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Theorem prarloclemarch2 7247
Description: Like prarloclemarch 7246 but the integer must be at least two, and there is also  B added to the right hand side. These details follow straightforwardly but are chosen to be helpful in the proof of prarloc 7331. (Contributed by Jim Kingdon, 25-Nov-2019.)
Assertion
Ref Expression
prarloclemarch2  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  E. x  e.  N.  ( 1o  <N  x  /\  A  <Q  ( B  +Q  ( [ <. x ,  1o >. ]  ~Q  .Q  C ) ) ) )
Distinct variable groups:    x, A    x, B    x, C

Proof of Theorem prarloclemarch2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 prarloclemarch 7246 . . 3  |-  ( ( A  e.  Q.  /\  C  e.  Q. )  ->  E. z  e.  N.  A  <Q  ( [ <. z ,  1o >. ]  ~Q  .Q  C ) )
213adant2 1001 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  E. z  e.  N.  A  <Q  ( [ <. z ,  1o >. ]  ~Q  .Q  C
) )
3 pinn 7137 . . . . . . . 8  |-  ( z  e.  N.  ->  z  e.  om )
4 1pi 7143 . . . . . . . . . . . 12  |-  1o  e.  N.
54elexi 2699 . . . . . . . . . . 11  |-  1o  e.  _V
65sucid 4343 . . . . . . . . . 10  |-  1o  e.  suc  1o
7 df-2o 6318 . . . . . . . . . 10  |-  2o  =  suc  1o
86, 7eleqtrri 2216 . . . . . . . . 9  |-  1o  e.  2o
9 2onn 6421 . . . . . . . . . . 11  |-  2o  e.  om
10 nnaword2 6414 . . . . . . . . . . 11  |-  ( ( 2o  e.  om  /\  z  e.  om )  ->  2o  C_  ( z  +o  2o ) )
119, 10mpan 421 . . . . . . . . . 10  |-  ( z  e.  om  ->  2o  C_  ( z  +o  2o ) )
1211sseld 3097 . . . . . . . . 9  |-  ( z  e.  om  ->  ( 1o  e.  2o  ->  1o  e.  ( z  +o  2o ) ) )
138, 12mpi 15 . . . . . . . 8  |-  ( z  e.  om  ->  1o  e.  ( z  +o  2o ) )
143, 13syl 14 . . . . . . 7  |-  ( z  e.  N.  ->  1o  e.  ( z  +o  2o ) )
15 o1p1e2 6368 . . . . . . . . 9  |-  ( 1o 
+o  1o )  =  2o
16 addpiord 7144 . . . . . . . . . . 11  |-  ( ( 1o  e.  N.  /\  1o  e.  N. )  -> 
( 1o  +N  1o )  =  ( 1o  +o  1o ) )
174, 4, 16mp2an 423 . . . . . . . . . 10  |-  ( 1o 
+N  1o )  =  ( 1o  +o  1o )
18 addclpi 7155 . . . . . . . . . . 11  |-  ( ( 1o  e.  N.  /\  1o  e.  N. )  -> 
( 1o  +N  1o )  e.  N. )
194, 4, 18mp2an 423 . . . . . . . . . 10  |-  ( 1o 
+N  1o )  e. 
N.
2017, 19eqeltrri 2214 . . . . . . . . 9  |-  ( 1o 
+o  1o )  e. 
N.
2115, 20eqeltrri 2214 . . . . . . . 8  |-  2o  e.  N.
22 addpiord 7144 . . . . . . . 8  |-  ( ( z  e.  N.  /\  2o  e.  N. )  -> 
( z  +N  2o )  =  ( z  +o  2o ) )
2321, 22mpan2 422 . . . . . . 7  |-  ( z  e.  N.  ->  (
z  +N  2o )  =  ( z  +o  2o ) )
2414, 23eleqtrrd 2220 . . . . . 6  |-  ( z  e.  N.  ->  1o  e.  ( z  +N  2o ) )
25 addclpi 7155 . . . . . . . 8  |-  ( ( z  e.  N.  /\  2o  e.  N. )  -> 
( z  +N  2o )  e.  N. )
2621, 25mpan2 422 . . . . . . 7  |-  ( z  e.  N.  ->  (
z  +N  2o )  e.  N. )
27 ltpiord 7147 . . . . . . . 8  |-  ( ( 1o  e.  N.  /\  ( z  +N  2o )  e.  N. )  ->  ( 1o  <N  (
z  +N  2o )  <-> 
1o  e.  ( z  +N  2o ) ) )
284, 27mpan 421 . . . . . . 7  |-  ( ( z  +N  2o )  e.  N.  ->  ( 1o  <N  ( z  +N  2o )  <->  1o  e.  ( z  +N  2o ) ) )
2926, 28syl 14 . . . . . 6  |-  ( z  e.  N.  ->  ( 1o  <N  ( z  +N  2o )  <->  1o  e.  ( z  +N  2o ) ) )
3024, 29mpbird 166 . . . . 5  |-  ( z  e.  N.  ->  1o  <N  ( z  +N  2o ) )
3130adantl 275 . . . 4  |-  ( ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  /\  z  e.  N. )  ->  1o  <N  (
z  +N  2o ) )
3231adantrr 471 . . 3  |-  ( ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  /\  ( z  e.  N.  /\  A  <Q  ( [ <. z ,  1o >. ]  ~Q  .Q  C ) ) )  ->  1o  <N  ( z  +N  2o ) )
33 nna0 6374 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  om  ->  (
z  +o  (/) )  =  z )
34 0lt1o 6341 . . . . . . . . . . . . . . . . . . . 20  |-  (/)  e.  1o
35 1on 6324 . . . . . . . . . . . . . . . . . . . . . 22  |-  1o  e.  On
3635onsuci 4436 . . . . . . . . . . . . . . . . . . . . 21  |-  suc  1o  e.  On
37 ontr1 4315 . . . . . . . . . . . . . . . . . . . . 21  |-  ( suc 
1o  e.  On  ->  ( ( (/)  e.  1o  /\  1o  e.  suc  1o )  ->  (/)  e.  suc  1o ) )
3836, 37ax-mp 5 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
(/)  e.  1o  /\  1o  e.  suc  1o )  ->  (/) 
e.  suc  1o )
3934, 6, 38mp2an 423 . . . . . . . . . . . . . . . . . . 19  |-  (/)  e.  suc  1o
4039, 7eleqtrri 2216 . . . . . . . . . . . . . . . . . 18  |-  (/)  e.  2o
41 nnaordi 6408 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 2o  e.  om  /\  z  e.  om )  ->  ( (/)  e.  2o  ->  ( z  +o  (/) )  e.  ( z  +o  2o ) ) )
429, 41mpan 421 . . . . . . . . . . . . . . . . . 18  |-  ( z  e.  om  ->  ( (/) 
e.  2o  ->  ( z  +o  (/) )  e.  ( z  +o  2o ) ) )
4340, 42mpi 15 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  om  ->  (
z  +o  (/) )  e.  ( z  +o  2o ) )
4433, 43eqeltrrd 2218 . . . . . . . . . . . . . . . 16  |-  ( z  e.  om  ->  z  e.  ( z  +o  2o ) )
453, 44syl 14 . . . . . . . . . . . . . . 15  |-  ( z  e.  N.  ->  z  e.  ( z  +o  2o ) )
4645, 23eleqtrrd 2220 . . . . . . . . . . . . . 14  |-  ( z  e.  N.  ->  z  e.  ( z  +N  2o ) )
47 ltpiord 7147 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  N.  /\  ( z  +N  2o )  e.  N. )  ->  ( z  <N  (
z  +N  2o )  <-> 
z  e.  ( z  +N  2o ) ) )
4826, 47mpdan 418 . . . . . . . . . . . . . 14  |-  ( z  e.  N.  ->  (
z  <N  ( z  +N  2o )  <->  z  e.  ( z  +N  2o ) ) )
4946, 48mpbird 166 . . . . . . . . . . . . 13  |-  ( z  e.  N.  ->  z  <N  ( z  +N  2o ) )
50 mulidpi 7146 . . . . . . . . . . . . 13  |-  ( z  e.  N.  ->  (
z  .N  1o )  =  z )
51 mulcompig 7159 . . . . . . . . . . . . . . . 16  |-  ( ( ( z  +N  2o )  e.  N.  /\  1o  e.  N. )  ->  (
( z  +N  2o )  .N  1o )  =  ( 1o  .N  (
z  +N  2o ) ) )
524, 51mpan2 422 . . . . . . . . . . . . . . 15  |-  ( ( z  +N  2o )  e.  N.  ->  (
( z  +N  2o )  .N  1o )  =  ( 1o  .N  (
z  +N  2o ) ) )
5326, 52syl 14 . . . . . . . . . . . . . 14  |-  ( z  e.  N.  ->  (
( z  +N  2o )  .N  1o )  =  ( 1o  .N  (
z  +N  2o ) ) )
54 mulidpi 7146 . . . . . . . . . . . . . . 15  |-  ( ( z  +N  2o )  e.  N.  ->  (
( z  +N  2o )  .N  1o )  =  ( z  +N  2o ) )
5526, 54syl 14 . . . . . . . . . . . . . 14  |-  ( z  e.  N.  ->  (
( z  +N  2o )  .N  1o )  =  ( z  +N  2o ) )
5653, 55eqtr3d 2175 . . . . . . . . . . . . 13  |-  ( z  e.  N.  ->  ( 1o  .N  ( z  +N  2o ) )  =  ( z  +N  2o ) )
5749, 50, 563brtr4d 3964 . . . . . . . . . . . 12  |-  ( z  e.  N.  ->  (
z  .N  1o ) 
<N  ( 1o  .N  (
z  +N  2o ) ) )
58 ordpipqqs 7202 . . . . . . . . . . . . . . 15  |-  ( ( ( z  e.  N.  /\  1o  e.  N. )  /\  ( ( z  +N  2o )  e.  N.  /\  1o  e.  N. )
)  ->  ( [ <. z ,  1o >. ]  ~Q  <Q  [ <. (
z  +N  2o ) ,  1o >. ]  ~Q  <->  ( z  .N  1o ) 
<N  ( 1o  .N  (
z  +N  2o ) ) ) )
594, 58mpanl2 432 . . . . . . . . . . . . . 14  |-  ( ( z  e.  N.  /\  ( ( z  +N  2o )  e.  N.  /\  1o  e.  N. )
)  ->  ( [ <. z ,  1o >. ]  ~Q  <Q  [ <. (
z  +N  2o ) ,  1o >. ]  ~Q  <->  ( z  .N  1o ) 
<N  ( 1o  .N  (
z  +N  2o ) ) ) )
604, 59mpanr2 435 . . . . . . . . . . . . 13  |-  ( ( z  e.  N.  /\  ( z  +N  2o )  e.  N. )  ->  ( [ <. z ,  1o >. ]  ~Q  <Q  [
<. ( z  +N  2o ) ,  1o >. ]  ~Q  <->  ( z  .N  1o ) 
<N  ( 1o  .N  (
z  +N  2o ) ) ) )
6126, 60mpdan 418 . . . . . . . . . . . 12  |-  ( z  e.  N.  ->  ( [ <. z ,  1o >. ]  ~Q  <Q  [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  <->  ( z  .N  1o ) 
<N  ( 1o  .N  (
z  +N  2o ) ) ) )
6257, 61mpbird 166 . . . . . . . . . . 11  |-  ( z  e.  N.  ->  [ <. z ,  1o >. ]  ~Q  <Q  [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  )
6362adantl 275 . . . . . . . . . 10  |-  ( ( C  e.  Q.  /\  z  e.  N. )  ->  [ <. z ,  1o >. ]  ~Q  <Q  [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  )
64 opelxpi 4575 . . . . . . . . . . . . . . . 16  |-  ( ( ( z  +N  2o )  e.  N.  /\  1o  e.  N. )  ->  <. (
z  +N  2o ) ,  1o >.  e.  ( N.  X.  N. )
)
654, 64mpan2 422 . . . . . . . . . . . . . . 15  |-  ( ( z  +N  2o )  e.  N.  ->  <. (
z  +N  2o ) ,  1o >.  e.  ( N.  X.  N. )
)
66 enqex 7188 . . . . . . . . . . . . . . . 16  |-  ~Q  e.  _V
6766ecelqsi 6487 . . . . . . . . . . . . . . 15  |-  ( <.
( z  +N  2o ) ,  1o >.  e.  ( N.  X.  N. )  ->  [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  e.  ( ( N.  X.  N. ) /.  ~Q  ) )
6826, 65, 673syl 17 . . . . . . . . . . . . . 14  |-  ( z  e.  N.  ->  [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  e.  ( ( N.  X.  N. ) /.  ~Q  )
)
69 df-nqqs 7176 . . . . . . . . . . . . . 14  |-  Q.  =  ( ( N.  X.  N. ) /.  ~Q  )
7068, 69eleqtrrdi 2234 . . . . . . . . . . . . 13  |-  ( z  e.  N.  ->  [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  e.  Q. )
71 opelxpi 4575 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  N.  /\  1o  e.  N. )  ->  <. z ,  1o >.  e.  ( N.  X.  N. ) )
724, 71mpan2 422 . . . . . . . . . . . . . . . 16  |-  ( z  e.  N.  ->  <. z ,  1o >.  e.  ( N.  X.  N. ) )
7366ecelqsi 6487 . . . . . . . . . . . . . . . 16  |-  ( <.
z ,  1o >.  e.  ( N.  X.  N. )  ->  [ <. z ,  1o >. ]  ~Q  e.  ( ( N.  X.  N. ) /.  ~Q  )
)
7472, 73syl 14 . . . . . . . . . . . . . . 15  |-  ( z  e.  N.  ->  [ <. z ,  1o >. ]  ~Q  e.  ( ( N.  X.  N. ) /.  ~Q  )
)
7574, 69eleqtrrdi 2234 . . . . . . . . . . . . . 14  |-  ( z  e.  N.  ->  [ <. z ,  1o >. ]  ~Q  e.  Q. )
76 ltmnqg 7229 . . . . . . . . . . . . . 14  |-  ( ( [ <. z ,  1o >. ]  ~Q  e.  Q.  /\ 
[ <. ( z  +N  2o ) ,  1o >. ]  ~Q  e.  Q.  /\  C  e.  Q. )  ->  ( [ <. z ,  1o >. ]  ~Q  <Q  [
<. ( z  +N  2o ) ,  1o >. ]  ~Q  <->  ( C  .Q  [ <. z ,  1o >. ]  ~Q  )  <Q  ( C  .Q  [
<. ( z  +N  2o ) ,  1o >. ]  ~Q  ) ) )
7775, 76syl3an1 1250 . . . . . . . . . . . . 13  |-  ( ( z  e.  N.  /\  [
<. ( z  +N  2o ) ,  1o >. ]  ~Q  e.  Q.  /\  C  e. 
Q. )  ->  ( [ <. z ,  1o >. ]  ~Q  <Q  [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  <->  ( C  .Q  [ <. z ,  1o >. ]  ~Q  )  <Q  ( C  .Q  [
<. ( z  +N  2o ) ,  1o >. ]  ~Q  ) ) )
7870, 77syl3an2 1251 . . . . . . . . . . . 12  |-  ( ( z  e.  N.  /\  z  e.  N.  /\  C  e.  Q. )  ->  ( [ <. z ,  1o >. ]  ~Q  <Q  [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  <->  ( C  .Q  [ <. z ,  1o >. ]  ~Q  )  <Q  ( C  .Q  [
<. ( z  +N  2o ) ,  1o >. ]  ~Q  ) ) )
79783anidm12 1274 . . . . . . . . . . 11  |-  ( ( z  e.  N.  /\  C  e.  Q. )  ->  ( [ <. z ,  1o >. ]  ~Q  <Q  [
<. ( z  +N  2o ) ,  1o >. ]  ~Q  <->  ( C  .Q  [ <. z ,  1o >. ]  ~Q  )  <Q  ( C  .Q  [
<. ( z  +N  2o ) ,  1o >. ]  ~Q  ) ) )
8079ancoms 266 . . . . . . . . . 10  |-  ( ( C  e.  Q.  /\  z  e.  N. )  ->  ( [ <. z ,  1o >. ]  ~Q  <Q  [
<. ( z  +N  2o ) ,  1o >. ]  ~Q  <->  ( C  .Q  [ <. z ,  1o >. ]  ~Q  )  <Q  ( C  .Q  [
<. ( z  +N  2o ) ,  1o >. ]  ~Q  ) ) )
8163, 80mpbid 146 . . . . . . . . 9  |-  ( ( C  e.  Q.  /\  z  e.  N. )  ->  ( C  .Q  [ <. z ,  1o >. ]  ~Q  )  <Q  ( C  .Q  [ <. (
z  +N  2o ) ,  1o >. ]  ~Q  ) )
82 mulcomnqg 7211 . . . . . . . . . 10  |-  ( ( C  e.  Q.  /\  [
<. z ,  1o >. ]  ~Q  e.  Q. )  ->  ( C  .Q  [ <. z ,  1o >. ]  ~Q  )  =  ( [ <. z ,  1o >. ]  ~Q  .Q  C
) )
8375, 82sylan2 284 . . . . . . . . 9  |-  ( ( C  e.  Q.  /\  z  e.  N. )  ->  ( C  .Q  [ <. z ,  1o >. ]  ~Q  )  =  ( [ <. z ,  1o >. ]  ~Q  .Q  C
) )
84 mulcomnqg 7211 . . . . . . . . . 10  |-  ( ( C  e.  Q.  /\  [
<. ( z  +N  2o ) ,  1o >. ]  ~Q  e.  Q. )  ->  ( C  .Q  [ <. (
z  +N  2o ) ,  1o >. ]  ~Q  )  =  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C ) )
8570, 84sylan2 284 . . . . . . . . 9  |-  ( ( C  e.  Q.  /\  z  e.  N. )  ->  ( C  .Q  [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  )  =  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C ) )
8681, 83, 853brtr3d 3963 . . . . . . . 8  |-  ( ( C  e.  Q.  /\  z  e.  N. )  ->  ( [ <. z ,  1o >. ]  ~Q  .Q  C )  <Q  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C
) )
87863ad2antl3 1146 . . . . . . 7  |-  ( ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  /\  z  e.  N. )  ->  ( [ <. z ,  1o >. ]  ~Q  .Q  C )  <Q  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C
) )
8887adantrr 471 . . . . . 6  |-  ( ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  /\  ( z  e.  N.  /\  A  <Q  ( [ <. z ,  1o >. ]  ~Q  .Q  C ) ) )  ->  ( [ <. z ,  1o >. ]  ~Q  .Q  C
)  <Q  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C ) )
89 ltsonq 7226 . . . . . . . . . 10  |-  <Q  Or  Q.
90 ltrelnq 7193 . . . . . . . . . 10  |-  <Q  C_  ( Q.  X.  Q. )
9189, 90sotri 4938 . . . . . . . . 9  |-  ( ( A  <Q  ( [ <. z ,  1o >. ]  ~Q  .Q  C )  /\  ( [ <. z ,  1o >. ]  ~Q  .Q  C )  <Q  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C
) )  ->  A  <Q  ( [ <. (
z  +N  2o ) ,  1o >. ]  ~Q  .Q  C ) )
9291ex 114 . . . . . . . 8  |-  ( A 
<Q  ( [ <. z ,  1o >. ]  ~Q  .Q  C )  ->  (
( [ <. z ,  1o >. ]  ~Q  .Q  C )  <Q  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C
)  ->  A  <Q  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C
) ) )
9392adantl 275 . . . . . . 7  |-  ( ( z  e.  N.  /\  A  <Q  ( [ <. z ,  1o >. ]  ~Q  .Q  C ) )  -> 
( ( [ <. z ,  1o >. ]  ~Q  .Q  C )  <Q  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C
)  ->  A  <Q  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C
) ) )
9493adantl 275 . . . . . 6  |-  ( ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  /\  ( z  e.  N.  /\  A  <Q  ( [ <. z ,  1o >. ]  ~Q  .Q  C ) ) )  ->  (
( [ <. z ,  1o >. ]  ~Q  .Q  C )  <Q  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C
)  ->  A  <Q  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C
) ) )
9588, 94mpd 13 . . . . 5  |-  ( ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  /\  ( z  e.  N.  /\  A  <Q  ( [ <. z ,  1o >. ]  ~Q  .Q  C ) ) )  ->  A  <Q  ( [ <. (
z  +N  2o ) ,  1o >. ]  ~Q  .Q  C ) )
96 mulclnq 7204 . . . . . . . . . 10  |-  ( ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  e.  Q.  /\  C  e.  Q. )  ->  ( [ <. (
z  +N  2o ) ,  1o >. ]  ~Q  .Q  C )  e.  Q. )
9770, 96sylan 281 . . . . . . . . 9  |-  ( ( z  e.  N.  /\  C  e.  Q. )  ->  ( [ <. (
z  +N  2o ) ,  1o >. ]  ~Q  .Q  C )  e.  Q. )
9897ancoms 266 . . . . . . . 8  |-  ( ( C  e.  Q.  /\  z  e.  N. )  ->  ( [ <. (
z  +N  2o ) ,  1o >. ]  ~Q  .Q  C )  e.  Q. )
99983ad2antl3 1146 . . . . . . 7  |-  ( ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  /\  z  e.  N. )  ->  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C )  e.  Q. )
100 simpl2 986 . . . . . . 7  |-  ( ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  /\  z  e.  N. )  ->  B  e.  Q. )
101 ltaddnq 7235 . . . . . . 7  |-  ( ( ( [ <. (
z  +N  2o ) ,  1o >. ]  ~Q  .Q  C )  e.  Q.  /\  B  e.  Q. )  ->  ( [ <. (
z  +N  2o ) ,  1o >. ]  ~Q  .Q  C )  <Q  (
( [ <. (
z  +N  2o ) ,  1o >. ]  ~Q  .Q  C )  +Q  B
) )
10299, 100, 101syl2anc 409 . . . . . 6  |-  ( ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  /\  z  e.  N. )  ->  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C )  <Q  (
( [ <. (
z  +N  2o ) ,  1o >. ]  ~Q  .Q  C )  +Q  B
) )
103102adantrr 471 . . . . 5  |-  ( ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  /\  ( z  e.  N.  /\  A  <Q  ( [ <. z ,  1o >. ]  ~Q  .Q  C ) ) )  ->  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C
)  <Q  ( ( [
<. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C )  +Q  B
) )
10489, 90sotri 4938 . . . . 5  |-  ( ( A  <Q  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C )  /\  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C
)  <Q  ( ( [
<. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C )  +Q  B
) )  ->  A  <Q  ( ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C )  +Q  B
) )
10595, 103, 104syl2anc 409 . . . 4  |-  ( ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  /\  ( z  e.  N.  /\  A  <Q  ( [ <. z ,  1o >. ]  ~Q  .Q  C ) ) )  ->  A  <Q  ( ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C )  +Q  B
) )
106 addcomnqg 7209 . . . . . . 7  |-  ( ( ( [ <. (
z  +N  2o ) ,  1o >. ]  ~Q  .Q  C )  e.  Q.  /\  B  e.  Q. )  ->  ( ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C )  +Q  B
)  =  ( B  +Q  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C ) ) )
10799, 100, 106syl2anc 409 . . . . . 6  |-  ( ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  /\  z  e.  N. )  ->  ( ( [
<. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C )  +Q  B
)  =  ( B  +Q  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C ) ) )
108107breq2d 3945 . . . . 5  |-  ( ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  /\  z  e.  N. )  ->  ( A  <Q  ( ( [ <. (
z  +N  2o ) ,  1o >. ]  ~Q  .Q  C )  +Q  B
)  <->  A  <Q  ( B  +Q  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C ) ) ) )
109108adantrr 471 . . . 4  |-  ( ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  /\  ( z  e.  N.  /\  A  <Q  ( [ <. z ,  1o >. ]  ~Q  .Q  C ) ) )  ->  ( A  <Q  ( ( [
<. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C )  +Q  B
)  <->  A  <Q  ( B  +Q  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C ) ) ) )
110105, 109mpbid 146 . . 3  |-  ( ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  /\  ( z  e.  N.  /\  A  <Q  ( [ <. z ,  1o >. ]  ~Q  .Q  C ) ) )  ->  A  <Q  ( B  +Q  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C
) ) )
111 simpr 109 . . . . 5  |-  ( ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  /\  z  e.  N. )  ->  z  e.  N. )
112 breq2 3937 . . . . . . . 8  |-  ( x  =  ( z  +N  2o )  ->  ( 1o  <N  x  <->  1o  <N  ( z  +N  2o ) ) )
113 opeq1 3709 . . . . . . . . . . . 12  |-  ( x  =  ( z  +N  2o )  ->  <. x ,  1o >.  =  <. ( z  +N  2o ) ,  1o >. )
114113eceq1d 6469 . . . . . . . . . . 11  |-  ( x  =  ( z  +N  2o )  ->  [ <. x ,  1o >. ]  ~Q  =  [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  )
115114oveq1d 5793 . . . . . . . . . 10  |-  ( x  =  ( z  +N  2o )  ->  ( [ <. x ,  1o >. ]  ~Q  .Q  C
)  =  ( [
<. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C ) )
116115oveq2d 5794 . . . . . . . . 9  |-  ( x  =  ( z  +N  2o )  ->  ( B  +Q  ( [ <. x ,  1o >. ]  ~Q  .Q  C ) )  =  ( B  +Q  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C
) ) )
117116breq2d 3945 . . . . . . . 8  |-  ( x  =  ( z  +N  2o )  ->  ( A  <Q  ( B  +Q  ( [ <. x ,  1o >. ]  ~Q  .Q  C
) )  <->  A  <Q  ( B  +Q  ( [
<. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C ) ) ) )
118112, 117anbi12d 465 . . . . . . 7  |-  ( x  =  ( z  +N  2o )  ->  (
( 1o  <N  x  /\  A  <Q  ( B  +Q  ( [ <. x ,  1o >. ]  ~Q  .Q  C ) ) )  <-> 
( 1o  <N  (
z  +N  2o )  /\  A  <Q  ( B  +Q  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C ) ) ) ) )
119118rspcev 2790 . . . . . 6  |-  ( ( ( z  +N  2o )  e.  N.  /\  ( 1o  <N  ( z  +N  2o )  /\  A  <Q  ( B  +Q  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C
) ) ) )  ->  E. x  e.  N.  ( 1o  <N  x  /\  A  <Q  ( B  +Q  ( [ <. x ,  1o >. ]  ~Q  .Q  C
) ) ) )
120119ex 114 . . . . 5  |-  ( ( z  +N  2o )  e.  N.  ->  (
( 1o  <N  (
z  +N  2o )  /\  A  <Q  ( B  +Q  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C ) ) )  ->  E. x  e.  N.  ( 1o  <N  x  /\  A  <Q  ( B  +Q  ( [ <. x ,  1o >. ]  ~Q  .Q  C
) ) ) ) )
121111, 26, 1203syl 17 . . . 4  |-  ( ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  /\  z  e.  N. )  ->  ( ( 1o 
<N  ( z  +N  2o )  /\  A  <Q  ( B  +Q  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C ) ) )  ->  E. x  e.  N.  ( 1o  <N  x  /\  A  <Q  ( B  +Q  ( [ <. x ,  1o >. ]  ~Q  .Q  C
) ) ) ) )
122121adantrr 471 . . 3  |-  ( ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  /\  ( z  e.  N.  /\  A  <Q  ( [ <. z ,  1o >. ]  ~Q  .Q  C ) ) )  ->  (
( 1o  <N  (
z  +N  2o )  /\  A  <Q  ( B  +Q  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C ) ) )  ->  E. x  e.  N.  ( 1o  <N  x  /\  A  <Q  ( B  +Q  ( [ <. x ,  1o >. ]  ~Q  .Q  C
) ) ) ) )
12332, 110, 122mp2and 430 . 2  |-  ( ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  /\  ( z  e.  N.  /\  A  <Q  ( [ <. z ,  1o >. ]  ~Q  .Q  C ) ) )  ->  E. x  e.  N.  ( 1o  <N  x  /\  A  <Q  ( B  +Q  ( [ <. x ,  1o >. ]  ~Q  .Q  C ) ) ) )
1242, 123rexlimddv 2555 1  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  E. x  e.  N.  ( 1o  <N  x  /\  A  <Q  ( B  +Q  ( [ <. x ,  1o >. ]  ~Q  .Q  C ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 963    = wceq 1332    e. wcel 1481   E.wrex 2418    C_ wss 3072   (/)c0 3364   <.cop 3531   class class class wbr 3933   Oncon0 4289   suc csuc 4291   omcom 4508    X. cxp 4541  (class class class)co 5778   1oc1o 6310   2oc2o 6311    +o coa 6314   [cec 6431   /.cqs 6432   N.cnpi 7100    +N cpli 7101    .N cmi 7102    <N clti 7103    ~Q ceq 7107   Q.cnq 7108    +Q cplq 7110    .Q cmq 7111    <Q cltq 7113
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4047  ax-sep 4050  ax-nul 4058  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456  ax-iinf 4506
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2689  df-sbc 2911  df-csb 3005  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-int 3776  df-iun 3819  df-br 3934  df-opab 3994  df-mpt 3995  df-tr 4031  df-eprel 4215  df-id 4219  df-po 4222  df-iso 4223  df-iord 4292  df-on 4294  df-suc 4297  df-iom 4509  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135  df-ov 5781  df-oprab 5782  df-mpo 5783  df-1st 6042  df-2nd 6043  df-recs 6206  df-irdg 6271  df-1o 6317  df-2o 6318  df-oadd 6321  df-omul 6322  df-er 6433  df-ec 6435  df-qs 6439  df-ni 7132  df-pli 7133  df-mi 7134  df-lti 7135  df-plpq 7172  df-mpq 7173  df-enq 7175  df-nqqs 7176  df-plqqs 7177  df-mqqs 7178  df-1nqqs 7179  df-rq 7180  df-ltnqqs 7181
This theorem is referenced by:  prarloc  7331
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