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Theorem recexprlemss1u 7193
Description: The upper cut of  A  .P.  B is a subset of the upper cut of one. Lemma for recexpr 7195. (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 7189 . . . . 5  |-  ( A  e.  P.  ->  B  e.  P. )
3 df-imp 7026 . . . . . 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 6933 . . . . . 6  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  .Q  g
)  e.  Q. )
53, 4genpelvu 7070 . . . . 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 412 . . . 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 7180 . . . . . . . 8  |-  ( q  e.  ( 2nd `  B
)  <->  E. y ( y 
<Q  q  /\  ( *Q `  y )  e.  ( 1st `  A
) ) )
8 ltrelnq 6922 . . . . . . . . . . . . . 14  |-  <Q  C_  ( Q.  X.  Q. )
98brel 4490 . . . . . . . . . . . . 13  |-  ( y 
<Q  q  ->  ( y  e.  Q.  /\  q  e.  Q. ) )
109simpld 110 . . . . . . . . . . . 12  |-  ( y 
<Q  q  ->  y  e. 
Q. )
11 prop 7032 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
12 elprnqu 7039 . . . . . . . . . . . . . . . . . 18  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
z  e.  Q. )
1311, 12sylan 277 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
z  e.  Q. )
14 ltmnqi 6960 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  <Q  q  /\  z  e.  Q. )  ->  ( z  .Q  y
)  <Q  ( z  .Q  q ) )
1514expcom 114 . . . . . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  /\  y  e.  Q. )  ->  ( y  <Q  q  ->  ( z  .Q  y
)  <Q  ( z  .Q  q ) ) )
18 prltlu 7044 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  ( *Q `  y
)  e.  ( 1st `  A )  /\  z  e.  ( 2nd `  A
) )  ->  ( *Q `  y )  <Q 
z )
1911, 18syl3an1 1207 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  P.  /\  ( *Q `  y )  e.  ( 1st `  A
)  /\  z  e.  ( 2nd `  A ) )  ->  ( *Q `  y )  <Q  z
)
20193com23 1149 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  P.  /\  z  e.  ( 2nd `  A )  /\  ( *Q `  y )  e.  ( 1st `  A
) )  ->  ( *Q `  y )  <Q 
z )
21203expia 1145 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
( ( *Q `  y )  e.  ( 1st `  A )  ->  ( *Q `  y )  <Q  z
) )
2221adantr 270 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  /\  y  e.  Q. )  ->  ( ( *Q `  y )  e.  ( 1st `  A )  ->  ( *Q `  y )  <Q  z
) )
23 ltmnqi 6960 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( *Q `  y
)  <Q  z  /\  y  e.  Q. )  ->  (
y  .Q  ( *Q
`  y ) ) 
<Q  ( y  .Q  z
) )
2423expcom 114 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  Q.  ->  (
( *Q `  y
)  <Q  z  ->  (
y  .Q  ( *Q
`  y ) ) 
<Q  ( y  .Q  z
) ) )
2524adantr 270 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( ( *Q `  y )  <Q  z  ->  ( y  .Q  ( *Q `  y ) ) 
<Q  ( y  .Q  z
) ) )
26 recidnq 6950 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  e.  Q.  ->  (
y  .Q  ( *Q
`  y ) )  =  1Q )
2726adantr 270 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  ( *Q `  y ) )  =  1Q )
28 mulcomnqg 6940 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  z
)  =  ( z  .Q  y ) )
2927, 28breq12d 3858 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( ( y  .Q  ( *Q `  y
) )  <Q  (
y  .Q  z )  <-> 
1Q  <Q  ( z  .Q  y ) ) )
3025, 29sylibd 147 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( ( *Q `  y )  <Q  z  ->  1Q  <Q  ( z  .Q  y ) ) )
3130ancoms 264 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  Q.  /\  y  e.  Q. )  ->  ( ( *Q `  y )  <Q  z  ->  1Q  <Q  ( z  .Q  y ) ) )
3213, 31sylan 277 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  /\  y  e.  Q. )  ->  ( ( *Q `  y )  <Q  z  ->  1Q  <Q  ( z  .Q  y ) ) )
3322, 32syld 44 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  /\  y  e.  Q. )  ->  ( ( *Q `  y )  e.  ( 1st `  A )  ->  1Q  <Q  (
z  .Q  y ) ) )
3417, 33anim12d 328 . . . . . . . . . . . . . 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 6955 . . . . . . . . . . . . . . . 16  |-  <Q  Or  Q.
3635, 8sotri 4827 . . . . . . . . . . . . . . 15  |-  ( ( 1Q  <Q  ( z  .Q  y )  /\  (
z  .Q  y ) 
<Q  ( z  .Q  q
) )  ->  1Q  <Q  ( z  .Q  q
) )
3736ancoms 264 . . . . . . . . . . . . . 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 359 . . . . . . . . . . . 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 49 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
( y  <Q  q  ->  ( ( *Q `  y )  e.  ( 1st `  A )  ->  1Q  <Q  (
z  .Q  q ) ) ) )
4241impd 251 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
( ( y  <Q 
q  /\  ( *Q `  y )  e.  ( 1st `  A ) )  ->  1Q  <Q  ( z  .Q  q ) ) )
4342exlimdv 1747 . . . . . . . 8  |-  ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
( E. y ( y  <Q  q  /\  ( *Q `  y )  e.  ( 1st `  A
) )  ->  1Q  <Q  ( z  .Q  q
) ) )
447, 43syl5bi 150 . . . . . . 7  |-  ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
( q  e.  ( 2nd `  B )  ->  1Q  <Q  (
z  .Q  q ) ) )
45 breq2 3849 . . . . . . . 8  |-  ( w  =  ( z  .Q  q )  ->  ( 1Q  <Q  w  <->  1Q  <Q  ( z  .Q  q ) ) )
4645biimprcd 158 . . . . . . 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 355 . . . . 5  |-  ( A  e.  P.  ->  (
( z  e.  ( 2nd `  A )  /\  q  e.  ( 2nd `  B ) )  ->  ( w  =  ( z  .Q  q )  ->  1Q  <Q  w ) ) )
4948rexlimdvv 2495 . . . 4  |-  ( A  e.  P.  ->  ( E. z  e.  ( 2nd `  A ) E. q  e.  ( 2nd `  B ) w  =  ( z  .Q  q
)  ->  1Q  <Q  w ) )
506, 49sylbid 148 . . 3  |-  ( A  e.  P.  ->  (
w  e.  ( 2nd `  ( A  .P.  B
) )  ->  1Q  <Q  w ) )
51 1pru 7113 . . . 4  |-  ( 2nd `  1P )  =  {
w  |  1Q  <Q  w }
5251abeq2i 2198 . . 3  |-  ( w  e.  ( 2nd `  1P ) 
<->  1Q  <Q  w )
5350, 52syl6ibr 160 . 2  |-  ( A  e.  P.  ->  (
w  e.  ( 2nd `  ( A  .P.  B
) )  ->  w  e.  ( 2nd `  1P ) ) )
5453ssrdv 3031 1  |-  ( A  e.  P.  ->  ( 2nd `  ( A  .P.  B ) )  C_  ( 2nd `  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 6837   1Qc1q 6838    .Q cmq 6840   *Qcrq 6841    <Q cltq 6842   P.cnp 6848   1Pc1p 6849    .P. cmp 6851
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 6290  df-ec 6292  df-qs 6296  df-ni 6861  df-pli 6862  df-mi 6863  df-lti 6864  df-plpq 6901  df-mpq 6902  df-enq 6904  df-nqqs 6905  df-plqqs 6906  df-mqqs 6907  df-1nqqs 6908  df-rq 6909  df-ltnqqs 6910  df-inp 7023  df-i1p 7024  df-imp 7026
This theorem is referenced by:  recexprlemex  7194
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