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Theorem recexprlemss1l 7665
Description: The lower cut of  A  .P.  B is a subset of the lower cut of one. Lemma for recexpr 7668. (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 7662 . . . . 5  |-  ( A  e.  P.  ->  B  e.  P. )
3 df-imp 7499 . . . . . 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 7406 . . . . . 6  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  .Q  g
)  e.  Q. )
53, 4genpelvl 7542 . . . . 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 421 . . . 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 7652 . . . . . . . 8  |-  ( q  e.  ( 1st `  B
)  <->  E. y ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
8 ltrelnq 7395 . . . . . . . . . . . . . 14  |-  <Q  C_  ( Q.  X.  Q. )
98brel 4696 . . . . . . . . . . . . 13  |-  ( q 
<Q  y  ->  ( q  e.  Q.  /\  y  e.  Q. ) )
109simprd 114 . . . . . . . . . . . 12  |-  ( q 
<Q  y  ->  y  e. 
Q. )
11 prop 7505 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
12 elprnql 7511 . . . . . . . . . . . . . . . . . 18  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
1311, 12sylan 283 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
14 ltmnqi 7433 . . . . . . . . . . . . . . . . . 18  |-  ( ( q  <Q  y  /\  z  e.  Q. )  ->  ( z  .Q  q
)  <Q  ( z  .Q  y ) )
1514expcom 116 . . . . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  /\  y  e.  Q. )  ->  ( q  <Q  y  ->  ( z  .Q  q
)  <Q  ( z  .Q  y ) ) )
18 prltlu 7517 . . . . . . . . . . . . . . . . . . 19  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A )  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  z  <Q  ( *Q `  y
) )
1911, 18syl3an1 1282 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A )  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  z  <Q  ( *Q `  y
) )
20193expia 1207 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
( ( *Q `  y )  e.  ( 2nd `  A )  ->  z  <Q  ( *Q `  y ) ) )
2120adantr 276 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  /\  y  e.  Q. )  ->  ( ( *Q `  y )  e.  ( 2nd `  A )  ->  z  <Q  ( *Q `  y ) ) )
22 ltmnqi 7433 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( z  <Q  ( *Q `  y )  /\  y  e.  Q. )  ->  (
y  .Q  z ) 
<Q  ( y  .Q  ( *Q `  y ) ) )
2322expcom 116 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  Q.  ->  (
z  <Q  ( *Q `  y )  ->  (
y  .Q  z ) 
<Q  ( y  .Q  ( *Q `  y ) ) ) )
2423adantr 276 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( z  <Q  ( *Q `  y )  -> 
( y  .Q  z
)  <Q  ( y  .Q  ( *Q `  y
) ) ) )
25 mulcomnqg 7413 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  z
)  =  ( z  .Q  y ) )
26 recidnq 7423 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  e.  Q.  ->  (
y  .Q  ( *Q
`  y ) )  =  1Q )
2726adantr 276 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  ( *Q `  y ) )  =  1Q )
2825, 27breq12d 4031 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( ( y  .Q  z )  <Q  (
y  .Q  ( *Q
`  y ) )  <-> 
( z  .Q  y
)  <Q  1Q ) )
2924, 28sylibd 149 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( z  <Q  ( *Q `  y )  -> 
( z  .Q  y
)  <Q  1Q ) )
3029ancoms 268 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  Q.  /\  y  e.  Q. )  ->  ( z  <Q  ( *Q `  y )  -> 
( z  .Q  y
)  <Q  1Q ) )
3113, 30sylan 283 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  /\  y  e.  Q. )  ->  ( z  <Q  ( *Q `  y )  -> 
( z  .Q  y
)  <Q  1Q ) )
3221, 31syld 45 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  /\  y  e.  Q. )  ->  ( ( *Q `  y )  e.  ( 2nd `  A )  ->  ( z  .Q  y )  <Q  1Q ) )
3317, 32anim12d 335 . . . . . . . . . . . . . 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 7428 . . . . . . . . . . . . . . 15  |-  <Q  Or  Q.
3534, 8sotri 5042 . . . . . . . . . . . . . 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 367 . . . . . . . . . . . 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 50 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
( q  <Q  y  ->  ( ( *Q `  y )  e.  ( 2nd `  A )  ->  ( z  .Q  q )  <Q  1Q ) ) )
4039impd 254 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
( ( q  <Q 
y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  ( z  .Q  q )  <Q  1Q ) )
4140exlimdv 1830 . . . . . . . 8  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
( E. y ( q  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  (
z  .Q  q ) 
<Q  1Q ) )
427, 41biimtrid 152 . . . . . . 7  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
( q  e.  ( 1st `  B )  ->  ( z  .Q  q )  <Q  1Q ) )
43 breq1 4021 . . . . . . . 8  |-  ( w  =  ( z  .Q  q )  ->  (
w  <Q  1Q  <->  ( z  .Q  q )  <Q  1Q ) )
4443biimprcd 160 . . . . . . 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 363 . . . . 5  |-  ( A  e.  P.  ->  (
( z  e.  ( 1st `  A )  /\  q  e.  ( 1st `  B ) )  ->  ( w  =  ( z  .Q  q )  ->  w  <Q  1Q ) ) )
4746rexlimdvv 2614 . . . 4  |-  ( A  e.  P.  ->  ( E. z  e.  ( 1st `  A ) E. q  e.  ( 1st `  B ) w  =  ( z  .Q  q
)  ->  w  <Q  1Q ) )
486, 47sylbid 150 . . 3  |-  ( A  e.  P.  ->  (
w  e.  ( 1st `  ( A  .P.  B
) )  ->  w  <Q  1Q ) )
49 1prl 7585 . . . 4  |-  ( 1st `  1P )  =  {
w  |  w  <Q  1Q }
5049abeq2i 2300 . . 3  |-  ( w  e.  ( 1st `  1P ) 
<->  w  <Q  1Q )
5148, 50imbitrrdi 162 . 2  |-  ( A  e.  P.  ->  (
w  e.  ( 1st `  ( A  .P.  B
) )  ->  w  e.  ( 1st `  1P ) ) )
5251ssrdv 3176 1  |-  ( A  e.  P.  ->  ( 1st `  ( A  .P.  B ) )  C_  ( 1st `  1P ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364   E.wex 1503    e. wcel 2160   {cab 2175   E.wrex 2469    C_ wss 3144   <.cop 3610   class class class wbr 4018   ` cfv 5235  (class class class)co 5897   1stc1st 6164   2ndc2nd 6165   Q.cnq 7310   1Qc1q 7311    .Q cmq 7313   *Qcrq 7314    <Q cltq 7315   P.cnp 7321   1Pc1p 7322    .P. cmp 7324
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 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
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 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-1o 6442  df-oadd 6446  df-omul 6447  df-er 6560  df-ec 6562  df-qs 6566  df-ni 7334  df-pli 7335  df-mi 7336  df-lti 7337  df-plpq 7374  df-mpq 7375  df-enq 7377  df-nqqs 7378  df-plqqs 7379  df-mqqs 7380  df-1nqqs 7381  df-rq 7382  df-ltnqqs 7383  df-inp 7496  df-i1p 7497  df-imp 7499
This theorem is referenced by:  recexprlemex  7667
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