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Theorem prarloclemcalc 7274
Description: Some calculations for prarloc 7275. (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 509 . . . . 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 7226 . . . . 5  |-  ( ( Q  e.  Q.  /\  Q  e.  Q. )  ->  ( Q  +Q  Q
)  =  ( Q +Q0  Q
) )
31, 1, 2syl2anc 406 . . . 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 5756 . . 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 501 . . . . 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 511 . . . . . 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 512 . . . . . . . 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 7087 . . . . . . . . . . 11  |-  1o  e.  N.
9 opelxpi 4539 . . . . . . . . . . 11  |-  ( ( M  e.  om  /\  1o  e.  N. )  ->  <. M ,  1o >.  e.  ( om  X.  N. ) )
108, 9mpan2 419 . . . . . . . . . 10  |-  ( M  e.  om  ->  <. M ,  1o >.  e.  ( om 
X.  N. ) )
11 enq0ex 7211 . . . . . . . . . . 11  |- ~Q0  e.  _V
1211ecelqsi 6449 . . . . . . . . . 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 7197 . . . . . . . . 9  |- Q0  =  ( ( om 
X.  N. ) /. ~Q0  )
1513, 14syl6eleqr 2209 . . . . . . . 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 7213 . . . . . . . 8  |-  Q.  C_ Q0
1817, 1sseldi 3063 . . . . . . 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 7224 . . . . . . 7  |-  ( ( [ <. M ,  1o >. ] ~Q0  e. Q0  /\  Q  e. Q0 )  ->  ( [ <. M ,  1o >. ] ~Q0 ·Q0 
Q )  e. Q0 )
2016, 18, 19syl2anc 406 . . . . . 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 7225 . . . . . 6  |-  ( ( X  e.  Q.  /\  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q )  e. Q0 )  ->  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q ) )  e.  Q. )
226, 20, 21syl2anc 406 . . . . 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 2192 . . . 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 7147 . . . . 5  |-  ( ( Q  e.  Q.  /\  Q  e.  Q. )  ->  ( Q  +Q  Q
)  e.  Q. )
251, 1, 24syl2anc 406 . . . 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 7226 . . . 4  |-  ( ( A  e.  Q.  /\  ( Q  +Q  Q
)  e.  Q. )  ->  ( A  +Q  ( Q  +Q  Q ) )  =  ( A +Q0  ( Q  +Q  Q
) ) )
2723, 25, 26syl2anc 406 . . 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 502 . . . . . 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 6383 . . . . . . . . . . . . . 14  |-  2o  e.  om
30 2on0 6289 . . . . . . . . . . . . . 14  |-  2o  =/=  (/)
31 elni 7080 . . . . . . . . . . . . . 14  |-  ( 2o  e.  N.  <->  ( 2o  e.  om  /\  2o  =/=  (/) ) )
3229, 30, 31mpbir2an 909 . . . . . . . . . . . . 13  |-  2o  e.  N.
33 nnppipi 7115 . . . . . . . . . . . . 13  |-  ( ( M  e.  om  /\  2o  e.  N. )  -> 
( M  +o  2o )  e.  N. )
3432, 33mpan2 419 . . . . . . . . . . . 12  |-  ( M  e.  om  ->  ( M  +o  2o )  e. 
N. )
35 opelxpi 4539 . . . . . . . . . . . 12  |-  ( ( ( M  +o  2o )  e.  N.  /\  1o  e.  N. )  ->  <. ( M  +o  2o ) ,  1o >.  e.  ( N.  X.  N. ) )
3634, 8, 35sylancl 407 . . . . . . . . . . 11  |-  ( M  e.  om  ->  <. ( M  +o  2o ) ,  1o >.  e.  ( N.  X.  N. ) )
37 enqex 7132 . . . . . . . . . . . 12  |-  ~Q  e.  _V
3837ecelqsi 6449 . . . . . . . . . . 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 7120 . . . . . . . . . 10  |-  Q.  =  ( ( N.  X.  N. ) /.  ~Q  )
4139, 40syl6eleqr 2209 . . . . . . . . 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 7148 . . . . . . . 8  |-  ( ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  e.  Q.  /\  Q  e.  Q. )  ->  ( [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  .Q  Q )  e.  Q. )
4442, 1, 43syl2anc 406 . . . . . . 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 7226 . . . . . . 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 406 . . . . . 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 7227 . . . . . . . . 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 406 . . . . . . . 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 7210 . . . . . . . . . . 11  |-  ( ( ( M  +o  2o )  e.  N.  /\  1o  e.  N. )  ->  [ <. ( M  +o  2o ) ,  1o >. ] ~Q0  =  [ <. ( M  +o  2o ) ,  1o >. ]  ~Q  )
5034, 8, 49sylancl 407 . . . . . . . . . 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 5755 . . . . . . . 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 2151 . . . . . . 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 5756 . . . . . 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 2152 . . . . 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 7230 . . . . . . . . . 10  |-  ( ( M  e.  om  /\  2o  e.  om  /\  1o  e.  N. )  ->  [ <. ( M  +o  2o ) ,  1o >. ] ~Q0  =  ( [ <. M ,  1o >. ] ~Q0 +Q0  [ <. 2o ,  1o >. ] ~Q0  )
)
578, 56mp3an3 1287 . . . . . . . . 9  |-  ( ( M  e.  om  /\  2o  e.  om )  ->  [ <. ( M  +o  2o ) ,  1o >. ] ~Q0  =  ( [ <. M ,  1o >. ] ~Q0 +Q0  [ <. 2o ,  1o >. ] ~Q0  ) )
587, 29, 57sylancl 407 . . . . . . . 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 5755 . . . . . . 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 4539 . . . . . . . . . . . 12  |-  ( ( 2o  e.  om  /\  1o  e.  N. )  ->  <. 2o ,  1o >.  e.  ( om  X.  N. ) )
6129, 8, 60mp2an 420 . . . . . . . . . . 11  |-  <. 2o ,  1o >.  e.  ( om 
X.  N. )
6211ecelqsi 6449 . . . . . . . . . . 11  |-  ( <. 2o ,  1o >.  e.  ( om  X.  N. )  ->  [ <. 2o ,  1o >. ] ~Q0  e.  ( ( om  X.  N. ) /. ~Q0  ) )
6361, 62ax-mp 5 . . . . . . . . . 10  |-  [ <. 2o ,  1o >. ] ~Q0  e.  ( ( om 
X.  N. ) /. ~Q0  )
6463, 14eleqtrri 2191 . . . . . . . . 9  |-  [ <. 2o ,  1o >. ] ~Q0  e. Q0
65 distnq0r 7235 . . . . . . . . 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 1287 . . . . . . . 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 406 . . . . . . 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 2148 . . . . . 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 5756 . . . . 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 7237 . . . . . . . . 9  |-  ( Q  e. Q0  ->  ( [ <. 2o ,  1o >. ] ~Q0 ·Q0  Q )  =  ( Q +Q0  Q ) )
7170oveq2d 5756 . . . . . . . 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 5756 . . . . . . 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 3063 . . . . . . 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 7223 . . . . . . . 8  |-  ( ( Q  e. Q0  /\  Q  e. Q0 )  ->  ( Q +Q0  Q )  e. Q0 )
7618, 18, 75syl2anc 406 . . . . . . 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 7234 . . . . . . 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 1199 . . . . . 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 2151 . . . . 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 2152 . . . 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 5747 . . . . . 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 2127 . . . . 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 166 . . 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 2159 . 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 510 . . 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 7137 . . . . . 6  |-  <Q  C_  ( Q.  X.  Q. )
8887brel 4559 . . . . 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 7172 . . . . 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 1164 . . . 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 406 . . 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 146 . 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 3918 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 103    <-> wb 104    = wceq 1314    e. wcel 1463    =/= wne 2283   (/)c0 3331   <.cop 3498   class class class wbr 3897   omcom 4472    X. cxp 4505  (class class class)co 5740   1oc1o 6272   2oc2o 6273    +o coa 6276   [cec 6393   /.cqs 6394   N.cnpi 7044    ~Q ceq 7051   Q.cnq 7052    +Q cplq 7054    .Q cmq 7055    <Q cltq 7057   ~Q0 ceq0 7058  Q0cnq0 7059   +Q0 cplq0 7061   ·Q0 cmq0 7062
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-2o 6280  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-ltnqqs 7125  df-enq0 7196  df-nq0 7197  df-plq0 7199  df-mq0 7200
This theorem is referenced by:  prarloc  7275
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