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Theorem recexprlemss1l 7194
Description: The lower cut of  A  .P.  B is a subset of the lower cut of one. Lemma for recexpr 7197. (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 7191 . . . . 5  |-  ( A  e.  P.  ->  B  e.  P. )
3 df-imp 7028 . . . . . 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 6935 . . . . . 6  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  .Q  g
)  e.  Q. )
53, 4genpelvl 7071 . . . . 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 412 . . . 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 7181 . . . . . . . 8  |-  ( q  e.  ( 1st `  B
)  <->  E. y ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
8 ltrelnq 6924 . . . . . . . . . . . . . 14  |-  <Q  C_  ( Q.  X.  Q. )
98brel 4490 . . . . . . . . . . . . 13  |-  ( q 
<Q  y  ->  ( q  e.  Q.  /\  y  e.  Q. ) )
109simprd 112 . . . . . . . . . . . 12  |-  ( q 
<Q  y  ->  y  e. 
Q. )
11 prop 7034 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
12 elprnql 7040 . . . . . . . . . . . . . . . . . 18  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
1311, 12sylan 277 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
14 ltmnqi 6962 . . . . . . . . . . . . . . . . . 18  |-  ( ( q  <Q  y  /\  z  e.  Q. )  ->  ( z  .Q  q
)  <Q  ( z  .Q  y ) )
1514expcom 114 . . . . . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  /\  y  e.  Q. )  ->  ( q  <Q  y  ->  ( z  .Q  q
)  <Q  ( z  .Q  y ) ) )
18 prltlu 7046 . . . . . . . . . . . . . . . . . . 19  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A )  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  z  <Q  ( *Q `  y
) )
1911, 18syl3an1 1207 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A )  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  z  <Q  ( *Q `  y
) )
20193expia 1145 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
( ( *Q `  y )  e.  ( 2nd `  A )  ->  z  <Q  ( *Q `  y ) ) )
2120adantr 270 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  /\  y  e.  Q. )  ->  ( ( *Q `  y )  e.  ( 2nd `  A )  ->  z  <Q  ( *Q `  y ) ) )
22 ltmnqi 6962 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( z  <Q  ( *Q `  y )  /\  y  e.  Q. )  ->  (
y  .Q  z ) 
<Q  ( y  .Q  ( *Q `  y ) ) )
2322expcom 114 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  Q.  ->  (
z  <Q  ( *Q `  y )  ->  (
y  .Q  z ) 
<Q  ( y  .Q  ( *Q `  y ) ) ) )
2423adantr 270 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( z  <Q  ( *Q `  y )  -> 
( y  .Q  z
)  <Q  ( y  .Q  ( *Q `  y
) ) ) )
25 mulcomnqg 6942 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  z
)  =  ( z  .Q  y ) )
26 recidnq 6952 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  e.  Q.  ->  (
y  .Q  ( *Q
`  y ) )  =  1Q )
2726adantr 270 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  ( *Q `  y ) )  =  1Q )
2825, 27breq12d 3858 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( ( y  .Q  z )  <Q  (
y  .Q  ( *Q
`  y ) )  <-> 
( z  .Q  y
)  <Q  1Q ) )
2924, 28sylibd 147 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( z  <Q  ( *Q `  y )  -> 
( z  .Q  y
)  <Q  1Q ) )
3029ancoms 264 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  Q.  /\  y  e.  Q. )  ->  ( z  <Q  ( *Q `  y )  -> 
( z  .Q  y
)  <Q  1Q ) )
3113, 30sylan 277 . . . . . . . . . . . . . . . 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 328 . . . . . . . . . . . . . 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 6957 . . . . . . . . . . . . . . 15  |-  <Q  Or  Q.
3534, 8sotri 4827 . . . . . . . . . . . . . 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 359 . . . . . . . . . . . 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 49 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
( q  <Q  y  ->  ( ( *Q `  y )  e.  ( 2nd `  A )  ->  ( z  .Q  q )  <Q  1Q ) ) )
4039impd 251 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
( ( q  <Q 
y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  ( z  .Q  q )  <Q  1Q ) )
4140exlimdv 1747 . . . . . . . 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 150 . . . . . . 7  |-  ( ( A  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
( q  e.  ( 1st `  B )  ->  ( z  .Q  q )  <Q  1Q ) )
43 breq1 3848 . . . . . . . 8  |-  ( w  =  ( z  .Q  q )  ->  (
w  <Q  1Q  <->  ( z  .Q  q )  <Q  1Q ) )
4443biimprcd 158 . . . . . . 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 355 . . . . 5  |-  ( A  e.  P.  ->  (
( z  e.  ( 1st `  A )  /\  q  e.  ( 1st `  B ) )  ->  ( w  =  ( z  .Q  q )  ->  w  <Q  1Q ) ) )
4746rexlimdvv 2495 . . . 4  |-  ( A  e.  P.  ->  ( E. z  e.  ( 1st `  A ) E. q  e.  ( 1st `  B ) w  =  ( z  .Q  q
)  ->  w  <Q  1Q ) )
486, 47sylbid 148 . . 3  |-  ( A  e.  P.  ->  (
w  e.  ( 1st `  ( A  .P.  B
) )  ->  w  <Q  1Q ) )
49 1prl 7114 . . . 4  |-  ( 1st `  1P )  =  {
w  |  w  <Q  1Q }
5049abeq2i 2198 . . 3  |-  ( w  e.  ( 1st `  1P ) 
<->  w  <Q  1Q )
5148, 50syl6ibr 160 . 2  |-  ( A  e.  P.  ->  (
w  e.  ( 1st `  ( A  .P.  B
) )  ->  w  e.  ( 1st `  1P ) ) )
5251ssrdv 3031 1  |-  ( A  e.  P.  ->  ( 1st `  ( A  .P.  B ) )  C_  ( 1st `  1P ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   {cab 2074   E.wrex 2360    C_ wss 2999   <.cop 3449   class class class wbr 3845   ` cfv 5015  (class class class)co 5652   1stc1st 5909   2ndc2nd 5910   Q.cnq 6839   1Qc1q 6840    .Q cmq 6842   *Qcrq 6843    <Q cltq 6844   P.cnp 6850   1Pc1p 6851    .P. cmp 6853
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-1o 6181  df-oadd 6185  df-omul 6186  df-er 6292  df-ec 6294  df-qs 6298  df-ni 6863  df-pli 6864  df-mi 6865  df-lti 6866  df-plpq 6903  df-mpq 6904  df-enq 6906  df-nqqs 6907  df-plqqs 6908  df-mqqs 6909  df-1nqqs 6910  df-rq 6911  df-ltnqqs 6912  df-inp 7025  df-i1p 7026  df-imp 7028
This theorem is referenced by:  recexprlemex  7196
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