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Theorem prarloclem5 7276
Description: A substitution of zero for  y and  N minus two for  x. Lemma for prarloc 7279. (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 7275 . . . 4  |-  ( ( N  e.  N.  /\  1o  <N  N )  ->  E. x  e.  om  ( 2o  +o  x
)  =  N )
213adant2 985 . . 3  |-  ( ( N  e.  N.  /\  P  e.  Q.  /\  1o  <N  N )  ->  E. x  e.  om  ( 2o  +o  x )  =  N )
323ad2ant2 988 . 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 7257 . . . . . . 7  |-  ( (
<. L ,  U >.  e. 
P.  /\  A  e.  L )  ->  A  e.  Q. )
543ad2ant1 987 . . . . . 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 1000 . . . . . 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 7217 . . . . . . . . 9  |-  Q.  C_ Q0
87sseli 3063 . . . . . . . 8  |-  ( A  e.  Q.  ->  A  e. Q0 )
9 nq0a0 7233 . . . . . . . 8  |-  ( A  e. Q0  ->  ( A +Q0 0Q0 )  =  A )
108, 9syl 14 . . . . . . 7  |-  ( A  e.  Q.  ->  ( A +Q0 0Q0 )  =  A )
11 df-0nq0 7202 . . . . . . . . . 10  |- 0Q0  =  [ <. (/) ,  1o >. ] ~Q0
1211oveq1i 5752 . . . . . . . . 9  |-  (0Q0 ·Q0 
P )  =  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0 
P )
137sseli 3063 . . . . . . . . . 10  |-  ( P  e.  Q.  ->  P  e. Q0 )
14 nq0m0r 7232 . . . . . . . . . 10  |-  ( P  e. Q0  ->  (0Q0 ·Q0  P )  = 0Q0 )
1513, 14syl 14 . . . . . . . . 9  |-  ( P  e.  Q.  ->  (0Q0 ·Q0 
P )  = 0Q0 )
1612, 15syl5reqr 2165 . . . . . . . 8  |-  ( P  e.  Q.  -> 0Q0  =  ( [ <. (/)
,  1o >. ] ~Q0 ·Q0  P ) )
1716oveq2d 5758 . . . . . . 7  |-  ( P  e.  Q.  ->  ( A +Q0 0Q0 )  =  ( A +Q0  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0 
P ) ) )
1810, 17sylan9req 2171 . . . . . 6  |-  ( ( A  e.  Q.  /\  P  e.  Q. )  ->  A  =  ( A +Q0  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0 
P ) ) )
195, 6, 18syl2anc 408 . . . . 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 991 . . . . 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 2195 . . . 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 6385 . . . . . . . . . . . . . . 15  |-  2o  e.  om
23 nna0r 6342 . . . . . . . . . . . . . . 15  |-  ( 2o  e.  om  ->  ( (/) 
+o  2o )  =  2o )
2422, 23ax-mp 5 . . . . . . . . . . . . . 14  |-  ( (/)  +o  2o )  =  2o
2524oveq1i 5752 . . . . . . . . . . . . 13  |-  ( (
(/)  +o  2o )  +o  x )  =  ( 2o  +o  x )
2625eqeq1i 2125 . . . . . . . . . . . 12  |-  ( ( ( (/)  +o  2o )  +o  x )  =  N  <->  ( 2o  +o  x )  =  N )
2726biimpri 132 . . . . . . . . . . 11  |-  ( ( 2o  +o  x )  =  N  ->  (
( (/)  +o  2o )  +o  x )  =  N )
2827opeq1d 3681 . . . . . . . . . 10  |-  ( ( 2o  +o  x )  =  N  ->  <. (
( (/)  +o  2o )  +o  x ) ,  1o >.  =  <. N ,  1o >. )
2928eceq1d 6433 . . . . . . . . 9  |-  ( ( 2o  +o  x )  =  N  ->  [ <. ( ( (/)  +o  2o )  +o  x ) ,  1o >. ]  ~Q  =  [ <. N ,  1o >. ]  ~Q  )
3029oveq1d 5757 . . . . . . . 8  |-  ( ( 2o  +o  x )  =  N  ->  ( [ <. ( ( (/)  +o  2o )  +o  x
) ,  1o >. ]  ~Q  .Q  P )  =  ( [ <. N ,  1o >. ]  ~Q  .Q  P ) )
3130oveq2d 5758 . . . . . . 7  |-  ( ( 2o  +o  x )  =  N  ->  ( A  +Q  ( [ <. ( ( (/)  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P ) )  =  ( A  +Q  ( [ <. N ,  1o >. ]  ~Q  .Q  P
) ) )
3231eleq1d 2186 . . . . . 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 159 . . . . 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 989 . . . 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 4478 . . . . 5  |-  (/)  e.  om
36 opeq1 3675 . . . . . . . . . . 11  |-  ( y  =  (/)  ->  <. y ,  1o >.  =  <. (/)
,  1o >. )
3736eceq1d 6433 . . . . . . . . . 10  |-  ( y  =  (/)  ->  [ <. y ,  1o >. ] ~Q0  =  [ <. (/) ,  1o >. ] ~Q0  )
3837oveq1d 5757 . . . . . . . . 9  |-  ( y  =  (/)  ->  ( [
<. y ,  1o >. ] ~Q0 ·Q0 
P )  =  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0 
P ) )
3938oveq2d 5758 . . . . . . . 8  |-  ( y  =  (/)  ->  ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P )
)  =  ( A +Q0  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0 
P ) ) )
4039eleq1d 2186 . . . . . . 7  |-  ( y  =  (/)  ->  ( ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P )
)  e.  L  <->  ( A +Q0  ( [
<. (/) ,  1o >. ] ~Q0 ·Q0 
P ) )  e.  L ) )
41 oveq1 5749 . . . . . . . . . . . . 13  |-  ( y  =  (/)  ->  ( y  +o  2o )  =  ( (/)  +o  2o ) )
4241oveq1d 5757 . . . . . . . . . . . 12  |-  ( y  =  (/)  ->  ( ( y  +o  2o )  +o  x )  =  ( ( (/)  +o  2o )  +o  x ) )
4342opeq1d 3681 . . . . . . . . . . 11  |-  ( y  =  (/)  ->  <. (
( y  +o  2o )  +o  x ) ,  1o >.  =  <. ( ( (/)  +o  2o )  +o  x ) ,  1o >. )
4443eceq1d 6433 . . . . . . . . . 10  |-  ( y  =  (/)  ->  [ <. ( ( y  +o  2o )  +o  x ) ,  1o >. ]  ~Q  =  [ <. ( ( (/)  +o  2o )  +o  x
) ,  1o >. ]  ~Q  )
4544oveq1d 5757 . . . . . . . . 9  |-  ( y  =  (/)  ->  ( [
<. ( ( y  +o  2o )  +o  x
) ,  1o >. ]  ~Q  .Q  P )  =  ( [ <. ( ( (/)  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P ) )
4645oveq2d 5758 . . . . . . . 8  |-  ( y  =  (/)  ->  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P ) )  =  ( A  +Q  ( [ <. ( ( (/)  +o  2o )  +o  x
) ,  1o >. ]  ~Q  .Q  P ) ) )
4746eleq1d 2186 . . . . . . 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 464 . . . . . 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 2763 . . . . 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 420 . . . 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 1395 . . 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 2510 . 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 103    /\ w3a 947    = wceq 1316    e. wcel 1465   E.wrex 2394   (/)c0 3333   <.cop 3500   class class class wbr 3899   omcom 4474  (class class class)co 5742   1oc1o 6274   2oc2o 6275    +o coa 6278   [cec 6395   N.cnpi 7048    <N clti 7051    ~Q ceq 7055   Q.cnq 7056    +Q cplq 7058    .Q cmq 7059   ~Q0 ceq0 7062  Q0cnq0 7063  0Q0c0q0 7064   +Q0 cplq0 7065   ·Q0 cmq0 7066   P.cnp 7067
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-2o 6282  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-mi 7082  df-lti 7083  df-enq 7123  df-nqqs 7124  df-enq0 7200  df-nq0 7201  df-0nq0 7202  df-plq0 7203  df-mq0 7204  df-inp 7242
This theorem is referenced by:  prarloclem  7277
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