ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  recexprlemss1l Unicode version

Theorem recexprlemss1l 6791
Description: The lower cut of  A  .P.  B is a subset of the lower cut of one. Lemma for recexpr 6794. (Contributed by Jim Kingdon, 27-Dec-2019.)
Hypothesis
Ref Expression
recexpr.1  |-  B  = 
<. { x  |  E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  {
x  |  E. y
( y  <Q  x  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) }
>.
Assertion
Ref Expression
recexprlemss1l  |-  ( A  e.  P.  ->  ( 1st `  ( A  .P.  B ) )  C_  ( 1st `  1P ) )
Distinct variable groups:    x, y, A   
x, B, y

Proof of Theorem recexprlemss1l
Dummy variables  q  z  w  u  f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 recexpr.1 . . . . . 6  |-  B  = 
<. { x  |  E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  {
x  |  E. y
( y  <Q  x  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) }
>.
21recexprlempr 6788 . . . . 5  |-  ( A  e.  P.  ->  B  e.  P. )
3 df-imp 6625 . . . . . 6  |-  .P.  =  ( y  e.  P. ,  w  e.  P.  |->  <. { u  e.  Q.  |  E. f  e.  Q.  E. g  e.  Q.  (
f  e.  ( 1st `  y )  /\  g  e.  ( 1st `  w
)  /\  u  =  ( f  .Q  g
) ) } ,  { u  e.  Q.  |  E. f  e.  Q.  E. g  e.  Q.  (
f  e.  ( 2nd `  y )  /\  g  e.  ( 2nd `  w
)  /\  u  =  ( f  .Q  g
) ) } >. )
4 mulclnq 6532 . . . . . 6  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  .Q  g
)  e.  Q. )
53, 4genpelvl 6668 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( w  e.  ( 1st `  ( A  .P.  B ) )  <->  E. z  e.  ( 1st `  A ) E. q  e.  ( 1st `  B ) w  =  ( z  .Q  q
) ) )
62, 5mpdan 406 . . . 4  |-  ( A  e.  P.  ->  (
w  e.  ( 1st `  ( A  .P.  B
) )  <->  E. z  e.  ( 1st `  A
) E. q  e.  ( 1st `  B
) w  =  ( z  .Q  q ) ) )
71recexprlemell 6778 . . . . . . . 8  |-  ( q  e.  ( 1st `  B
)  <->  E. y ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
8 ltrelnq 6521 . . . . . . . . . . . . . 14  |-  <Q  C_  ( Q.  X.  Q. )
98brel 4420 . . . . . . . . . . . . 13  |-  ( q 
<Q  y  ->  ( q  e.  Q.  /\  y  e.  Q. ) )
109simprd 111 . . . . . . . . . . . 12  |-  ( q 
<Q  y  ->  y  e. 
Q. )
11 prop 6631 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
12 elprnql 6637 . . . . . . . . . . . . . . . . . 18  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
1311, 12sylan 271 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
14 ltmnqi 6559 . . . . . . . . . . . . . . . . . 18  |-  ( ( q  <Q  y  /\  z  e.  Q. )  ->  ( z  .Q  q
)  <Q  ( z  .Q  y ) )
1514expcom 113 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  Q.  ->  (
q  <Q  y  ->  (
z  .Q  q ) 
<Q  ( z  .Q  y
) ) )
1613, 15syl 14 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
( q  <Q  y  ->  ( z  .Q  q
)  <Q  ( z  .Q  y ) ) )
1716adantr 265 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  /\  y  e.  Q. )  ->  ( q  <Q  y  ->  ( z  .Q  q
)  <Q  ( z  .Q  y ) ) )
18 prltlu 6643 . . . . . . . . . . . . . . . . . . 19  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A )  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  z  <Q  ( *Q `  y
) )
1911, 18syl3an1 1179 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A )  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  z  <Q  ( *Q `  y
) )
20193expia 1117 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
( ( *Q `  y )  e.  ( 2nd `  A )  ->  z  <Q  ( *Q `  y ) ) )
2120adantr 265 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  /\  y  e.  Q. )  ->  ( ( *Q `  y )  e.  ( 2nd `  A )  ->  z  <Q  ( *Q `  y ) ) )
22 ltmnqi 6559 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( z  <Q  ( *Q `  y )  /\  y  e.  Q. )  ->  (
y  .Q  z ) 
<Q  ( y  .Q  ( *Q `  y ) ) )
2322expcom 113 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  Q.  ->  (
z  <Q  ( *Q `  y )  ->  (
y  .Q  z ) 
<Q  ( y  .Q  ( *Q `  y ) ) ) )
2423adantr 265 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( z  <Q  ( *Q `  y )  -> 
( y  .Q  z
)  <Q  ( y  .Q  ( *Q `  y
) ) ) )
25 mulcomnqg 6539 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  z
)  =  ( z  .Q  y ) )
26 recidnq 6549 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  e.  Q.  ->  (
y  .Q  ( *Q
`  y ) )  =  1Q )
2726adantr 265 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  ( *Q `  y ) )  =  1Q )
2825, 27breq12d 3805 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( ( y  .Q  z )  <Q  (
y  .Q  ( *Q
`  y ) )  <-> 
( z  .Q  y
)  <Q  1Q ) )
2924, 28sylibd 142 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( z  <Q  ( *Q `  y )  -> 
( z  .Q  y
)  <Q  1Q ) )
3029ancoms 259 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  Q.  /\  y  e.  Q. )  ->  ( z  <Q  ( *Q `  y )  -> 
( z  .Q  y
)  <Q  1Q ) )
3113, 30sylan 271 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  /\  y  e.  Q. )  ->  ( z  <Q  ( *Q `  y )  -> 
( z  .Q  y
)  <Q  1Q ) )
3221, 31syld 44 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  /\  y  e.  Q. )  ->  ( ( *Q `  y )  e.  ( 2nd `  A )  ->  ( z  .Q  y )  <Q  1Q ) )
3317, 32anim12d 322 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  /\  y  e.  Q. )  ->  ( ( q  <Q 
y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  ( (
z  .Q  q ) 
<Q  ( z  .Q  y
)  /\  ( z  .Q  y )  <Q  1Q ) ) )
34 ltsonq 6554 . . . . . . . . . . . . . . 15  |-  <Q  Or  Q.
3534, 8sotri 4748 . . . . . . . . . . . . . 14  |-  ( ( ( z  .Q  q
)  <Q  ( z  .Q  y )  /\  (
z  .Q  y ) 
<Q  1Q )  ->  (
z  .Q  q ) 
<Q  1Q )
3633, 35syl6 33 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  /\  y  e.  Q. )  ->  ( ( q  <Q 
y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  ( z  .Q  q )  <Q  1Q ) )
3736exp4b 353 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
( y  e.  Q.  ->  ( q  <Q  y  ->  ( ( *Q `  y )  e.  ( 2nd `  A )  ->  ( z  .Q  q )  <Q  1Q ) ) ) )
3810, 37syl5 32 . . . . . . . . . . 11  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
( q  <Q  y  ->  ( q  <Q  y  ->  ( ( *Q `  y )  e.  ( 2nd `  A )  ->  ( z  .Q  q )  <Q  1Q ) ) ) )
3938pm2.43d 48 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
( q  <Q  y  ->  ( ( *Q `  y )  e.  ( 2nd `  A )  ->  ( z  .Q  q )  <Q  1Q ) ) )
4039impd 246 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
( ( q  <Q 
y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  ( z  .Q  q )  <Q  1Q ) )
4140exlimdv 1716 . . . . . . . 8  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
( E. y ( q  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  (
z  .Q  q ) 
<Q  1Q ) )
427, 41syl5bi 145 . . . . . . 7  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
( q  e.  ( 1st `  B )  ->  ( z  .Q  q )  <Q  1Q ) )
43 breq1 3795 . . . . . . . 8  |-  ( w  =  ( z  .Q  q )  ->  (
w  <Q  1Q  <->  ( z  .Q  q )  <Q  1Q ) )
4443biimprcd 153 . . . . . . 7  |-  ( ( z  .Q  q ) 
<Q  1Q  ->  ( w  =  ( z  .Q  q )  ->  w  <Q  1Q ) )
4542, 44syl6 33 . . . . . 6  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
( q  e.  ( 1st `  B )  ->  ( w  =  ( z  .Q  q
)  ->  w  <Q  1Q ) ) )
4645expimpd 349 . . . . 5  |-  ( A  e.  P.  ->  (
( z  e.  ( 1st `  A )  /\  q  e.  ( 1st `  B ) )  ->  ( w  =  ( z  .Q  q )  ->  w  <Q  1Q ) ) )
4746rexlimdvv 2456 . . . 4  |-  ( A  e.  P.  ->  ( E. z  e.  ( 1st `  A ) E. q  e.  ( 1st `  B ) w  =  ( z  .Q  q
)  ->  w  <Q  1Q ) )
486, 47sylbid 143 . . 3  |-  ( A  e.  P.  ->  (
w  e.  ( 1st `  ( A  .P.  B
) )  ->  w  <Q  1Q ) )
49 1prl 6711 . . . 4  |-  ( 1st `  1P )  =  {
w  |  w  <Q  1Q }
5049abeq2i 2164 . . 3  |-  ( w  e.  ( 1st `  1P ) 
<->  w  <Q  1Q )
5148, 50syl6ibr 155 . 2  |-  ( A  e.  P.  ->  (
w  e.  ( 1st `  ( A  .P.  B
) )  ->  w  e.  ( 1st `  1P ) ) )
5251ssrdv 2979 1  |-  ( A  e.  P.  ->  ( 1st `  ( A  .P.  B ) )  C_  ( 1st `  1P ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259   E.wex 1397    e. wcel 1409   {cab 2042   E.wrex 2324    C_ wss 2945   <.cop 3406   class class class wbr 3792   ` cfv 4930  (class class class)co 5540   1stc1st 5793   2ndc2nd 5794   Q.cnq 6436   1Qc1q 6437    .Q cmq 6439   *Qcrq 6440    <Q cltq 6441   P.cnp 6447   1Pc1p 6448    .P. cmp 6450
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 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
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 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-inp 6622  df-i1p 6623  df-imp 6625
This theorem is referenced by:  recexprlemex  6793
  Copyright terms: Public domain W3C validator