . . . . 7  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  .Q  h
)  e.  Q. )
53, 4genpelvl 6841 . . . . . 6  |-  ( (
<. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  e.  P.  /\ 
<. { l  |  l 
<Q  B } ,  {
u  |  B  <Q  u } >.  e.  P. )  ->  ( r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
)  <->  E. s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. ) E. t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
r  =  ( s  .Q  t ) ) )
61, 2, 5syl2an 283 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
)  <->  E. s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. ) E. t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
r  =  ( s  .Q  t ) ) )
76biimpa 290 . . . 4  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) ) )  ->  E. s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. ) E. t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
r  =  ( s  .Q  t ) )
8 vex 2614 . . . . . . . . . . . . 13  |-  s  e. 
_V
9 breq1 3809 . . . . . . . . . . . . 13  |-  ( l  =  s  ->  (
l  <Q  A  <->  s  <Q  A ) )
10 ltnqex 6878 . . . . . . . . . . . . . 14  |-  { l  |  l  <Q  A }  e.  _V
11 gtnqex 6879 . . . . . . . . . . . . . 14  |-  { u  |  A  <Q  u }  e.  _V
1210, 11op1st 5826 . . . . . . . . . . . . 13  |-  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  =  { l  |  l 
<Q  A }
138, 9, 12elab2 2750 . . . . . . . . . . . 12  |-  ( s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  <->  s  <Q  A )
1413biimpi 118 . . . . . . . . . . 11  |-  ( s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  ->  s  <Q  A )
1514ad2antrl 474 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  ->  s  <Q  A )
1615adantr 270 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  .Q  t ) )  -> 
s  <Q  A )
17 vex 2614 . . . . . . . . . . . . 13  |-  t  e. 
_V
18 breq1 3809 . . . . . . . . . . . . 13  |-  ( l  =  t  ->  (
l  <Q  B  <->  t  <Q  B ) )
19 ltnqex 6878 . . . . . . . . . . . . . 14  |-  { l  |  l  <Q  B }  e.  _V
20 gtnqex 6879 . . . . . . . . . . . . . 14  |-  { u  |  B  <Q  u }  e.  _V
2119, 20op1st 5826 . . . . . . . . . . . . 13  |-  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )  =  { l  |  l 
<Q  B }
2217, 18, 21elab2 2750 . . . . . . . . . . . 12  |-  ( t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )  <->  t  <Q  B )
2322biimpi 118 . . . . . . . . . . 11  |-  ( t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )  ->  t  <Q  B )
2423ad2antll 475 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  ->  t  <Q  B )
2524adantr 270 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  .Q  t ) )  -> 
t  <Q  B )
26 ltrelnq 6694 . . . . . . . . . . . 12  |-  <Q  C_  ( Q.  X.  Q. )
2726brel 4439 . . . . . . . . . . 11  |-  ( s 
<Q  A  ->  ( s  e.  Q.  /\  A  e.  Q. ) )
2816, 27syl 14 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  .Q  t ) )  -> 
( s  e.  Q.  /\  A  e.  Q. )
)
2926brel 4439 . . . . . . . . . . 11  |-  ( t 
<Q  B  ->  ( t  e.  Q.  /\  B  e.  Q. ) )
3025, 29syl 14 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  .Q  t ) )  -> 
( t  e.  Q.  /\  B  e.  Q. )
)
31 lt2mulnq 6734 . . . . . . . . . 10  |-  ( ( ( s  e.  Q.  /\  A  e.  Q. )  /\  ( t  e.  Q.  /\  B  e.  Q. )
)  ->  ( (
s  <Q  A  /\  t  <Q  B )  ->  (
s  .Q  t ) 
<Q  ( A  .Q  B
) ) )
3228, 30, 31syl2anc 403 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  .Q  t ) )  -> 
( ( s  <Q  A  /\  t  <Q  B )  ->  ( s  .Q  t )  <Q  ( A  .Q  B ) ) )
3316, 25, 32mp2and 424 . . . . . . . 8  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  .Q  t ) )  -> 
( s  .Q  t
)  <Q  ( A  .Q  B ) )
34 breq1 3809 . . . . . . . . 9  |-  ( r  =  ( s  .Q  t )  ->  (
r  <Q  ( A  .Q  B )  <->  ( s  .Q  t )  <Q  ( A  .Q  B ) ) )
3534adantl 271 . . . . . . . 8  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  .Q  t ) )  -> 
( r  <Q  ( A  .Q  B )  <->  ( s  .Q  t )  <Q  ( A  .Q  B ) ) )
3633, 35mpbird 165 . . . . . . 7  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  .Q  t ) )  -> 
r  <Q  ( A  .Q  B ) )
37 vex 2614 . . . . . . . 8  |-  r  e. 
_V
38 breq1 3809 . . . . . . . 8  |-  ( l  =  r  ->  (
l  <Q  ( A  .Q  B )  <->  r  <Q  ( A  .Q  B ) ) )
39 ltnqex 6878 . . . . . . . . 9  |-  { l  |  l  <Q  ( A  .Q  B ) }  e.  _V
40 gtnqex 6879 . . . . . . . . 9  |-  { u  |  ( A  .Q  B )  <Q  u }  e.  _V
4139, 40op1st 5826 . . . . . . . 8  |-  ( 1st `  <. { l  |  l  <Q  ( A  .Q  B ) } ,  { u  |  ( A  .Q  B )  <Q  u } >. )  =  {
l  |  l  <Q 
( A  .Q  B
) }
4237, 38, 41elab2 2750 . . . . . . 7  |-  ( r  e.  ( 1st `  <. { l  |  l  <Q 
( A  .Q  B
) } ,  {
u  |  ( A  .Q  B )  <Q  u } >. )  <->  r  <Q  ( A  .Q  B ) )
4336, 42sylibr 132 . . . . . 6  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  .Q  t ) )  -> 
r  e.  ( 1st `  <. { l  |  l  <Q  ( A  .Q  B ) } ,  { u  |  ( A  .Q  B )  <Q  u } >. ) )
4443ex 113 . . . . 5  |-  ( ( ( ( A  e. 
Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  ->  (
r  =  ( s  .Q  t )  -> 
r  e.  ( 1st `  <. { l  |  l  <Q  ( A  .Q  B ) } ,  { u  |  ( A  .Q  B )  <Q  u } >. ) ) )
4544rexlimdvva 2490 . . . 4  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) ) )  -> 
( E. s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. ) E. t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) r  =  ( s  .Q  t )  ->  r  e.  ( 1st `  <. { l  |  l  <Q  ( A  .Q  B ) } ,  { u  |  ( A  .Q  B
)  <Q  u } >. ) ) )
467, 45mpd 13 . . 3  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) ) )  -> 
r  e.  ( 1st `  <. { l  |  l  <Q  ( A  .Q  B ) } ,  { u  |  ( A  .Q  B )  <Q  u } >. ) )
4746ex 113 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
)  ->  r  e.  ( 1st `  <. { l  |  l  <Q  ( A  .Q  B ) } ,  { u  |  ( A  .Q  B
)  <Q  u } >. ) ) )
4847ssrdv 3015 1  |-  ( ( 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 } >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   {cab 2069   E.wrex 2354    C_ wss 2983   <.cop 3420   class class class wbr 3806   ` cfv 4953  (class class class)co 5565   1stc1st 5818   Q.cnq 6609    .Q cmq 6612    <Q cltq 6614   P.cnp 6620    .P. cmp 6623
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3914  ax-sep 3917  ax-nul 3925  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-iinf 4358
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2613  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-tr 3897  df-eprel 4073  df-id 4077  df-po 4080  df-iso 4081  df-iord 4150  df-on 4152  df-suc 4155  df-iom 4361  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-ov 5568  df-oprab 5569  df-mpt2 5570  df-1st 5820  df-2nd 5821  df-recs 5976  df-irdg 6041  df-1o 6087  df-oadd 6091  df-omul 6092  df-er 6195  df-ec 6197  df-qs 6201  df-ni 6633  df-pli 6634  df-mi 6635  df-lti 6636  df-plpq 6673  df-mpq 6674  df-enq 6676  df-nqqs 6677  df-plqqs 6678  df-mqqs 6679  df-1nqqs 6680  df-rq 6681  df-ltnqqs 6682  df-inp 6795  df-imp 6798
This theorem is referenced by:  mulnqprlemfu  6905  mulnqpr  6906
  Copyright terms: Public domain W3C validator