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Theorem prarloclemcalc 7059
Description: Some calculations for prarloc 7060. (Contributed by Jim Kingdon, 26-Oct-2019.)
Assertion
Ref Expression
prarloclemcalc  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  B  <Q  ( A  +Q  P
) )

Proof of Theorem prarloclemcalc
StepHypRef Expression
1 simprll 504 . . . . 5  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  Q  e.  Q. )
2 nqnq0a 7011 . . . . 5  |-  ( ( Q  e.  Q.  /\  Q  e.  Q. )  ->  ( Q  +Q  Q
)  =  ( Q +Q0  Q
) )
31, 1, 2syl2anc 403 . . . 4  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  ( Q  +Q  Q )  =  ( Q +Q0  Q ) )
43oveq2d 5668 . . 3  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  ( A +Q0  ( Q  +Q  Q ) )  =  ( A +Q0  ( Q +Q0  Q ) ) )
5 simpll 496 . . . . 5  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q ) ) )
6 simprrl 506 . . . . . 6  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  X  e.  Q. )
7 simprrr 507 . . . . . . . 8  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  M  e.  om )
8 1pi 6872 . . . . . . . . . . 11  |-  1o  e.  N.
9 opelxpi 4469 . . . . . . . . . . 11  |-  ( ( M  e.  om  /\  1o  e.  N. )  ->  <. M ,  1o >.  e.  ( om  X.  N. ) )
108, 9mpan2 416 . . . . . . . . . 10  |-  ( M  e.  om  ->  <. M ,  1o >.  e.  ( om 
X.  N. ) )
11 enq0ex 6996 . . . . . . . . . . 11  |- ~Q0  e.  _V
1211ecelqsi 6344 . . . . . . . . . 10  |-  ( <. M ,  1o >.  e.  ( om  X.  N. )  ->  [ <. M ,  1o >. ] ~Q0  e.  ( ( om  X.  N. ) /. ~Q0  ) )
1310, 12syl 14 . . . . . . . . 9  |-  ( M  e.  om  ->  [ <. M ,  1o >. ] ~Q0  e.  ( ( om 
X.  N. ) /. ~Q0  ) )
14 df-nq0 6982 . . . . . . . . 9  |- Q0  =  ( ( om 
X.  N. ) /. ~Q0  )
1513, 14syl6eleqr 2181 . . . . . . . 8  |-  ( M  e.  om  ->  [ <. M ,  1o >. ] ~Q0  e. Q0 )
167, 15syl 14 . . . . . . 7  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  [ <. M ,  1o >. ] ~Q0  e. Q0 )
17 nqnq0 6998 . . . . . . . 8  |-  Q.  C_ Q0
1817, 1sseldi 3023 . . . . . . 7  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  Q  e. Q0 )
19 mulclnq0 7009 . . . . . . 7  |-  ( ( [ <. M ,  1o >. ] ~Q0  e. Q0  /\  Q  e. Q0 )  ->  ( [ <. M ,  1o >. ] ~Q0 ·Q0 
Q )  e. Q0 )
2016, 18, 19syl2anc 403 . . . . . 6  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )  e. Q0 )
21 nqpnq0nq 7010 . . . . . 6  |-  ( ( X  e.  Q.  /\  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )  e. Q0 )  ->  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q ) )  e.  Q. )
226, 20, 21syl2anc 403 . . . . 5  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  e.  Q. )
235, 22eqeltrd 2164 . . . 4  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  A  e.  Q. )
24 addclnq 6932 . . . . 5  |-  ( ( Q  e.  Q.  /\  Q  e.  Q. )  ->  ( Q  +Q  Q
)  e.  Q. )
251, 1, 24syl2anc 403 . . . 4  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  ( Q  +Q  Q )  e. 
Q. )
26 nqnq0a 7011 . . . 4  |-  ( ( A  e.  Q.  /\  ( Q  +Q  Q
)  e.  Q. )  ->  ( A  +Q  ( Q  +Q  Q ) )  =  ( A +Q0  ( Q  +Q  Q
) ) )
2723, 25, 26syl2anc 403 . . 3  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  ( A  +Q  ( Q  +Q  Q ) )  =  ( A +Q0  ( Q  +Q  Q
) ) )
28 simplr 497 . . . . . 6  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )
29 2onn 6278 . . . . . . . . . . . . . 14  |-  2o  e.  om
30 2on0 6191 . . . . . . . . . . . . . 14  |-  2o  =/=  (/)
31 elni 6865 . . . . . . . . . . . . . 14  |-  ( 2o  e.  N.  <->  ( 2o  e.  om  /\  2o  =/=  (/) ) )
3229, 30, 31mpbir2an 888 . . . . . . . . . . . . 13  |-  2o  e.  N.
33 nnppipi 6900 . . . . . . . . . . . . 13  |-  ( ( M  e.  om  /\  2o  e.  N. )  -> 
( M  +o  2o )  e.  N. )
3432, 33mpan2 416 . . . . . . . . . . . 12  |-  ( M  e.  om  ->  ( M  +o  2o )  e. 
N. )
35 opelxpi 4469 . . . . . . . . . . . 12  |-  ( ( ( M  +o  2o )  e.  N.  /\  1o  e.  N. )  ->  <. ( M  +o  2o ) ,  1o >.  e.  ( N.  X.  N. ) )
3634, 8, 35sylancl 404 . . . . . . . . . . 11  |-  ( M  e.  om  ->  <. ( M  +o  2o ) ,  1o >.  e.  ( N.  X.  N. ) )
37 enqex 6917 . . . . . . . . . . . 12  |-  ~Q  e.  _V
3837ecelqsi 6344 . . . . . . . . . . 11  |-  ( <.
( M  +o  2o ) ,  1o >.  e.  ( N.  X.  N. )  ->  [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  e.  ( ( N.  X.  N. ) /.  ~Q  ) )
3936, 38syl 14 . . . . . . . . . 10  |-  ( M  e.  om  ->  [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  e.  ( ( N.  X.  N. ) /.  ~Q  )
)
40 df-nqqs 6905 . . . . . . . . . 10  |-  Q.  =  ( ( N.  X.  N. ) /.  ~Q  )
4139, 40syl6eleqr 2181 . . . . . . . . 9  |-  ( M  e.  om  ->  [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  e.  Q. )
427, 41syl 14 . . . . . . . 8  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  e.  Q. )
43 mulclnq 6933 . . . . . . . 8  |-  ( ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  e.  Q.  /\  Q  e.  Q. )  ->  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q )  e.  Q. )
4442, 1, 43syl2anc 403 . . . . . . 7  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q )  e.  Q. )
45 nqnq0a 7011 . . . . . . 7  |-  ( ( X  e.  Q.  /\  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q )  e.  Q. )  -> 
( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) )  =  ( X +Q0  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )
466, 44, 45syl2anc 403 . . . . . 6  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) )  =  ( X +Q0  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )
47 nqnq0m 7012 . . . . . . . . 9  |-  ( ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  e.  Q.  /\  Q  e.  Q. )  ->  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q )  =  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q ·Q0  Q ) )
4842, 1, 47syl2anc 403 . . . . . . . 8  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q )  =  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q ·Q0  Q )
)
49 nqnq0pi 6995 . . . . . . . . . . 11  |-  ( ( ( M  +o  2o )  e.  N.  /\  1o  e.  N. )  ->  [ <. ( M  +o  2o ) ,  1o >. ] ~Q0  =  [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  )
5034, 8, 49sylancl 404 . . . . . . . . . 10  |-  ( M  e.  om  ->  [ <. ( M  +o  2o ) ,  1o >. ] ~Q0  =  [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  )
517, 50syl 14 . . . . . . . . 9  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  [ <. ( M  +o  2o ) ,  1o >. ] ~Q0  =  [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  )
5251oveq1d 5667 . . . . . . . 8  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  ( [ <. ( M  +o  2o ) ,  1o >. ] ~Q0 ·Q0 
Q )  =  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q ·Q0  Q ) )
5348, 52eqtr4d 2123 . . . . . . 7  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q )  =  ( [ <. ( M  +o  2o ) ,  1o >. ] ~Q0 ·Q0  Q ) )
5453oveq2d 5668 . . . . . 6  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  ( X +Q0  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) )  =  ( X +Q0  ( [ <. ( M  +o  2o ) ,  1o >. ] ~Q0 ·Q0 
Q ) ) )
5528, 46, 543eqtrd 2124 . . . . 5  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  B  =  ( X +Q0  ( [ <. ( M  +o  2o ) ,  1o >. ] ~Q0 ·Q0  Q ) ) )
56 nnanq0 7015 . . . . . . . . . 10  |-  ( ( M  e.  om  /\  2o  e.  om  /\  1o  e.  N. )  ->  [ <. ( M  +o  2o ) ,  1o >. ] ~Q0  =  ( [ <. M ,  1o >. ] ~Q0 +Q0  [ <. 2o ,  1o >. ] ~Q0  )
)
578, 56mp3an3 1262 . . . . . . . . 9  |-  ( ( M  e.  om  /\  2o  e.  om )  ->  [ <. ( M  +o  2o ) ,  1o >. ] ~Q0  =  ( [ <. M ,  1o >. ] ~Q0 +Q0  [ <. 2o ,  1o >. ] ~Q0  ) )
587, 29, 57sylancl 404 . . . . . . . 8  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  [ <. ( M  +o  2o ) ,  1o >. ] ~Q0  =  ( [ <. M ,  1o >. ] ~Q0 +Q0  [ <. 2o ,  1o >. ] ~Q0  )
)
5958oveq1d 5667 . . . . . . 7  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  ( [ <. ( M  +o  2o ) ,  1o >. ] ~Q0 ·Q0 
Q )  =  ( ( [ <. M ,  1o >. ] ~Q0 +Q0  [ <. 2o ,  1o >. ] ~Q0  ) ·Q0  Q ) )
60 opelxpi 4469 . . . . . . . . . . . 12  |-  ( ( 2o  e.  om  /\  1o  e.  N. )  ->  <. 2o ,  1o >.  e.  ( om  X.  N. ) )
6129, 8, 60mp2an 417 . . . . . . . . . . 11  |-  <. 2o ,  1o >.  e.  ( om 
X.  N. )
6211ecelqsi 6344 . . . . . . . . . . 11  |-  ( <. 2o ,  1o >.  e.  ( om  X.  N. )  ->  [ <. 2o ,  1o >. ] ~Q0  e.  ( ( om  X.  N. ) /. ~Q0  ) )
6361, 62ax-mp 7 . . . . . . . . . 10  |-  [ <. 2o ,  1o >. ] ~Q0  e.  ( ( om 
X.  N. ) /. ~Q0  )
6463, 14eleqtrri 2163 . . . . . . . . 9  |-  [ <. 2o ,  1o >. ] ~Q0  e. Q0
65 distnq0r 7020 . . . . . . . . 9  |-  ( ( Q  e. Q0  /\  [ <. M ,  1o >. ] ~Q0  e. Q0  /\  [ <. 2o ,  1o >. ] ~Q0  e. Q0 )  ->  ( ( [
<. M ,  1o >. ] ~Q0 +Q0  [ <. 2o ,  1o >. ] ~Q0  ) ·Q0  Q
)  =  ( ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q ) +Q0  ( [
<. 2o ,  1o >. ] ~Q0 ·Q0 
Q ) ) )
6664, 65mp3an3 1262 . . . . . . . 8  |-  ( ( Q  e. Q0  /\  [ <. M ,  1o >. ] ~Q0  e. Q0 )  ->  ( ( [
<. M ,  1o >. ] ~Q0 +Q0  [ <. 2o ,  1o >. ] ~Q0  ) ·Q0  Q
)  =  ( ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q ) +Q0  ( [
<. 2o ,  1o >. ] ~Q0 ·Q0 
Q ) ) )
6718, 16, 66syl2anc 403 . . . . . . 7  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  (
( [ <. M ,  1o >. ] ~Q0 +Q0  [ <. 2o ,  1o >. ] ~Q0  ) ·Q0  Q )  =  ( ( [
<. M ,  1o >. ] ~Q0 ·Q0 
Q ) +Q0  ( [ <. 2o ,  1o >. ] ~Q0 ·Q0  Q ) ) )
6859, 67eqtrd 2120 . . . . . 6  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  ( [ <. ( M  +o  2o ) ,  1o >. ] ~Q0 ·Q0 
Q )  =  ( ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q ) +Q0  ( [ <. 2o ,  1o >. ] ~Q0 ·Q0  Q )
) )
6968oveq2d 5668 . . . . 5  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  ( X +Q0  ( [ <. ( M  +o  2o ) ,  1o >. ] ~Q0 ·Q0 
Q ) )  =  ( X +Q0  ( ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q ) +Q0  ( [ <. 2o ,  1o >. ] ~Q0 ·Q0  Q )
) ) )
70 nq02m 7022 . . . . . . . . 9  |-  ( Q  e. Q0  ->  ( [ <. 2o ,  1o >. ] ~Q0 ·Q0  Q )  =  ( Q +Q0  Q ) )
7170oveq2d 5668 . . . . . . . 8  |-  ( Q  e. Q0  ->  ( ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q ) +Q0  ( [ <. 2o ,  1o >. ] ~Q0 ·Q0  Q )
)  =  ( ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q ) +Q0  ( Q +Q0  Q
) ) )
7271oveq2d 5668 . . . . . . 7  |-  ( Q  e. Q0  ->  ( X +Q0  ( ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q ) +Q0  ( [ <. 2o ,  1o >. ] ~Q0 ·Q0  Q )
) )  =  ( X +Q0  ( ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q ) +Q0  ( Q +Q0  Q ) ) ) )
7318, 72syl 14 . . . . . 6  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  ( X +Q0  ( ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q ) +Q0  ( [ <. 2o ,  1o >. ] ~Q0 ·Q0  Q )
) )  =  ( X +Q0  ( ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q ) +Q0  ( Q +Q0  Q ) ) ) )
7417, 6sseldi 3023 . . . . . . 7  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  X  e. Q0 )
75 addclnq0 7008 . . . . . . . 8  |-  ( ( Q  e. Q0  /\  Q  e. Q0 )  ->  ( Q +Q0  Q )  e. Q0 )
7618, 18, 75syl2anc 403 . . . . . . 7  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  ( Q +Q0  Q )  e. Q0 )
77 addassnq0 7019 . . . . . . 7  |-  ( ( X  e. Q0  /\  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )  e. Q0  /\  ( Q +Q0  Q )  e. Q0 )  ->  ( ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
) +Q0  ( Q +Q0  Q ) )  =  ( X +Q0  ( ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q ) +Q0  ( Q +Q0  Q ) ) ) )
7874, 20, 76, 77syl3anc 1174 . . . . . 6  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  (
( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q ) ) +Q0  ( Q +Q0  Q ) )  =  ( X +Q0  ( ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q ) +Q0  ( Q +Q0  Q ) ) ) )
7973, 78eqtr4d 2123 . . . . 5  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  ( X +Q0  ( ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q ) +Q0  ( [ <. 2o ,  1o >. ] ~Q0 ·Q0  Q )
) )  =  ( ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q ) ) +Q0  ( Q +Q0  Q ) ) )
8055, 69, 793eqtrd 2124 . . . 4  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  B  =  ( ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
) +Q0  ( Q +Q0  Q ) ) )
81 oveq1 5659 . . . . . 6  |-  ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q ) )  ->  ( A +Q0  ( Q +Q0  Q
) )  =  ( ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q ) ) +Q0  ( Q +Q0  Q ) ) )
8281eqeq2d 2099 . . . . 5  |-  ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q ) )  ->  ( B  =  ( A +Q0  ( Q +Q0  Q ) )  <->  B  =  ( ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q ) ) +Q0  ( Q +Q0  Q ) ) ) )
835, 82syl 14 . . . 4  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  ( B  =  ( A +Q0  ( Q +Q0  Q
) )  <->  B  =  ( ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q ) ) +Q0  ( Q +Q0  Q ) ) ) )
8480, 83mpbird 165 . . 3  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  B  =  ( A +Q0  ( Q +Q0  Q ) ) )
854, 27, 843eqtr4rd 2131 . 2  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  B  =  ( A  +Q  ( Q  +Q  Q
) ) )
86 simprlr 505 . . 3  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  ( Q  +Q  Q )  <Q  P )
87 ltrelnq 6922 . . . . . 6  |-  <Q  C_  ( Q.  X.  Q. )
8887brel 4490 . . . . 5  |-  ( ( Q  +Q  Q ) 
<Q  P  ->  ( ( Q  +Q  Q )  e.  Q.  /\  P  e.  Q. ) )
8986, 88syl 14 . . . 4  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  (
( Q  +Q  Q
)  e.  Q.  /\  P  e.  Q. )
)
90 ltanqg 6957 . . . . 5  |-  ( ( ( Q  +Q  Q
)  e.  Q.  /\  P  e.  Q.  /\  A  e.  Q. )  ->  (
( Q  +Q  Q
)  <Q  P  <->  ( A  +Q  ( Q  +Q  Q
) )  <Q  ( A  +Q  P ) ) )
91903expa 1143 . . . 4  |-  ( ( ( ( Q  +Q  Q )  e.  Q.  /\  P  e.  Q. )  /\  A  e.  Q. )  ->  ( ( Q  +Q  Q )  <Q  P 
<->  ( A  +Q  ( Q  +Q  Q ) ) 
<Q  ( A  +Q  P
) ) )
9289, 23, 91syl2anc 403 . . 3  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  (
( Q  +Q  Q
)  <Q  P  <->  ( A  +Q  ( Q  +Q  Q
) )  <Q  ( A  +Q  P ) ) )
9386, 92mpbid 145 . 2  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  ( A  +Q  ( Q  +Q  Q ) )  <Q 
( A  +Q  P
) )
9485, 93eqbrtrd 3865 1  |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )
)  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q ) ) )  /\  (
( Q  e.  Q.  /\  ( Q  +Q  Q
)  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
) )  ->  B  <Q  ( A  +Q  P
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438    =/= wne 2255   (/)c0 3286   <.cop 3449   class class class wbr 3845   omcom 4405    X. cxp 4436  (class class class)co 5652   1oc1o 6174   2oc2o 6175    +o coa 6178   [cec 6288   /.cqs 6289   N.cnpi 6829    ~Q ceq 6836   Q.cnq 6837    +Q cplq 6839    .Q cmq 6840    <Q cltq 6842   ~Q0 ceq0 6843  Q0cnq0 6844   +Q0 cplq0 6846   ·Q0 cmq0 6847
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-1o 6181  df-2o 6182  df-oadd 6185  df-omul 6186  df-er 6290  df-ec 6292  df-qs 6296  df-ni 6861  df-pli 6862  df-mi 6863  df-lti 6864  df-plpq 6901  df-mpq 6902  df-enq 6904  df-nqqs 6905  df-plqqs 6906  df-mqqs 6907  df-ltnqqs 6910  df-enq0 6981  df-nq0 6982  df-plq0 6984  df-mq0 6985
This theorem is referenced by:  prarloc  7060
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