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Theorem prarloclem5 7498
Description: A substitution of zero for  y and  N minus two for  x. Lemma for prarloc 7501. (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 7497 . . . 4  |-  ( ( N  e.  N.  /\  1o  <N  N )  ->  E. x  e.  om  ( 2o  +o  x
)  =  N )
213adant2 1016 . . 3  |-  ( ( N  e.  N.  /\  P  e.  Q.  /\  1o  <N  N )  ->  E. x  e.  om  ( 2o  +o  x )  =  N )
323ad2ant2 1019 . 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 7479 . . . . . . 7  |-  ( (
<. L ,  U >.  e. 
P.  /\  A  e.  L )  ->  A  e.  Q. )
543ad2ant1 1018 . . . . . 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 1031 . . . . . 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 7439 . . . . . . . . 9  |-  Q.  C_ Q0
87sseli 3151 . . . . . . . 8  |-  ( A  e.  Q.  ->  A  e. Q0 )
9 nq0a0 7455 . . . . . . . 8  |-  ( A  e. Q0  ->  ( A +Q0 0Q0 )  =  A )
108, 9syl 14 . . . . . . 7  |-  ( A  e.  Q.  ->  ( A +Q0 0Q0 )  =  A )
117sseli 3151 . . . . . . . . . 10  |-  ( P  e.  Q.  ->  P  e. Q0 )
12 nq0m0r 7454 . . . . . . . . . 10  |-  ( P  e. Q0  ->  (0Q0 ·Q0  P )  = 0Q0 )
1311, 12syl 14 . . . . . . . . 9  |-  ( P  e.  Q.  ->  (0Q0 ·Q0 
P )  = 0Q0 )
14 df-0nq0 7424 . . . . . . . . . 10  |- 0Q0  =  [ <. (/) ,  1o >. ] ~Q0
1514oveq1i 5884 . . . . . . . . 9  |-  (0Q0 ·Q0 
P )  =  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0 
P )
1613, 15eqtr3di 2225 . . . . . . . 8  |-  ( P  e.  Q.  -> 0Q0  =  ( [ <. (/)
,  1o >. ] ~Q0 ·Q0  P ) )
1716oveq2d 5890 . . . . . . 7  |-  ( P  e.  Q.  ->  ( A +Q0 0Q0 )  =  ( A +Q0  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0 
P ) ) )
1810, 17sylan9req 2231 . . . . . 6  |-  ( ( A  e.  Q.  /\  P  e.  Q. )  ->  A  =  ( A +Q0  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0 
P ) ) )
195, 6, 18syl2anc 411 . . . . 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 1022 . . . . 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 2255 . . . 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 6521 . . . . . . . . . . . . . . 15  |-  2o  e.  om
23 nna0r 6478 . . . . . . . . . . . . . . 15  |-  ( 2o  e.  om  ->  ( (/) 
+o  2o )  =  2o )
2422, 23ax-mp 5 . . . . . . . . . . . . . 14  |-  ( (/)  +o  2o )  =  2o
2524oveq1i 5884 . . . . . . . . . . . . 13  |-  ( (
(/)  +o  2o )  +o  x )  =  ( 2o  +o  x )
2625eqeq1i 2185 . . . . . . . . . . . 12  |-  ( ( ( (/)  +o  2o )  +o  x )  =  N  <->  ( 2o  +o  x )  =  N )
2726biimpri 133 . . . . . . . . . . 11  |-  ( ( 2o  +o  x )  =  N  ->  (
( (/)  +o  2o )  +o  x )  =  N )
2827opeq1d 3784 . . . . . . . . . 10  |-  ( ( 2o  +o  x )  =  N  ->  <. (
( (/)  +o  2o )  +o  x ) ,  1o >.  =  <. N ,  1o >. )
2928eceq1d 6570 . . . . . . . . 9  |-  ( ( 2o  +o  x )  =  N  ->  [ <. ( ( (/)  +o  2o )  +o  x ) ,  1o >. ]  ~Q  =  [ <. N ,  1o >. ]  ~Q  )
3029oveq1d 5889 . . . . . . . 8  |-  ( ( 2o  +o  x )  =  N  ->  ( [ <. ( ( (/)  +o  2o )  +o  x
) ,  1o >. ]  ~Q  .Q  P )  =  ( [ <. N ,  1o >. ]  ~Q  .Q  P ) )
3130oveq2d 5890 . . . . . . 7  |-  ( ( 2o  +o  x )  =  N  ->  ( A  +Q  ( [ <. ( ( (/)  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P ) )  =  ( A  +Q  ( [ <. N ,  1o >. ]  ~Q  .Q  P
) ) )
3231eleq1d 2246 . . . . . 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 160 . . . . 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 1020 . . . 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 4593 . . . . 5  |-  (/)  e.  om
36 opeq1 3778 . . . . . . . . . . 11  |-  ( y  =  (/)  ->  <. y ,  1o >.  =  <. (/)
,  1o >. )
3736eceq1d 6570 . . . . . . . . . 10  |-  ( y  =  (/)  ->  [ <. y ,  1o >. ] ~Q0  =  [ <. (/) ,  1o >. ] ~Q0  )
3837oveq1d 5889 . . . . . . . . 9  |-  ( y  =  (/)  ->  ( [
<. y ,  1o >. ] ~Q0 ·Q0 
P )  =  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0 
P ) )
3938oveq2d 5890 . . . . . . . 8  |-  ( y  =  (/)  ->  ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P )
)  =  ( A +Q0  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0 
P ) ) )
4039eleq1d 2246 . . . . . . 7  |-  ( y  =  (/)  ->  ( ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P )
)  e.  L  <->  ( A +Q0  ( [
<. (/) ,  1o >. ] ~Q0 ·Q0 
P ) )  e.  L ) )
41 oveq1 5881 . . . . . . . . . . . . 13  |-  ( y  =  (/)  ->  ( y  +o  2o )  =  ( (/)  +o  2o ) )
4241oveq1d 5889 . . . . . . . . . . . 12  |-  ( y  =  (/)  ->  ( ( y  +o  2o )  +o  x )  =  ( ( (/)  +o  2o )  +o  x ) )
4342opeq1d 3784 . . . . . . . . . . 11  |-  ( y  =  (/)  ->  <. (
( y  +o  2o )  +o  x ) ,  1o >.  =  <. ( ( (/)  +o  2o )  +o  x ) ,  1o >. )
4443eceq1d 6570 . . . . . . . . . 10  |-  ( y  =  (/)  ->  [ <. ( ( y  +o  2o )  +o  x ) ,  1o >. ]  ~Q  =  [ <. ( ( (/)  +o  2o )  +o  x
) ,  1o >. ]  ~Q  )
4544oveq1d 5889 . . . . . . . . 9  |-  ( y  =  (/)  ->  ( [
<. ( ( y  +o  2o )  +o  x
) ,  1o >. ]  ~Q  .Q  P )  =  ( [ <. ( ( (/)  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P ) )
4645oveq2d 5890 . . . . . . . 8  |-  ( y  =  (/)  ->  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P ) )  =  ( A  +Q  ( [ <. ( ( (/)  +o  2o )  +o  x
) ,  1o >. ]  ~Q  .Q  P ) ) )
4746eleq1d 2246 . . . . . . 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 473 . . . . . 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 2841 . . . . 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 424 . . . 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 1434 . . 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 2578 . 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 104    /\ w3a 978    = wceq 1353    e. wcel 2148   E.wrex 2456   (/)c0 3422   <.cop 3595   class class class wbr 4003   omcom 4589  (class class class)co 5874   1oc1o 6409   2oc2o 6410    +o coa 6413   [cec 6532   N.cnpi 7270    <N clti 7273    ~Q ceq 7277   Q.cnq 7278    +Q cplq 7280    .Q cmq 7281   ~Q0 ceq0 7284  Q0cnq0 7285  0Q0c0q0 7286   +Q0 cplq0 7287   ·Q0 cmq0 7288   P.cnp 7289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-2o 6417  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-mi 7304  df-lti 7305  df-enq 7345  df-nqqs 7346  df-enq0 7422  df-nq0 7423  df-0nq0 7424  df-plq0 7425  df-mq0 7426  df-inp 7464
This theorem is referenced by:  prarloclem  7499
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