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Theorem recexprlemss1u 7408
Description: The upper cut of  A  .P.  B is a subset of the upper cut of one. Lemma for recexpr 7410. (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 7404 . . . . 5  |-  ( A  e.  P.  ->  B  e.  P. )
3 df-imp 7241 . . . . . 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 7148 . . . . . 6  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  .Q  g
)  e.  Q. )
53, 4genpelvu 7285 . . . . 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 415 . . . 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 7395 . . . . . . . 8  |-  ( q  e.  ( 2nd `  B
)  <->  E. y ( y 
<Q  q  /\  ( *Q `  y )  e.  ( 1st `  A
) ) )
8 ltrelnq 7137 . . . . . . . . . . . . . 14  |-  <Q  C_  ( Q.  X.  Q. )
98brel 4559 . . . . . . . . . . . . 13  |-  ( y 
<Q  q  ->  ( y  e.  Q.  /\  q  e.  Q. ) )
109simpld 111 . . . . . . . . . . . 12  |-  ( y 
<Q  q  ->  y  e. 
Q. )
11 prop 7247 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
12 elprnqu 7254 . . . . . . . . . . . . . . . . . 18  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
z  e.  Q. )
1311, 12sylan 279 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
z  e.  Q. )
14 ltmnqi 7175 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  <Q  q  /\  z  e.  Q. )  ->  ( z  .Q  y
)  <Q  ( z  .Q  q ) )
1514expcom 115 . . . . . . . . . . . . . . . . 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 272 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  /\  y  e.  Q. )  ->  ( y  <Q  q  ->  ( z  .Q  y
)  <Q  ( z  .Q  q ) ) )
18 prltlu 7259 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  ( *Q `  y
)  e.  ( 1st `  A )  /\  z  e.  ( 2nd `  A
) )  ->  ( *Q `  y )  <Q 
z )
1911, 18syl3an1 1232 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  P.  /\  ( *Q `  y )  e.  ( 1st `  A
)  /\  z  e.  ( 2nd `  A ) )  ->  ( *Q `  y )  <Q  z
)
20193com23 1170 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  P.  /\  z  e.  ( 2nd `  A )  /\  ( *Q `  y )  e.  ( 1st `  A
) )  ->  ( *Q `  y )  <Q 
z )
21203expia 1166 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
( ( *Q `  y )  e.  ( 1st `  A )  ->  ( *Q `  y )  <Q  z
) )
2221adantr 272 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  /\  y  e.  Q. )  ->  ( ( *Q `  y )  e.  ( 1st `  A )  ->  ( *Q `  y )  <Q  z
) )
23 ltmnqi 7175 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( *Q `  y
)  <Q  z  /\  y  e.  Q. )  ->  (
y  .Q  ( *Q
`  y ) ) 
<Q  ( y  .Q  z
) )
2423expcom 115 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  Q.  ->  (
( *Q `  y
)  <Q  z  ->  (
y  .Q  ( *Q
`  y ) ) 
<Q  ( y  .Q  z
) ) )
2524adantr 272 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( ( *Q `  y )  <Q  z  ->  ( y  .Q  ( *Q `  y ) ) 
<Q  ( y  .Q  z
) ) )
26 recidnq 7165 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  e.  Q.  ->  (
y  .Q  ( *Q
`  y ) )  =  1Q )
2726adantr 272 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  ( *Q `  y ) )  =  1Q )
28 mulcomnqg 7155 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  z
)  =  ( z  .Q  y ) )
2927, 28breq12d 3910 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( ( y  .Q  ( *Q `  y
) )  <Q  (
y  .Q  z )  <-> 
1Q  <Q  ( z  .Q  y ) ) )
3025, 29sylibd 148 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( ( *Q `  y )  <Q  z  ->  1Q  <Q  ( z  .Q  y ) ) )
3130ancoms 266 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  Q.  /\  y  e.  Q. )  ->  ( ( *Q `  y )  <Q  z  ->  1Q  <Q  ( z  .Q  y ) ) )
3213, 31sylan 279 . . . . . . . . . . . . . . . 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 331 . . . . . . . . . . . . . 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 7170 . . . . . . . . . . . . . . . 16  |-  <Q  Or  Q.
3635, 8sotri 4902 . . . . . . . . . . . . . . 15  |-  ( ( 1Q  <Q  ( z  .Q  y )  /\  (
z  .Q  y ) 
<Q  ( z  .Q  q
) )  ->  1Q  <Q  ( z  .Q  q
) )
3736ancoms 266 . . . . . . . . . . . . . 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 362 . . . . . . . . . . . 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 252 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
( ( y  <Q 
q  /\  ( *Q `  y )  e.  ( 1st `  A ) )  ->  1Q  <Q  ( z  .Q  q ) ) )
4342exlimdv 1773 . . . . . . . 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 151 . . . . . . 7  |-  ( ( A  e.  P.  /\  z  e.  ( 2nd `  A ) )  -> 
( q  e.  ( 2nd `  B )  ->  1Q  <Q  (
z  .Q  q ) ) )
45 breq2 3901 . . . . . . . 8  |-  ( w  =  ( z  .Q  q )  ->  ( 1Q  <Q  w  <->  1Q  <Q  ( z  .Q  q ) ) )
4645biimprcd 159 . . . . . . 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 358 . . . . 5  |-  ( A  e.  P.  ->  (
( z  e.  ( 2nd `  A )  /\  q  e.  ( 2nd `  B ) )  ->  ( w  =  ( z  .Q  q )  ->  1Q  <Q  w ) ) )
4948rexlimdvv 2531 . . . 4  |-  ( A  e.  P.  ->  ( E. z  e.  ( 2nd `  A ) E. q  e.  ( 2nd `  B ) w  =  ( z  .Q  q
)  ->  1Q  <Q  w ) )
506, 49sylbid 149 . . 3  |-  ( A  e.  P.  ->  (
w  e.  ( 2nd `  ( A  .P.  B
) )  ->  1Q  <Q  w ) )
51 1pru 7328 . . . 4  |-  ( 2nd `  1P )  =  {
w  |  1Q  <Q  w }
5251abeq2i 2226 . . 3  |-  ( w  e.  ( 2nd `  1P ) 
<->  1Q  <Q  w )
5350, 52syl6ibr 161 . 2  |-  ( A  e.  P.  ->  (
w  e.  ( 2nd `  ( A  .P.  B
) )  ->  w  e.  ( 2nd `  1P ) ) )
5453ssrdv 3071 1  |-  ( A  e.  P.  ->  ( 2nd `  ( A  .P.  B ) )  C_  ( 2nd `  1P ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314   E.wex 1451    e. wcel 1463   {cab 2101   E.wrex 2392    C_ wss 3039   <.cop 3498   class class class wbr 3897   ` cfv 5091  (class class class)co 5740   1stc1st 6002   2ndc2nd 6003   Q.cnq 7052   1Qc1q 7053    .Q cmq 7055   *Qcrq 7056    <Q cltq 7057   P.cnp 7063   1Pc1p 7064    .P. cmp 7066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-inp 7238  df-i1p 7239  df-imp 7241
This theorem is referenced by:  recexprlemex  7409
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