ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  prarloclem5 Unicode version

Theorem prarloclem5 7433
Description: A substitution of zero for  y and  N minus two for  x. Lemma for prarloc 7436. (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 7432 . . . 4  |-  ( ( N  e.  N.  /\  1o  <N  N )  ->  E. x  e.  om  ( 2o  +o  x
)  =  N )
213adant2 1005 . . 3  |-  ( ( N  e.  N.  /\  P  e.  Q.  /\  1o  <N  N )  ->  E. x  e.  om  ( 2o  +o  x )  =  N )
323ad2ant2 1008 . 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 7414 . . . . . . 7  |-  ( (
<. L ,  U >.  e. 
P.  /\  A  e.  L )  ->  A  e.  Q. )
543ad2ant1 1007 . . . . . 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 1020 . . . . . 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 7374 . . . . . . . . 9  |-  Q.  C_ Q0
87sseli 3134 . . . . . . . 8  |-  ( A  e.  Q.  ->  A  e. Q0 )
9 nq0a0 7390 . . . . . . . 8  |-  ( A  e. Q0  ->  ( A +Q0 0Q0 )  =  A )
108, 9syl 14 . . . . . . 7  |-  ( A  e.  Q.  ->  ( A +Q0 0Q0 )  =  A )
117sseli 3134 . . . . . . . . . 10  |-  ( P  e.  Q.  ->  P  e. Q0 )
12 nq0m0r 7389 . . . . . . . . . 10  |-  ( P  e. Q0  ->  (0Q0 ·Q0  P )  = 0Q0 )
1311, 12syl 14 . . . . . . . . 9  |-  ( P  e.  Q.  ->  (0Q0 ·Q0 
P )  = 0Q0 )
14 df-0nq0 7359 . . . . . . . . . 10  |- 0Q0  =  [ <. (/) ,  1o >. ] ~Q0
1514oveq1i 5847 . . . . . . . . 9  |-  (0Q0 ·Q0 
P )  =  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0 
P )
1613, 15eqtr3di 2212 . . . . . . . 8  |-  ( P  e.  Q.  -> 0Q0  =  ( [ <. (/)
,  1o >. ] ~Q0 ·Q0  P ) )
1716oveq2d 5853 . . . . . . 7  |-  ( P  e.  Q.  ->  ( A +Q0 0Q0 )  =  ( A +Q0  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0 
P ) ) )
1810, 17sylan9req 2218 . . . . . 6  |-  ( ( A  e.  Q.  /\  P  e.  Q. )  ->  A  =  ( A +Q0  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0 
P ) ) )
195, 6, 18syl2anc 409 . . . . 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 1011 . . . . 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 2242 . . . 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 6481 . . . . . . . . . . . . . . 15  |-  2o  e.  om
23 nna0r 6438 . . . . . . . . . . . . . . 15  |-  ( 2o  e.  om  ->  ( (/) 
+o  2o )  =  2o )
2422, 23ax-mp 5 . . . . . . . . . . . . . 14  |-  ( (/)  +o  2o )  =  2o
2524oveq1i 5847 . . . . . . . . . . . . 13  |-  ( (
(/)  +o  2o )  +o  x )  =  ( 2o  +o  x )
2625eqeq1i 2172 . . . . . . . . . . . 12  |-  ( ( ( (/)  +o  2o )  +o  x )  =  N  <->  ( 2o  +o  x )  =  N )
2726biimpri 132 . . . . . . . . . . 11  |-  ( ( 2o  +o  x )  =  N  ->  (
( (/)  +o  2o )  +o  x )  =  N )
2827opeq1d 3759 . . . . . . . . . 10  |-  ( ( 2o  +o  x )  =  N  ->  <. (
( (/)  +o  2o )  +o  x ) ,  1o >.  =  <. N ,  1o >. )
2928eceq1d 6529 . . . . . . . . 9  |-  ( ( 2o  +o  x )  =  N  ->  [ <. ( ( (/)  +o  2o )  +o  x ) ,  1o >. ]  ~Q  =  [ <. N ,  1o >. ]  ~Q  )
3029oveq1d 5852 . . . . . . . 8  |-  ( ( 2o  +o  x )  =  N  ->  ( [ <. ( ( (/)  +o  2o )  +o  x
) ,  1o >. ]  ~Q  .Q  P )  =  ( [ <. N ,  1o >. ]  ~Q  .Q  P ) )
3130oveq2d 5853 . . . . . . 7  |-  ( ( 2o  +o  x )  =  N  ->  ( A  +Q  ( [ <. ( ( (/)  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P ) )  =  ( A  +Q  ( [ <. N ,  1o >. ]  ~Q  .Q  P
) ) )
3231eleq1d 2233 . . . . . 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 1009 . . . 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 4566 . . . . 5  |-  (/)  e.  om
36 opeq1 3753 . . . . . . . . . . 11  |-  ( y  =  (/)  ->  <. y ,  1o >.  =  <. (/)
,  1o >. )
3736eceq1d 6529 . . . . . . . . . 10  |-  ( y  =  (/)  ->  [ <. y ,  1o >. ] ~Q0  =  [ <. (/) ,  1o >. ] ~Q0  )
3837oveq1d 5852 . . . . . . . . 9  |-  ( y  =  (/)  ->  ( [
<. y ,  1o >. ] ~Q0 ·Q0 
P )  =  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0 
P ) )
3938oveq2d 5853 . . . . . . . 8  |-  ( y  =  (/)  ->  ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P )
)  =  ( A +Q0  ( [ <. (/) ,  1o >. ] ~Q0 ·Q0 
P ) ) )
4039eleq1d 2233 . . . . . . 7  |-  ( y  =  (/)  ->  ( ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P )
)  e.  L  <->  ( A +Q0  ( [
<. (/) ,  1o >. ] ~Q0 ·Q0 
P ) )  e.  L ) )
41 oveq1 5844 . . . . . . . . . . . . 13  |-  ( y  =  (/)  ->  ( y  +o  2o )  =  ( (/)  +o  2o ) )
4241oveq1d 5852 . . . . . . . . . . . 12  |-  ( y  =  (/)  ->  ( ( y  +o  2o )  +o  x )  =  ( ( (/)  +o  2o )  +o  x ) )
4342opeq1d 3759 . . . . . . . . . . 11  |-  ( y  =  (/)  ->  <. (
( y  +o  2o )  +o  x ) ,  1o >.  =  <. ( ( (/)  +o  2o )  +o  x ) ,  1o >. )
4443eceq1d 6529 . . . . . . . . . 10  |-  ( y  =  (/)  ->  [ <. ( ( y  +o  2o )  +o  x ) ,  1o >. ]  ~Q  =  [ <. ( ( (/)  +o  2o )  +o  x
) ,  1o >. ]  ~Q  )
4544oveq1d 5852 . . . . . . . . 9  |-  ( y  =  (/)  ->  ( [
<. ( ( y  +o  2o )  +o  x
) ,  1o >. ]  ~Q  .Q  P )  =  ( [ <. ( ( (/)  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P ) )
4645oveq2d 5853 . . . . . . . 8  |-  ( y  =  (/)  ->  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P ) )  =  ( A  +Q  ( [ <. ( ( (/)  +o  2o )  +o  x
) ,  1o >. ]  ~Q  .Q  P ) ) )
4746eleq1d 2233 . . . . . . 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 465 . . . . . 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 2826 . . . . 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 421 . . . 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 1421 . . 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 2565 . 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 967    = wceq 1342    e. wcel 2135   E.wrex 2443   (/)c0 3405   <.cop 3574   class class class wbr 3977   omcom 4562  (class class class)co 5837   1oc1o 6369   2oc2o 6370    +o coa 6373   [cec 6491   N.cnpi 7205    <N clti 7208    ~Q ceq 7212   Q.cnq 7213    +Q cplq 7215    .Q cmq 7216   ~Q0 ceq0 7219  Q0cnq0 7220  0Q0c0q0 7221   +Q0 cplq0 7222   ·Q0 cmq0 7223   P.cnp 7224
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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4092  ax-sep 4095  ax-nul 4103  ax-pow 4148  ax-pr 4182  ax-un 4406  ax-setind 4509  ax-iinf 4560
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2724  df-sbc 2948  df-csb 3042  df-dif 3114  df-un 3116  df-in 3118  df-ss 3125  df-nul 3406  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-uni 3785  df-int 3820  df-iun 3863  df-br 3978  df-opab 4039  df-mpt 4040  df-tr 4076  df-eprel 4262  df-id 4266  df-iord 4339  df-on 4341  df-suc 4344  df-iom 4563  df-xp 4605  df-rel 4606  df-cnv 4607  df-co 4608  df-dm 4609  df-rn 4610  df-res 4611  df-ima 4612  df-iota 5148  df-fun 5185  df-fn 5186  df-f 5187  df-f1 5188  df-fo 5189  df-f1o 5190  df-fv 5191  df-ov 5840  df-oprab 5841  df-mpo 5842  df-1st 6101  df-2nd 6102  df-recs 6265  df-irdg 6330  df-1o 6376  df-2o 6377  df-oadd 6380  df-omul 6381  df-er 6493  df-ec 6495  df-qs 6499  df-ni 7237  df-mi 7239  df-lti 7240  df-enq 7280  df-nqqs 7281  df-enq0 7357  df-nq0 7358  df-0nq0 7359  df-plq0 7360  df-mq0 7361  df-inp 7399
This theorem is referenced by:  prarloclem  7434
  Copyright terms: Public domain W3C validator