ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  recexprlemss1u Unicode version

Theorem recexprlemss1u 7634
Description: The upper cut of  A  .P.  B is a subset of the upper cut of one. Lemma for recexpr 7636. (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
recexprlemss1u  |-  ( A  e.  P.  ->  ( 2nd `  ( A  .P.  B ) )  C_  ( 2nd `  1P ) )
Distinct variable groups:    x, y, A   
x, B, y

Proof of Theorem recexprlemss1u
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 7630 . . . . 5  |-  ( A  e.  P.  ->  B  e.  P. )
3 df-imp 7467 . . . . . 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 7374 . . . . . 6  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  .Q  g
)  e.  Q. )
53, 4genpelvu 7511 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( w  e.  ( 2nd `  ( A  .P.  B ) )  <->  E. z  e.  ( 2nd `  A ) E. q  e.  ( 2nd `  B ) w  =  ( z  .Q  q
) ) )
62, 5mpdan 421 . . . 4  |-  ( A  e.  P.  ->  (
w  e.  ( 2nd `  ( A  .P.  B
) )  <->  E. z  e.  ( 2nd `  A
) E. q  e.  ( 2nd `  B
) w  =  ( z  .Q  q ) ) )
71recexprlemelu 7621 . . . . . . . 8  |-  ( q  e.  ( 2nd `  B
)  <->  E. y ( y 
<Q  q  /\  ( *Q `  y )  e.  ( 1st `  A
) ) )
8 ltrelnq 7363 . . . . . . . . . . . . . 14  |-  <Q  C_  ( Q.  X.  Q. )
98brel 4678 . . . . . . . . . . . . 13  |-  ( y 
<Q  q  ->  ( y  e.  Q.  /\  q  e.  Q. ) )
109simpld 112 . . . . . . . . . . . 12  |-  ( y 
<Q  q  ->  y  e. 
Q. )
11 prop 7473 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
12 elprnqu 7480 . . . . . . . . . . . . . . . . . 18  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
z  e.  Q. )
1311, 12sylan 283 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
z  e.  Q. )
14 ltmnqi 7401 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  <Q  q  /\  z  e.  Q. )  ->  ( z  .Q  y
)  <Q  ( z  .Q  q ) )
1514expcom 116 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  Q.  ->  (
y  <Q  q  ->  (
z  .Q  y ) 
<Q  ( z  .Q  q
) ) )
1613, 15syl 14 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
( y  <Q  q  ->  ( z  .Q  y
)  <Q  ( z  .Q  q ) ) )
1716adantr 276 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  /\  y  e.  Q. )  ->  ( y  <Q  q  ->  ( z  .Q  y
)  <Q  ( z  .Q  q ) ) )
18 prltlu 7485 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  ( *Q `  y
)  e.  ( 1st `  A )  /\  z  e.  ( 2nd `  A
) )  ->  ( *Q `  y )  <Q 
z )
1911, 18syl3an1 1271 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  P.  /\  ( *Q `  y )  e.  ( 1st `  A
)  /\  z  e.  ( 2nd `  A ) )  ->  ( *Q `  y )  <Q  z
)
20193com23 1209 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  P.  /\  z  e.  ( 2nd `  A )  /\  ( *Q `  y )  e.  ( 1st `  A
) )  ->  ( *Q `  y )  <Q 
z )
21203expia 1205 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
( ( *Q `  y )  e.  ( 1st `  A )  ->  ( *Q `  y )  <Q  z
) )
2221adantr 276 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  /\  y  e.  Q. )  ->  ( ( *Q `  y )  e.  ( 1st `  A )  ->  ( *Q `  y )  <Q  z
) )
23 ltmnqi 7401 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( *Q `  y
)  <Q  z  /\  y  e.  Q. )  ->  (
y  .Q  ( *Q
`  y ) ) 
<Q  ( y  .Q  z
) )
2423expcom 116 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  Q.  ->  (
( *Q `  y
)  <Q  z  ->  (
y  .Q  ( *Q
`  y ) ) 
<Q  ( y  .Q  z
) ) )
2524adantr 276 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( ( *Q `  y )  <Q  z  ->  ( y  .Q  ( *Q `  y ) ) 
<Q  ( y  .Q  z
) ) )
26 recidnq 7391 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  e.  Q.  ->  (
y  .Q  ( *Q
`  y ) )  =  1Q )
2726adantr 276 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  ( *Q `  y ) )  =  1Q )
28 mulcomnqg 7381 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  z
)  =  ( z  .Q  y ) )
2927, 28breq12d 4016 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( ( y  .Q  ( *Q `  y
) )  <Q  (
y  .Q  z )  <-> 
1Q  <Q  ( z  .Q  y ) ) )
3025, 29sylibd 149 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( ( *Q `  y )  <Q  z  ->  1Q  <Q  ( z  .Q  y ) ) )
3130ancoms 268 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  Q.  /\  y  e.  Q. )  ->  ( ( *Q `  y )  <Q  z  ->  1Q  <Q  ( z  .Q  y ) ) )
3213, 31sylan 283 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  /\  y  e.  Q. )  ->  ( ( *Q `  y )  <Q  z  ->  1Q  <Q  ( z  .Q  y ) ) )
3322, 32syld 45 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  /\  y  e.  Q. )  ->  ( ( *Q `  y )  e.  ( 1st `  A )  ->  1Q  <Q  (
z  .Q  y ) ) )
3417, 33anim12d 335 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  /\  y  e.  Q. )  ->  ( ( y  <Q 
q  /\  ( *Q `  y )  e.  ( 1st `  A ) )  ->  ( (
z  .Q  y ) 
<Q  ( z  .Q  q
)  /\  1Q  <Q  ( z  .Q  y ) ) ) )
35 ltsonq 7396 . . . . . . . . . . . . . . . 16  |-  <Q  Or  Q.
3635, 8sotri 5024 . . . . . . . . . . . . . . 15  |-  ( ( 1Q  <Q  ( z  .Q  y )  /\  (
z  .Q  y ) 
<Q  ( z  .Q  q
) )  ->  1Q  <Q  ( z  .Q  q
) )
3736ancoms 268 . . . . . . . . . . . . . 14  |-  ( ( ( z  .Q  y
)  <Q  ( z  .Q  q )  /\  1Q  <Q  ( z  .Q  y
) )  ->  1Q  <Q  ( z  .Q  q
) )
3834, 37syl6 33 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  /\  y  e.  Q. )  ->  ( ( y  <Q 
q  /\  ( *Q `  y )  e.  ( 1st `  A ) )  ->  1Q  <Q  ( z  .Q  q ) ) )
3938exp4b 367 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
( y  e.  Q.  ->  ( y  <Q  q  ->  ( ( *Q `  y )  e.  ( 1st `  A )  ->  1Q  <Q  (
z  .Q  q ) ) ) ) )
4010, 39syl5 32 . . . . . . . . . . 11  |-  ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
( y  <Q  q  ->  ( y  <Q  q  ->  ( ( *Q `  y )  e.  ( 1st `  A )  ->  1Q  <Q  (
z  .Q  q ) ) ) ) )
4140pm2.43d 50 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
( y  <Q  q  ->  ( ( *Q `  y )  e.  ( 1st `  A )  ->  1Q  <Q  (
z  .Q  q ) ) ) )
4241impd 254 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
( ( y  <Q 
q  /\  ( *Q `  y )  e.  ( 1st `  A ) )  ->  1Q  <Q  ( z  .Q  q ) ) )
4342exlimdv 1819 . . . . . . . 8  |-  ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
( E. y ( y  <Q  q  /\  ( *Q `  y )  e.  ( 1st `  A
) )  ->  1Q  <Q  ( z  .Q  q
) ) )
447, 43biimtrid 152 . . . . . . 7  |-  ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
( q  e.  ( 2nd `  B )  ->  1Q  <Q  (
z  .Q  q ) ) )
45 breq2 4007 . . . . . . . 8  |-  ( w  =  ( z  .Q  q )  ->  ( 1Q  <Q  w  <->  1Q  <Q  ( z  .Q  q ) ) )
4645biimprcd 160 . . . . . . 7  |-  ( 1Q 
<Q  ( z  .Q  q
)  ->  ( w  =  ( z  .Q  q )  ->  1Q  <Q  w ) )
4744, 46syl6 33 . . . . . 6  |-  ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
( q  e.  ( 2nd `  B )  ->  ( w  =  ( z  .Q  q
)  ->  1Q  <Q  w ) ) )
4847expimpd 363 . . . . 5  |-  ( A  e.  P.  ->  (
( z  e.  ( 2nd `  A )  /\  q  e.  ( 2nd `  B ) )  ->  ( w  =  ( z  .Q  q )  ->  1Q  <Q  w ) ) )
4948rexlimdvv 2601 . . . 4  |-  ( A  e.  P.  ->  ( E. z  e.  ( 2nd `  A ) E. q  e.  ( 2nd `  B ) w  =  ( z  .Q  q
)  ->  1Q  <Q  w ) )
506, 49sylbid 150 . . 3  |-  ( A  e.  P.  ->  (
w  e.  ( 2nd `  ( A  .P.  B
) )  ->  1Q  <Q  w ) )
51 1pru 7554 . . . 4  |-  ( 2nd `  1P )  =  {
w  |  1Q  <Q  w }
5251abeq2i 2288 . . 3  |-  ( w  e.  ( 2nd `  1P ) 
<->  1Q  <Q  w )
5350, 52syl6ibr 162 . 2  |-  ( A  e.  P.  ->  (
w  e.  ( 2nd `  ( A  .P.  B
) )  ->  w  e.  ( 2nd `  1P ) ) )
5453ssrdv 3161 1  |-  ( A  e.  P.  ->  ( 2nd `  ( A  .P.  B ) )  C_  ( 2nd `  1P ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353   E.wex 1492    e. wcel 2148   {cab 2163   E.wrex 2456    C_ wss 3129   <.cop 3595   class class class wbr 4003   ` cfv 5216  (class class class)co 5874   1stc1st 6138   2ndc2nd 6139   Q.cnq 7278   1Qc1q 7279    .Q cmq 7281   *Qcrq 7282    <Q cltq 7283   P.cnp 7289   1Pc1p 7290    .P. cmp 7292
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-inp 7464  df-i1p 7465  df-imp 7467
This theorem is referenced by:  recexprlemex  7635
  Copyright terms: Public domain W3C validator