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Theorem recexprlemm 6865
Description:  B is inhabited. Lemma for recexpr 6879. (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
recexprlemm  |-  ( A  e.  P.  ->  ( E. q  e.  Q.  q  e.  ( 1st `  B )  /\  E. r  e.  Q.  r  e.  ( 2nd `  B
) ) )
Distinct variable groups:    r, q, x, y, A    B, q,
r, x, y

Proof of Theorem recexprlemm
StepHypRef Expression
1 prop 6716 . . 3  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 prmu 6719 . . 3  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  ->  E. x  e.  Q.  x  e.  ( 2nd `  A ) )
3 recclnq 6633 . . . . . . 7  |-  ( x  e.  Q.  ->  ( *Q `  x )  e. 
Q. )
4 nsmallnqq 6653 . . . . . . 7  |-  ( ( *Q `  x )  e.  Q.  ->  E. q  e.  Q.  q  <Q  ( *Q `  x ) )
53, 4syl 14 . . . . . 6  |-  ( x  e.  Q.  ->  E. q  e.  Q.  q  <Q  ( *Q `  x ) )
65adantr 270 . . . . 5  |-  ( ( x  e.  Q.  /\  x  e.  ( 2nd `  A ) )  ->  E. q  e.  Q.  q  <Q  ( *Q `  x ) )
7 recrecnq 6635 . . . . . . . . . . . 12  |-  ( x  e.  Q.  ->  ( *Q `  ( *Q `  x ) )  =  x )
87eleq1d 2148 . . . . . . . . . . 11  |-  ( x  e.  Q.  ->  (
( *Q `  ( *Q `  x ) )  e.  ( 2nd `  A
)  <->  x  e.  ( 2nd `  A ) ) )
98anbi2d 452 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  (
( q  <Q  ( *Q `  x )  /\  ( *Q `  ( *Q
`  x ) )  e.  ( 2nd `  A
) )  <->  ( q  <Q  ( *Q `  x
)  /\  x  e.  ( 2nd `  A ) ) ) )
10 breq2 3791 . . . . . . . . . . . . 13  |-  ( y  =  ( *Q `  x )  ->  (
q  <Q  y  <->  q  <Q  ( *Q `  x ) ) )
11 fveq2 5203 . . . . . . . . . . . . . 14  |-  ( y  =  ( *Q `  x )  ->  ( *Q `  y )  =  ( *Q `  ( *Q `  x ) ) )
1211eleq1d 2148 . . . . . . . . . . . . 13  |-  ( y  =  ( *Q `  x )  ->  (
( *Q `  y
)  e.  ( 2nd `  A )  <->  ( *Q `  ( *Q `  x
) )  e.  ( 2nd `  A ) ) )
1310, 12anbi12d 457 . . . . . . . . . . . 12  |-  ( y  =  ( *Q `  x )  ->  (
( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  <->  ( q  <Q  ( *Q `  x
)  /\  ( *Q `  ( *Q `  x
) )  e.  ( 2nd `  A ) ) ) )
1413spcegv 2687 . . . . . . . . . . 11  |-  ( ( *Q `  x )  e.  Q.  ->  (
( q  <Q  ( *Q `  x )  /\  ( *Q `  ( *Q
`  x ) )  e.  ( 2nd `  A
) )  ->  E. y
( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) ) ) )
153, 14syl 14 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  (
( q  <Q  ( *Q `  x )  /\  ( *Q `  ( *Q
`  x ) )  e.  ( 2nd `  A
) )  ->  E. y
( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) ) ) )
169, 15sylbird 168 . . . . . . . . 9  |-  ( x  e.  Q.  ->  (
( q  <Q  ( *Q `  x )  /\  x  e.  ( 2nd `  A ) )  ->  E. y ( q  <Q 
y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) ) )
17 recexpr.1 . . . . . . . . . 10  |-  B  = 
<. { x  |  E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  {
x  |  E. y
( y  <Q  x  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) }
>.
1817recexprlemell 6863 . . . . . . . . 9  |-  ( q  e.  ( 1st `  B
)  <->  E. y ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
1916, 18syl6ibr 160 . . . . . . . 8  |-  ( x  e.  Q.  ->  (
( q  <Q  ( *Q `  x )  /\  x  e.  ( 2nd `  A ) )  -> 
q  e.  ( 1st `  B ) ) )
2019expcomd 1371 . . . . . . 7  |-  ( x  e.  Q.  ->  (
x  e.  ( 2nd `  A )  ->  (
q  <Q  ( *Q `  x )  ->  q  e.  ( 1st `  B
) ) ) )
2120imp 122 . . . . . 6  |-  ( ( x  e.  Q.  /\  x  e.  ( 2nd `  A ) )  -> 
( q  <Q  ( *Q `  x )  -> 
q  e.  ( 1st `  B ) ) )
2221reximdv 2463 . . . . 5  |-  ( ( x  e.  Q.  /\  x  e.  ( 2nd `  A ) )  -> 
( E. q  e. 
Q.  q  <Q  ( *Q `  x )  ->  E. q  e.  Q.  q  e.  ( 1st `  B ) ) )
236, 22mpd 13 . . . 4  |-  ( ( x  e.  Q.  /\  x  e.  ( 2nd `  A ) )  ->  E. q  e.  Q.  q  e.  ( 1st `  B ) )
2423rexlimiva 2473 . . 3  |-  ( E. x  e.  Q.  x  e.  ( 2nd `  A
)  ->  E. q  e.  Q.  q  e.  ( 1st `  B ) )
251, 2, 243syl 17 . 2  |-  ( A  e.  P.  ->  E. q  e.  Q.  q  e.  ( 1st `  B ) )
26 prml 6718 . . 3  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  ->  E. x  e.  Q.  x  e.  ( 1st `  A ) )
27 1nq 6607 . . . . . . . 8  |-  1Q  e.  Q.
28 addclnq 6616 . . . . . . . 8  |-  ( ( ( *Q `  x
)  e.  Q.  /\  1Q  e.  Q. )  -> 
( ( *Q `  x )  +Q  1Q )  e.  Q. )
293, 27, 28sylancl 404 . . . . . . 7  |-  ( x  e.  Q.  ->  (
( *Q `  x
)  +Q  1Q )  e.  Q. )
30 ltaddnq 6648 . . . . . . . 8  |-  ( ( ( *Q `  x
)  e.  Q.  /\  1Q  e.  Q. )  -> 
( *Q `  x
)  <Q  ( ( *Q
`  x )  +Q  1Q ) )
313, 27, 30sylancl 404 . . . . . . 7  |-  ( x  e.  Q.  ->  ( *Q `  x )  <Q 
( ( *Q `  x )  +Q  1Q ) )
32 breq2 3791 . . . . . . . 8  |-  ( r  =  ( ( *Q
`  x )  +Q  1Q )  ->  (
( *Q `  x
)  <Q  r  <->  ( *Q `  x )  <Q  (
( *Q `  x
)  +Q  1Q ) ) )
3332rspcev 2702 . . . . . . 7  |-  ( ( ( ( *Q `  x )  +Q  1Q )  e.  Q.  /\  ( *Q `  x )  <Q 
( ( *Q `  x )  +Q  1Q ) )  ->  E. r  e.  Q.  ( *Q `  x )  <Q  r
)
3429, 31, 33syl2anc 403 . . . . . 6  |-  ( x  e.  Q.  ->  E. r  e.  Q.  ( *Q `  x )  <Q  r
)
3534adantr 270 . . . . 5  |-  ( ( x  e.  Q.  /\  x  e.  ( 1st `  A ) )  ->  E. r  e.  Q.  ( *Q `  x ) 
<Q  r )
367eleq1d 2148 . . . . . . . . . . 11  |-  ( x  e.  Q.  ->  (
( *Q `  ( *Q `  x ) )  e.  ( 1st `  A
)  <->  x  e.  ( 1st `  A ) ) )
3736anbi2d 452 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  (
( ( *Q `  x )  <Q  r  /\  ( *Q `  ( *Q `  x ) )  e.  ( 1st `  A
) )  <->  ( ( *Q `  x )  <Q 
r  /\  x  e.  ( 1st `  A ) ) ) )
38 breq1 3790 . . . . . . . . . . . . 13  |-  ( y  =  ( *Q `  x )  ->  (
y  <Q  r  <->  ( *Q `  x )  <Q  r
) )
3911eleq1d 2148 . . . . . . . . . . . . 13  |-  ( y  =  ( *Q `  x )  ->  (
( *Q `  y
)  e.  ( 1st `  A )  <->  ( *Q `  ( *Q `  x
) )  e.  ( 1st `  A ) ) )
4038, 39anbi12d 457 . . . . . . . . . . . 12  |-  ( y  =  ( *Q `  x )  ->  (
( y  <Q  r  /\  ( *Q `  y
)  e.  ( 1st `  A ) )  <->  ( ( *Q `  x )  <Q 
r  /\  ( *Q `  ( *Q `  x
) )  e.  ( 1st `  A ) ) ) )
4140spcegv 2687 . . . . . . . . . . 11  |-  ( ( *Q `  x )  e.  Q.  ->  (
( ( *Q `  x )  <Q  r  /\  ( *Q `  ( *Q `  x ) )  e.  ( 1st `  A
) )  ->  E. y
( y  <Q  r  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) ) )
423, 41syl 14 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  (
( ( *Q `  x )  <Q  r  /\  ( *Q `  ( *Q `  x ) )  e.  ( 1st `  A
) )  ->  E. y
( y  <Q  r  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) ) )
4337, 42sylbird 168 . . . . . . . . 9  |-  ( x  e.  Q.  ->  (
( ( *Q `  x )  <Q  r  /\  x  e.  ( 1st `  A ) )  ->  E. y ( y 
<Q  r  /\  ( *Q `  y )  e.  ( 1st `  A
) ) ) )
4417recexprlemelu 6864 . . . . . . . . 9  |-  ( r  e.  ( 2nd `  B
)  <->  E. y ( y 
<Q  r  /\  ( *Q `  y )  e.  ( 1st `  A
) ) )
4543, 44syl6ibr 160 . . . . . . . 8  |-  ( x  e.  Q.  ->  (
( ( *Q `  x )  <Q  r  /\  x  e.  ( 1st `  A ) )  ->  r  e.  ( 2nd `  B ) ) )
4645expcomd 1371 . . . . . . 7  |-  ( x  e.  Q.  ->  (
x  e.  ( 1st `  A )  ->  (
( *Q `  x
)  <Q  r  ->  r  e.  ( 2nd `  B
) ) ) )
4746imp 122 . . . . . 6  |-  ( ( x  e.  Q.  /\  x  e.  ( 1st `  A ) )  -> 
( ( *Q `  x )  <Q  r  ->  r  e.  ( 2nd `  B ) ) )
4847reximdv 2463 . . . . 5  |-  ( ( x  e.  Q.  /\  x  e.  ( 1st `  A ) )  -> 
( E. r  e. 
Q.  ( *Q `  x )  <Q  r  ->  E. r  e.  Q.  r  e.  ( 2nd `  B ) ) )
4935, 48mpd 13 . . . 4  |-  ( ( x  e.  Q.  /\  x  e.  ( 1st `  A ) )  ->  E. r  e.  Q.  r  e.  ( 2nd `  B ) )
5049rexlimiva 2473 . . 3  |-  ( E. x  e.  Q.  x  e.  ( 1st `  A
)  ->  E. r  e.  Q.  r  e.  ( 2nd `  B ) )
511, 26, 503syl 17 . 2  |-  ( A  e.  P.  ->  E. r  e.  Q.  r  e.  ( 2nd `  B ) )
5225, 51jca 300 1  |-  ( A  e.  P.  ->  ( E. q  e.  Q.  q  e.  ( 1st `  B )  /\  E. r  e.  Q.  r  e.  ( 2nd `  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285   E.wex 1422    e. wcel 1434   {cab 2068   E.wrex 2350   <.cop 3403   class class class wbr 3787   ` cfv 4926  (class class class)co 5537   1stc1st 5790   2ndc2nd 5791   Q.cnq 6521   1Qc1q 6522    +Q cplq 6523   *Qcrq 6525    <Q cltq 6526   P.cnp 6532
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3895  ax-sep 3898  ax-nul 3906  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-iinf 4331
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-tr 3878  df-eprel 4046  df-id 4050  df-iord 4123  df-on 4125  df-suc 4128  df-iom 4334  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 6013  df-1o 6059  df-oadd 6063  df-omul 6064  df-er 6165  df-ec 6167  df-qs 6171  df-ni 6545  df-pli 6546  df-mi 6547  df-lti 6548  df-plpq 6585  df-mpq 6586  df-enq 6588  df-nqqs 6589  df-plqqs 6590  df-mqqs 6591  df-1nqqs 6592  df-rq 6593  df-ltnqqs 6594  df-inp 6707
This theorem is referenced by:  recexprlempr  6873
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