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Theorem recexprlemss1l 6733
Description: The lower cut of  A  .P.  B is a subset of the lower cut of one. Lemma for recexpr 6736. (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 6730 . . . . 5  |-  ( A  e.  P.  ->  B  e.  P. )
3 df-imp 6567 . . . . . 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 6474 . . . . . 6  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  .Q  g
)  e.  Q. )
53, 4genpelvl 6610 . . . . 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 398 . . . 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 6720 . . . . . . . 8  |-  ( q  e.  ( 1st `  B
)  <->  E. y ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
8 ltrelnq 6463 . . . . . . . . . . . . . 14  |-  <Q  C_  ( Q.  X.  Q. )
98brel 4392 . . . . . . . . . . . . 13  |-  ( q 
<Q  y  ->  ( q  e.  Q.  /\  y  e.  Q. ) )
109simprd 107 . . . . . . . . . . . 12  |-  ( q 
<Q  y  ->  y  e. 
Q. )
11 prop 6573 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
12 elprnql 6579 . . . . . . . . . . . . . . . . . 18  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
1311, 12sylan 267 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
14 ltmnqi 6501 . . . . . . . . . . . . . . . . . 18  |-  ( ( q  <Q  y  /\  z  e.  Q. )  ->  ( z  .Q  q
)  <Q  ( z  .Q  y ) )
1514expcom 109 . . . . . . . . . . . . . . . . 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 261 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  /\  y  e.  Q. )  ->  ( q  <Q  y  ->  ( z  .Q  q
)  <Q  ( z  .Q  y ) ) )
18 prltlu 6585 . . . . . . . . . . . . . . . . . . 19  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A )  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  z  <Q  ( *Q `  y
) )
1911, 18syl3an1 1168 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A )  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  z  <Q  ( *Q `  y
) )
20193expia 1106 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
( ( *Q `  y )  e.  ( 2nd `  A )  ->  z  <Q  ( *Q `  y ) ) )
2120adantr 261 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  /\  y  e.  Q. )  ->  ( ( *Q `  y )  e.  ( 2nd `  A )  ->  z  <Q  ( *Q `  y ) ) )
22 ltmnqi 6501 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( z  <Q  ( *Q `  y )  /\  y  e.  Q. )  ->  (
y  .Q  z ) 
<Q  ( y  .Q  ( *Q `  y ) ) )
2322expcom 109 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  Q.  ->  (
z  <Q  ( *Q `  y )  ->  (
y  .Q  z ) 
<Q  ( y  .Q  ( *Q `  y ) ) ) )
2423adantr 261 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( z  <Q  ( *Q `  y )  -> 
( y  .Q  z
)  <Q  ( y  .Q  ( *Q `  y
) ) ) )
25 mulcomnqg 6481 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  z
)  =  ( z  .Q  y ) )
26 recidnq 6491 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  e.  Q.  ->  (
y  .Q  ( *Q
`  y ) )  =  1Q )
2726adantr 261 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  ( *Q `  y ) )  =  1Q )
2825, 27breq12d 3777 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( ( y  .Q  z )  <Q  (
y  .Q  ( *Q
`  y ) )  <-> 
( z  .Q  y
)  <Q  1Q ) )
2924, 28sylibd 138 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( z  <Q  ( *Q `  y )  -> 
( z  .Q  y
)  <Q  1Q ) )
3029ancoms 255 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  Q.  /\  y  e.  Q. )  ->  ( z  <Q  ( *Q `  y )  -> 
( z  .Q  y
)  <Q  1Q ) )
3113, 30sylan 267 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  /\  y  e.  Q. )  ->  ( z  <Q  ( *Q `  y )  -> 
( z  .Q  y
)  <Q  1Q ) )
3221, 31syld 40 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  /\  y  e.  Q. )  ->  ( ( *Q `  y )  e.  ( 2nd `  A )  ->  ( z  .Q  y )  <Q  1Q ) )
3317, 32anim12d 318 . . . . . . . . . . . . . 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 6496 . . . . . . . . . . . . . . 15  |-  <Q  Or  Q.
3534, 8sotri 4720 . . . . . . . . . . . . . 14  |-  ( ( ( z  .Q  q
)  <Q  ( z  .Q  y )  /\  (
z  .Q  y ) 
<Q  1Q )  ->  (
z  .Q  q ) 
<Q  1Q )
3633, 35syl6 29 . . . . . . . . . . . . 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 349 . . . . . . . . . . . 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 28 . . . . . . . . . . 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 44 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
( q  <Q  y  ->  ( ( *Q `  y )  e.  ( 2nd `  A )  ->  ( z  .Q  q )  <Q  1Q ) ) )
4039impd 242 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
( ( q  <Q 
y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  ( z  .Q  q )  <Q  1Q ) )
4140exlimdv 1700 . . . . . . . 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 141 . . . . . . 7  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
( q  e.  ( 1st `  B )  ->  ( z  .Q  q )  <Q  1Q ) )
43 breq1 3767 . . . . . . . 8  |-  ( w  =  ( z  .Q  q )  ->  (
w  <Q  1Q  <->  ( z  .Q  q )  <Q  1Q ) )
4443biimprcd 149 . . . . . . 7  |-  ( ( z  .Q  q ) 
<Q  1Q  ->  ( w  =  ( z  .Q  q )  ->  w  <Q  1Q ) )
4542, 44syl6 29 . . . . . 6  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
( q  e.  ( 1st `  B )  ->  ( w  =  ( z  .Q  q
)  ->  w  <Q  1Q ) ) )
4645expimpd 345 . . . . 5  |-  ( A  e.  P.  ->  (
( z  e.  ( 1st `  A )  /\  q  e.  ( 1st `  B ) )  ->  ( w  =  ( z  .Q  q )  ->  w  <Q  1Q ) ) )
4746rexlimdvv 2439 . . . 4  |-  ( A  e.  P.  ->  ( E. z  e.  ( 1st `  A ) E. q  e.  ( 1st `  B ) w  =  ( z  .Q  q
)  ->  w  <Q  1Q ) )
486, 47sylbid 139 . . 3  |-  ( A  e.  P.  ->  (
w  e.  ( 1st `  ( A  .P.  B
) )  ->  w  <Q  1Q ) )
49 1prl 6653 . . . 4  |-  ( 1st `  1P )  =  {
w  |  w  <Q  1Q }
5049abeq2i 2148 . . 3  |-  ( w  e.  ( 1st `  1P ) 
<->  w  <Q  1Q )
5148, 50syl6ibr 151 . 2  |-  ( A  e.  P.  ->  (
w  e.  ( 1st `  ( A  .P.  B
) )  ->  w  e.  ( 1st `  1P ) ) )
5251ssrdv 2951 1  |-  ( A  e.  P.  ->  ( 1st `  ( A  .P.  B ) )  C_  ( 1st `  1P ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243   E.wex 1381    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   1stc1st 5765   2ndc2nd 5766   Q.cnq 6378   1Qc1q 6379    .Q cmq 6381   *Qcrq 6382    <Q cltq 6383   P.cnp 6389   1Pc1p 6390    .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-i1p 6565  df-imp 6567
This theorem is referenced by:  recexprlemex  6735
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