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Theorem prarloclem5 6570
Description: A substitution of zero for  y and  N minus two for  x. Lemma for prarloc 6573. (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 6569 . . . 4  |-  ( ( N  e.  N.  /\  1o  <N  N )  ->  E. x  e.  om  ( 2o  +o  x
)  =  N )
213adant2 923 . . 3  |-  ( ( N  e.  N.  /\  P  e.  Q.  /\  1o  <N  N )  ->  E. x  e.  om  ( 2o  +o  x )  =  N )
323ad2ant2 926 . 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 6551 . . . . . . 7  |-  ( (
<. L ,  U >.  e. 
P.  /\  A  e.  L )  ->  A  e.  Q. )
543ad2ant1 925 . . . . . 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 938 . . . . . 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 6511 . . . . . . . . 9  |-  Q.  C_ Q0
87sseli 2938 . . . . . . . 8  |-  ( A  e.  Q.  ->  A  e. Q0 )
9 nq0a0 6527 . . . . . . . 8  |-  ( A  e. Q0  ->  ( A +Q0 0Q0 )  =  A )
108, 9syl 14 . . . . . . 7  |-  ( A  e.  Q.  ->  ( A +Q0 0Q0 )  =  A )
11 df-0nq0 6496 . . . . . . . . . 10  |- 0Q0  =  [ <. (/) ,  1o >. ] ~Q0
1211oveq1i 5500 . . . . . . . . 9  |-  (0Q0 ·Q0 
P )  =  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0 
P )
137sseli 2938 . . . . . . . . . 10  |-  ( P  e.  Q.  ->  P  e. Q0 )
14 nq0m0r 6526 . . . . . . . . . 10  |-  ( P  e. Q0  ->  (0Q0 ·Q0  P )  = 0Q0 )
1513, 14syl 14 . . . . . . . . 9  |-  ( P  e.  Q.  ->  (0Q0 ·Q0 
P )  = 0Q0 )
1612, 15syl5reqr 2087 . . . . . . . 8  |-  ( P  e.  Q.  -> 0Q0  =  ( [ <. (/)
,  1o >. ] ~Q0 ·Q0  P ) )
1716oveq2d 5506 . . . . . . 7  |-  ( P  e.  Q.  ->  ( A +Q0 0Q0 )  =  ( A +Q0  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0 
P ) ) )
1810, 17sylan9req 2093 . . . . . 6  |-  ( ( A  e.  Q.  /\  P  e.  Q. )  ->  A  =  ( A +Q0  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0 
P ) ) )
195, 6, 18syl2anc 391 . . . . 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 929 . . . . 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 2115 . . . 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 6072 . . . . . . . . . . . . . . 15  |-  2o  e.  om
23 nna0r 6035 . . . . . . . . . . . . . . 15  |-  ( 2o  e.  om  ->  ( (/) 
+o  2o )  =  2o )
2422, 23ax-mp 7 . . . . . . . . . . . . . 14  |-  ( (/)  +o  2o )  =  2o
2524oveq1i 5500 . . . . . . . . . . . . 13  |-  ( (
(/)  +o  2o )  +o  x )  =  ( 2o  +o  x )
2625eqeq1i 2047 . . . . . . . . . . . 12  |-  ( ( ( (/)  +o  2o )  +o  x )  =  N  <->  ( 2o  +o  x )  =  N )
2726biimpri 124 . . . . . . . . . . 11  |-  ( ( 2o  +o  x )  =  N  ->  (
( (/)  +o  2o )  +o  x )  =  N )
2827opeq1d 3552 . . . . . . . . . 10  |-  ( ( 2o  +o  x )  =  N  ->  <. (
( (/)  +o  2o )  +o  x ) ,  1o >.  =  <. N ,  1o >. )
2928eceq1d 6120 . . . . . . . . 9  |-  ( ( 2o  +o  x )  =  N  ->  [ <. ( ( (/)  +o  2o )  +o  x ) ,  1o >. ]  ~Q  =  [ <. N ,  1o >. ]  ~Q  )
3029oveq1d 5505 . . . . . . . 8  |-  ( ( 2o  +o  x )  =  N  ->  ( [ <. ( ( (/)  +o  2o )  +o  x
) ,  1o >. ]  ~Q  .Q  P )  =  ( [ <. N ,  1o >. ]  ~Q  .Q  P ) )
3130oveq2d 5506 . . . . . . 7  |-  ( ( 2o  +o  x )  =  N  ->  ( A  +Q  ( [ <. ( ( (/)  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P ) )  =  ( A  +Q  ( [ <. N ,  1o >. ]  ~Q  .Q  P
) ) )
3231eleq1d 2106 . . . . . 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 149 . . . . 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 927 . . . 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 4295 . . . . 5  |-  (/)  e.  om
36 opeq1 3546 . . . . . . . . . . 11  |-  ( y  =  (/)  ->  <. y ,  1o >.  =  <. (/)
,  1o >. )
3736eceq1d 6120 . . . . . . . . . 10  |-  ( y  =  (/)  ->  [ <. y ,  1o >. ] ~Q0  =  [ <. (/) ,  1o >. ] ~Q0  )
3837oveq1d 5505 . . . . . . . . 9  |-  ( y  =  (/)  ->  ( [
<. y ,  1o >. ] ~Q0 ·Q0 
P )  =  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0 
P ) )
3938oveq2d 5506 . . . . . . . 8  |-  ( y  =  (/)  ->  ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P )
)  =  ( A +Q0  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0 
P ) ) )
4039eleq1d 2106 . . . . . . 7  |-  ( y  =  (/)  ->  ( ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P )
)  e.  L  <->  ( A +Q0  ( [
<. (/) ,  1o >. ] ~Q0 ·Q0 
P ) )  e.  L ) )
41 oveq1 5497 . . . . . . . . . . . . 13  |-  ( y  =  (/)  ->  ( y  +o  2o )  =  ( (/)  +o  2o ) )
4241oveq1d 5505 . . . . . . . . . . . 12  |-  ( y  =  (/)  ->  ( ( y  +o  2o )  +o  x )  =  ( ( (/)  +o  2o )  +o  x ) )
4342opeq1d 3552 . . . . . . . . . . 11  |-  ( y  =  (/)  ->  <. (
( y  +o  2o )  +o  x ) ,  1o >.  =  <. ( ( (/)  +o  2o )  +o  x ) ,  1o >. )
4443eceq1d 6120 . . . . . . . . . 10  |-  ( y  =  (/)  ->  [ <. ( ( y  +o  2o )  +o  x ) ,  1o >. ]  ~Q  =  [ <. ( ( (/)  +o  2o )  +o  x
) ,  1o >. ]  ~Q  )
4544oveq1d 5505 . . . . . . . . 9  |-  ( y  =  (/)  ->  ( [
<. ( ( y  +o  2o )  +o  x
) ,  1o >. ]  ~Q  .Q  P )  =  ( [ <. ( ( (/)  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P ) )
4645oveq2d 5506 . . . . . . . 8  |-  ( y  =  (/)  ->  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P ) )  =  ( A  +Q  ( [ <. ( ( (/)  +o  2o )  +o  x
) ,  1o >. ]  ~Q  .Q  P ) ) )
4746eleq1d 2106 . . . . . . 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 442 . . . . . 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 2653 . . . . 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 400 . . . 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 1323 . . 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 2417 . 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 97    /\ w3a 885    = wceq 1243    e. wcel 1393   E.wrex 2304   (/)c0 3221   <.cop 3375   class class class wbr 3761   omcom 4291  (class class class)co 5490   1oc1o 5972   2oc2o 5973    +o coa 5976   [cec 6082   N.cnpi 6342    <N clti 6345    ~Q ceq 6349   Q.cnq 6350    +Q cplq 6352    .Q cmq 6353   ~Q0 ceq0 6356  Q0cnq0 6357  0Q0c0q0 6358   +Q0 cplq0 6359   ·Q0 cmq0 6360   P.cnp 6361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3869  ax-sep 3872  ax-nul 3880  ax-pow 3924  ax-pr 3941  ax-un 4157  ax-setind 4247  ax-iinf 4289
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-iun 3656  df-br 3762  df-opab 3816  df-mpt 3817  df-tr 3852  df-eprel 4023  df-id 4027  df-iord 4090  df-on 4092  df-suc 4095  df-iom 4292  df-xp 4329  df-rel 4330  df-cnv 4331  df-co 4332  df-dm 4333  df-rn 4334  df-res 4335  df-ima 4336  df-iota 4845  df-fun 4882  df-fn 4883  df-f 4884  df-f1 4885  df-fo 4886  df-f1o 4887  df-fv 4888  df-ov 5493  df-oprab 5494  df-mpt2 5495  df-1st 5745  df-2nd 5746  df-recs 5898  df-irdg 5935  df-1o 5979  df-2o 5980  df-oadd 5983  df-omul 5984  df-er 6084  df-ec 6086  df-qs 6090  df-ni 6374  df-mi 6376  df-lti 6377  df-enq 6417  df-nqqs 6418  df-enq0 6494  df-nq0 6495  df-0nq0 6496  df-plq0 6497  df-mq0 6498  df-inp 6536
This theorem is referenced by:  prarloclem  6571
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