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Theorem prarloclem5 6656
Description: A substitution of zero for  y and  N minus two for  x. Lemma for prarloc 6659. (Contributed by Jim Kingdon, 4-Nov-2019.)
Assertion
Ref Expression
prarloclem5  |-  ( ( ( <. L ,  U >.  e.  P.  /\  A  e.  L )  /\  ( N  e.  N.  /\  P  e.  Q.  /\  1o  <N  N )  /\  ( A  +Q  ( [ <. N ,  1o >. ]  ~Q  .Q  P ) )  e.  U )  ->  E. x  e.  om  E. y  e. 
om  ( ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P )
)  e.  L  /\  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P
) )  e.  U
) )
Distinct variable groups:    x, A, y   
x, L, y    x, N    x, P, y    x, U, y
Allowed substitution hint:    N( y)

Proof of Theorem prarloclem5
StepHypRef Expression
1 prarloclemn 6655 . . . 4  |-  ( ( N  e.  N.  /\  1o  <N  N )  ->  E. x  e.  om  ( 2o  +o  x
)  =  N )
213adant2 934 . . 3  |-  ( ( N  e.  N.  /\  P  e.  Q.  /\  1o  <N  N )  ->  E. x  e.  om  ( 2o  +o  x )  =  N )
323ad2ant2 937 . 2  |-  ( ( ( <. L ,  U >.  e.  P.  /\  A  e.  L )  /\  ( N  e.  N.  /\  P  e.  Q.  /\  1o  <N  N )  /\  ( A  +Q  ( [ <. N ,  1o >. ]  ~Q  .Q  P ) )  e.  U )  ->  E. x  e.  om  ( 2o  +o  x )  =  N )
4 elprnql 6637 . . . . . . 7  |-  ( (
<. L ,  U >.  e. 
P.  /\  A  e.  L )  ->  A  e.  Q. )
543ad2ant1 936 . . . . . 6  |-  ( ( ( <. L ,  U >.  e.  P.  /\  A  e.  L )  /\  ( N  e.  N.  /\  P  e.  Q.  /\  1o  <N  N )  /\  ( A  +Q  ( [ <. N ,  1o >. ]  ~Q  .Q  P ) )  e.  U )  ->  A  e.  Q. )
6 simp22 949 . . . . . 6  |-  ( ( ( <. L ,  U >.  e.  P.  /\  A  e.  L )  /\  ( N  e.  N.  /\  P  e.  Q.  /\  1o  <N  N )  /\  ( A  +Q  ( [ <. N ,  1o >. ]  ~Q  .Q  P ) )  e.  U )  ->  P  e.  Q. )
7 nqnq0 6597 . . . . . . . . 9  |-  Q.  C_ Q0
87sseli 2969 . . . . . . . 8  |-  ( A  e.  Q.  ->  A  e. Q0 )
9 nq0a0 6613 . . . . . . . 8  |-  ( A  e. Q0  ->  ( A +Q0 0Q0 )  =  A )
108, 9syl 14 . . . . . . 7  |-  ( A  e.  Q.  ->  ( A +Q0 0Q0 )  =  A )
11 df-0nq0 6582 . . . . . . . . . 10  |- 0Q0  =  [ <. (/) ,  1o >. ] ~Q0
1211oveq1i 5550 . . . . . . . . 9  |-  (0Q0 ·Q0 
P )  =  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0 
P )
137sseli 2969 . . . . . . . . . 10  |-  ( P  e.  Q.  ->  P  e. Q0 )
14 nq0m0r 6612 . . . . . . . . . 10  |-  ( P  e. Q0  ->  (0Q0 ·Q0  P )  = 0Q0 )
1513, 14syl 14 . . . . . . . . 9  |-  ( P  e.  Q.  ->  (0Q0 ·Q0 
P )  = 0Q0 )
1612, 15syl5reqr 2103 . . . . . . . 8  |-  ( P  e.  Q.  -> 0Q0  =  ( [ <. (/)
,  1o >. ] ~Q0 ·Q0  P ) )
1716oveq2d 5556 . . . . . . 7  |-  ( P  e.  Q.  ->  ( A +Q0 0Q0 )  =  ( A +Q0  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0 
P ) ) )
1810, 17sylan9req 2109 . . . . . 6  |-  ( ( A  e.  Q.  /\  P  e.  Q. )  ->  A  =  ( A +Q0  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0 
P ) ) )
195, 6, 18syl2anc 397 . . . . 5  |-  ( ( ( <. L ,  U >.  e.  P.  /\  A  e.  L )  /\  ( N  e.  N.  /\  P  e.  Q.  /\  1o  <N  N )  /\  ( A  +Q  ( [ <. N ,  1o >. ]  ~Q  .Q  P ) )  e.  U )  ->  A  =  ( A +Q0  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0  P )
) )
20 simp1r 940 . . . . 5  |-  ( ( ( <. L ,  U >.  e.  P.  /\  A  e.  L )  /\  ( N  e.  N.  /\  P  e.  Q.  /\  1o  <N  N )  /\  ( A  +Q  ( [ <. N ,  1o >. ]  ~Q  .Q  P ) )  e.  U )  ->  A  e.  L )
2119, 20eqeltrrd 2131 . . . 4  |-  ( ( ( <. L ,  U >.  e.  P.  /\  A  e.  L )  /\  ( N  e.  N.  /\  P  e.  Q.  /\  1o  <N  N )  /\  ( A  +Q  ( [ <. N ,  1o >. ]  ~Q  .Q  P ) )  e.  U )  ->  ( A +Q0  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0 
P ) )  e.  L )
22 2onn 6125 . . . . . . . . . . . . . . 15  |-  2o  e.  om
23 nna0r 6088 . . . . . . . . . . . . . . 15  |-  ( 2o  e.  om  ->  ( (/) 
+o  2o )  =  2o )
2422, 23ax-mp 7 . . . . . . . . . . . . . 14  |-  ( (/)  +o  2o )  =  2o
2524oveq1i 5550 . . . . . . . . . . . . 13  |-  ( (
(/)  +o  2o )  +o  x )  =  ( 2o  +o  x )
2625eqeq1i 2063 . . . . . . . . . . . 12  |-  ( ( ( (/)  +o  2o )  +o  x )  =  N  <->  ( 2o  +o  x )  =  N )
2726biimpri 128 . . . . . . . . . . 11  |-  ( ( 2o  +o  x )  =  N  ->  (
( (/)  +o  2o )  +o  x )  =  N )
2827opeq1d 3583 . . . . . . . . . 10  |-  ( ( 2o  +o  x )  =  N  ->  <. (
( (/)  +o  2o )  +o  x ) ,  1o >.  =  <. N ,  1o >. )
2928eceq1d 6173 . . . . . . . . 9  |-  ( ( 2o  +o  x )  =  N  ->  [ <. ( ( (/)  +o  2o )  +o  x ) ,  1o >. ]  ~Q  =  [ <. N ,  1o >. ]  ~Q  )
3029oveq1d 5555 . . . . . . . 8  |-  ( ( 2o  +o  x )  =  N  ->  ( [ <. ( ( (/)  +o  2o )  +o  x
) ,  1o >. ]  ~Q  .Q  P )  =  ( [ <. N ,  1o >. ]  ~Q  .Q  P ) )
3130oveq2d 5556 . . . . . . 7  |-  ( ( 2o  +o  x )  =  N  ->  ( A  +Q  ( [ <. ( ( (/)  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P ) )  =  ( A  +Q  ( [ <. N ,  1o >. ]  ~Q  .Q  P
) ) )
3231eleq1d 2122 . . . . . 6  |-  ( ( 2o  +o  x )  =  N  ->  (
( A  +Q  ( [ <. ( ( (/)  +o  2o )  +o  x
) ,  1o >. ]  ~Q  .Q  P ) )  e.  U  <->  ( A  +Q  ( [ <. N ,  1o >. ]  ~Q  .Q  P ) )  e.  U ) )
3332biimprcd 153 . . . . 5  |-  ( ( A  +Q  ( [
<. N ,  1o >. ]  ~Q  .Q  P ) )  e.  U  -> 
( ( 2o  +o  x )  =  N  ->  ( A  +Q  ( [ <. ( ( (/)  +o  2o )  +o  x
) ,  1o >. ]  ~Q  .Q  P ) )  e.  U ) )
34333ad2ant3 938 . . . 4  |-  ( ( ( <. L ,  U >.  e.  P.  /\  A  e.  L )  /\  ( N  e.  N.  /\  P  e.  Q.  /\  1o  <N  N )  /\  ( A  +Q  ( [ <. N ,  1o >. ]  ~Q  .Q  P ) )  e.  U )  ->  (
( 2o  +o  x
)  =  N  -> 
( A  +Q  ( [ <. ( ( (/)  +o  2o )  +o  x
) ,  1o >. ]  ~Q  .Q  P ) )  e.  U ) )
35 peano1 4345 . . . . 5  |-  (/)  e.  om
36 opeq1 3577 . . . . . . . . . . 11  |-  ( y  =  (/)  ->  <. y ,  1o >.  =  <. (/)
,  1o >. )
3736eceq1d 6173 . . . . . . . . . 10  |-  ( y  =  (/)  ->  [ <. y ,  1o >. ] ~Q0  =  [ <. (/) ,  1o >. ] ~Q0  )
3837oveq1d 5555 . . . . . . . . 9  |-  ( y  =  (/)  ->  ( [
<. y ,  1o >. ] ~Q0 ·Q0 
P )  =  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0 
P ) )
3938oveq2d 5556 . . . . . . . 8  |-  ( y  =  (/)  ->  ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P )
)  =  ( A +Q0  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0 
P ) ) )
4039eleq1d 2122 . . . . . . 7  |-  ( y  =  (/)  ->  ( ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P )
)  e.  L  <->  ( A +Q0  ( [
<. (/) ,  1o >. ] ~Q0 ·Q0 
P ) )  e.  L ) )
41 oveq1 5547 . . . . . . . . . . . . 13  |-  ( y  =  (/)  ->  ( y  +o  2o )  =  ( (/)  +o  2o ) )
4241oveq1d 5555 . . . . . . . . . . . 12  |-  ( y  =  (/)  ->  ( ( y  +o  2o )  +o  x )  =  ( ( (/)  +o  2o )  +o  x ) )
4342opeq1d 3583 . . . . . . . . . . 11  |-  ( y  =  (/)  ->  <. (
( y  +o  2o )  +o  x ) ,  1o >.  =  <. ( ( (/)  +o  2o )  +o  x ) ,  1o >. )
4443eceq1d 6173 . . . . . . . . . 10  |-  ( y  =  (/)  ->  [ <. ( ( y  +o  2o )  +o  x ) ,  1o >. ]  ~Q  =  [ <. ( ( (/)  +o  2o )  +o  x
) ,  1o >. ]  ~Q  )
4544oveq1d 5555 . . . . . . . . 9  |-  ( y  =  (/)  ->  ( [
<. ( ( y  +o  2o )  +o  x
) ,  1o >. ]  ~Q  .Q  P )  =  ( [ <. ( ( (/)  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P ) )
4645oveq2d 5556 . . . . . . . 8  |-  ( y  =  (/)  ->  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P ) )  =  ( A  +Q  ( [ <. ( ( (/)  +o  2o )  +o  x
) ,  1o >. ]  ~Q  .Q  P ) ) )
4746eleq1d 2122 . . . . . . 7  |-  ( y  =  (/)  ->  ( ( A  +Q  ( [
<. ( ( y  +o  2o )  +o  x
) ,  1o >. ]  ~Q  .Q  P ) )  e.  U  <->  ( A  +Q  ( [ <. (
( (/)  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P ) )  e.  U ) )
4840, 47anbi12d 450 . . . . . 6  |-  ( y  =  (/)  ->  ( ( ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P ) )  e.  L  /\  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P
) )  e.  U
)  <->  ( ( A +Q0  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0 
P ) )  e.  L  /\  ( A  +Q  ( [ <. ( ( (/)  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P ) )  e.  U ) ) )
4948rspcev 2673 . . . . 5  |-  ( (
(/)  e.  om  /\  (
( A +Q0  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0  P )
)  e.  L  /\  ( A  +Q  ( [ <. ( ( (/)  +o  2o )  +o  x
) ,  1o >. ]  ~Q  .Q  P ) )  e.  U ) )  ->  E. y  e.  om  ( ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P )
)  e.  L  /\  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P
) )  e.  U
) )
5035, 49mpan 408 . . . 4  |-  ( ( ( A +Q0  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0  P )
)  e.  L  /\  ( A  +Q  ( [ <. ( ( (/)  +o  2o )  +o  x
) ,  1o >. ]  ~Q  .Q  P ) )  e.  U )  ->  E. y  e.  om  ( ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P ) )  e.  L  /\  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P
) )  e.  U
) )
5121, 34, 50syl6an 1339 . . 3  |-  ( ( ( <. L ,  U >.  e.  P.  /\  A  e.  L )  /\  ( N  e.  N.  /\  P  e.  Q.  /\  1o  <N  N )  /\  ( A  +Q  ( [ <. N ,  1o >. ]  ~Q  .Q  P ) )  e.  U )  ->  (
( 2o  +o  x
)  =  N  ->  E. y  e.  om  ( ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P ) )  e.  L  /\  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P
) )  e.  U
) ) )
5251reximdv 2437 . 2  |-  ( ( ( <. L ,  U >.  e.  P.  /\  A  e.  L )  /\  ( N  e.  N.  /\  P  e.  Q.  /\  1o  <N  N )  /\  ( A  +Q  ( [ <. N ,  1o >. ]  ~Q  .Q  P ) )  e.  U )  ->  ( E. x  e.  om  ( 2o  +o  x
)  =  N  ->  E. x  e.  om  E. y  e.  om  (
( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P ) )  e.  L  /\  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P
) )  e.  U
) ) )
533, 52mpd 13 1  |-  ( ( ( <. L ,  U >.  e.  P.  /\  A  e.  L )  /\  ( N  e.  N.  /\  P  e.  Q.  /\  1o  <N  N )  /\  ( A  +Q  ( [ <. N ,  1o >. ]  ~Q  .Q  P ) )  e.  U )  ->  E. x  e.  om  E. y  e. 
om  ( ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P )
)  e.  L  /\  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P
) )  e.  U
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    /\ w3a 896    = wceq 1259    e. wcel 1409   E.wrex 2324   (/)c0 3252   <.cop 3406   class class class wbr 3792   omcom 4341  (class class class)co 5540   1oc1o 6025   2oc2o 6026    +o coa 6029   [cec 6135   N.cnpi 6428    <N clti 6431    ~Q ceq 6435   Q.cnq 6436    +Q cplq 6438    .Q cmq 6439   ~Q0 ceq0 6442  Q0cnq0 6443  0Q0c0q0 6444   +Q0 cplq0 6445   ·Q0 cmq0 6446   P.cnp 6447
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-mi 6462  df-lti 6463  df-enq 6503  df-nqqs 6504  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622
This theorem is referenced by:  prarloclem  6657
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