ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  prarloclemarch2 Unicode version

Theorem prarloclemarch2 7449
Description: Like prarloclemarch 7448 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 7533. (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 7448 . . 3  |-  ( ( A  e.  Q.  /\  C  e.  Q. )  ->  E. z  e.  N.  A  <Q  ( [ <. z ,  1o >. ]  ~Q  .Q  C ) )
213adant2 1018 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  E. z  e.  N.  A  <Q  ( [ <. z ,  1o >. ]  ~Q  .Q  C
) )
3 pinn 7339 . . . . . . . 8  |-  ( z  e.  N.  ->  z  e.  om )
4 1pi 7345 . . . . . . . . . . . 12  |-  1o  e.  N.
54elexi 2764 . . . . . . . . . . 11  |-  1o  e.  _V
65sucid 4435 . . . . . . . . . 10  |-  1o  e.  suc  1o
7 df-2o 6443 . . . . . . . . . 10  |-  2o  =  suc  1o
86, 7eleqtrri 2265 . . . . . . . . 9  |-  1o  e.  2o
9 2onn 6547 . . . . . . . . . . 11  |-  2o  e.  om
10 nnaword2 6540 . . . . . . . . . . 11  |-  ( ( 2o  e.  om  /\  z  e.  om )  ->  2o  C_  ( z  +o  2o ) )
119, 10mpan 424 . . . . . . . . . 10  |-  ( z  e.  om  ->  2o  C_  ( z  +o  2o ) )
1211sseld 3169 . . . . . . . . 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 6494 . . . . . . . . 9  |-  ( 1o 
+o  1o )  =  2o
16 addpiord 7346 . . . . . . . . . . 11  |-  ( ( 1o  e.  N.  /\  1o  e.  N. )  -> 
( 1o  +N  1o )  =  ( 1o  +o  1o ) )
174, 4, 16mp2an 426 . . . . . . . . . 10  |-  ( 1o 
+N  1o )  =  ( 1o  +o  1o )
18 addclpi 7357 . . . . . . . . . . 11  |-  ( ( 1o  e.  N.  /\  1o  e.  N. )  -> 
( 1o  +N  1o )  e.  N. )
194, 4, 18mp2an 426 . . . . . . . . . 10  |-  ( 1o 
+N  1o )  e. 
N.
2017, 19eqeltrri 2263 . . . . . . . . 9  |-  ( 1o 
+o  1o )  e. 
N.
2115, 20eqeltrri 2263 . . . . . . . 8  |-  2o  e.  N.
22 addpiord 7346 . . . . . . . 8  |-  ( ( z  e.  N.  /\  2o  e.  N. )  -> 
( z  +N  2o )  =  ( z  +o  2o ) )
2321, 22mpan2 425 . . . . . . 7  |-  ( z  e.  N.  ->  (
z  +N  2o )  =  ( z  +o  2o ) )
2414, 23eleqtrrd 2269 . . . . . 6  |-  ( z  e.  N.  ->  1o  e.  ( z  +N  2o ) )
25 addclpi 7357 . . . . . . . 8  |-  ( ( z  e.  N.  /\  2o  e.  N. )  -> 
( z  +N  2o )  e.  N. )
2621, 25mpan2 425 . . . . . . 7  |-  ( z  e.  N.  ->  (
z  +N  2o )  e.  N. )
27 ltpiord 7349 . . . . . . . 8  |-  ( ( 1o  e.  N.  /\  ( z  +N  2o )  e.  N. )  ->  ( 1o  <N  (
z  +N  2o )  <-> 
1o  e.  ( z  +N  2o ) ) )
284, 27mpan 424 . . . . . . 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 167 . . . . 5  |-  ( z  e.  N.  ->  1o  <N  ( z  +N  2o ) )
3130adantl 277 . . . 4  |-  ( ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  /\  z  e.  N. )  ->  1o  <N  (
z  +N  2o ) )
3231adantrr 479 . . 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 6500 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  om  ->  (
z  +o  (/) )  =  z )
34 0lt1o 6466 . . . . . . . . . . . . . . . . . . . 20  |-  (/)  e.  1o
35 1on 6449 . . . . . . . . . . . . . . . . . . . . . 22  |-  1o  e.  On
3635onsuci 4533 . . . . . . . . . . . . . . . . . . . . 21  |-  suc  1o  e.  On
37 ontr1 4407 . . . . . . . . . . . . . . . . . . . . 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 426 . . . . . . . . . . . . . . . . . . 19  |-  (/)  e.  suc  1o
4039, 7eleqtrri 2265 . . . . . . . . . . . . . . . . . 18  |-  (/)  e.  2o
41 nnaordi 6534 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 2o  e.  om  /\  z  e.  om )  ->  ( (/)  e.  2o  ->  ( z  +o  (/) )  e.  ( z  +o  2o ) ) )
429, 41mpan 424 . . . . . . . . . . . . . . . . . 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 2267 . . . . . . . . . . . . . . . 16  |-  ( z  e.  om  ->  z  e.  ( z  +o  2o ) )
453, 44syl 14 . . . . . . . . . . . . . . 15  |-  ( z  e.  N.  ->  z  e.  ( z  +o  2o ) )
4645, 23eleqtrrd 2269 . . . . . . . . . . . . . 14  |-  ( z  e.  N.  ->  z  e.  ( z  +N  2o ) )
47 ltpiord 7349 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  N.  /\  ( z  +N  2o )  e.  N. )  ->  ( z  <N  (
z  +N  2o )  <-> 
z  e.  ( z  +N  2o ) ) )
4826, 47mpdan 421 . . . . . . . . . . . . . 14  |-  ( z  e.  N.  ->  (
z  <N  ( z  +N  2o )  <->  z  e.  ( z  +N  2o ) ) )
4946, 48mpbird 167 . . . . . . . . . . . . 13  |-  ( z  e.  N.  ->  z  <N  ( z  +N  2o ) )
50 mulidpi 7348 . . . . . . . . . . . . 13  |-  ( z  e.  N.  ->  (
z  .N  1o )  =  z )
51 mulcompig 7361 . . . . . . . . . . . . . . . 16  |-  ( ( ( z  +N  2o )  e.  N.  /\  1o  e.  N. )  ->  (
( z  +N  2o )  .N  1o )  =  ( 1o  .N  (
z  +N  2o ) ) )
524, 51mpan2 425 . . . . . . . . . . . . . . 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 7348 . . . . . . . . . . . . . . 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 2224 . . . . . . . . . . . . 13  |-  ( z  e.  N.  ->  ( 1o  .N  ( z  +N  2o ) )  =  ( z  +N  2o ) )
5749, 50, 563brtr4d 4050 . . . . . . . . . . . 12  |-  ( z  e.  N.  ->  (
z  .N  1o ) 
<N  ( 1o  .N  (
z  +N  2o ) ) )
58 ordpipqqs 7404 . . . . . . . . . . . . . . 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 435 . . . . . . . . . . . . . 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 438 . . . . . . . . . . . . 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 421 . . . . . . . . . . . 12  |-  ( z  e.  N.  ->  ( [ <. z ,  1o >. ]  ~Q  <Q  [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  <->  ( z  .N  1o ) 
<N  ( 1o  .N  (
z  +N  2o ) ) ) )
6257, 61mpbird 167 . . . . . . . . . . 11  |-  ( z  e.  N.  ->  [ <. z ,  1o >. ]  ~Q  <Q  [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  )
6362adantl 277 . . . . . . . . . 10  |-  ( ( C  e.  Q.  /\  z  e.  N. )  ->  [ <. z ,  1o >. ]  ~Q  <Q  [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  )
64 opelxpi 4676 . . . . . . . . . . . . . . . 16  |-  ( ( ( z  +N  2o )  e.  N.  /\  1o  e.  N. )  ->  <. (
z  +N  2o ) ,  1o >.  e.  ( N.  X.  N. )
)
654, 64mpan2 425 . . . . . . . . . . . . . . 15  |-  ( ( z  +N  2o )  e.  N.  ->  <. (
z  +N  2o ) ,  1o >.  e.  ( N.  X.  N. )
)
66 enqex 7390 . . . . . . . . . . . . . . . 16  |-  ~Q  e.  _V
6766ecelqsi 6616 . . . . . . . . . . . . . . 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 7378 . . . . . . . . . . . . . 14  |-  Q.  =  ( ( N.  X.  N. ) /.  ~Q  )
7068, 69eleqtrrdi 2283 . . . . . . . . . . . . 13  |-  ( z  e.  N.  ->  [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  e.  Q. )
71 opelxpi 4676 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  N.  /\  1o  e.  N. )  ->  <. z ,  1o >.  e.  ( N.  X.  N. ) )
724, 71mpan2 425 . . . . . . . . . . . . . . . 16  |-  ( z  e.  N.  ->  <. z ,  1o >.  e.  ( N.  X.  N. ) )
7366ecelqsi 6616 . . . . . . . . . . . . . . . 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 2283 . . . . . . . . . . . . . 14  |-  ( z  e.  N.  ->  [ <. z ,  1o >. ]  ~Q  e.  Q. )
76 ltmnqg 7431 . . . . . . . . . . . . . 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 1282 . . . . . . . . . . . . 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 1283 . . . . . . . . . . . 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 1306 . . . . . . . . . . 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 268 . . . . . . . . . 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 147 . . . . . . . . 9  |-  ( ( C  e.  Q.  /\  z  e.  N. )  ->  ( C  .Q  [ <. z ,  1o >. ]  ~Q  )  <Q  ( C  .Q  [ <. (
z  +N  2o ) ,  1o >. ]  ~Q  ) )
82 mulcomnqg 7413 . . . . . . . . . 10  |-  ( ( C  e.  Q.  /\  [
<. z ,  1o >. ]  ~Q  e.  Q. )  ->  ( C  .Q  [ <. z ,  1o >. ]  ~Q  )  =  ( [ <. z ,  1o >. ]  ~Q  .Q  C
) )
8375, 82sylan2 286 . . . . . . . . 9  |-  ( ( C  e.  Q.  /\  z  e.  N. )  ->  ( C  .Q  [ <. z ,  1o >. ]  ~Q  )  =  ( [ <. z ,  1o >. ]  ~Q  .Q  C
) )
84 mulcomnqg 7413 . . . . . . . . . 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 286 . . . . . . . . 9  |-  ( ( C  e.  Q.  /\  z  e.  N. )  ->  ( C  .Q  [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  )  =  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C ) )
8681, 83, 853brtr3d 4049 . . . . . . . 8  |-  ( ( C  e.  Q.  /\  z  e.  N. )  ->  ( [ <. z ,  1o >. ]  ~Q  .Q  C )  <Q  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C
) )
87863ad2antl3 1163 . . . . . . 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 479 . . . . . 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 7428 . . . . . . . . . 10  |-  <Q  Or  Q.
90 ltrelnq 7395 . . . . . . . . . 10  |-  <Q  C_  ( Q.  X.  Q. )
9189, 90sotri 5042 . . . . . . . . 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 115 . . . . . . . 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 277 . . . . . . 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 277 . . . . . 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 7406 . . . . . . . . . 10  |-  ( ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  e.  Q.  /\  C  e.  Q. )  ->  ( [ <. (
z  +N  2o ) ,  1o >. ]  ~Q  .Q  C )  e.  Q. )
9770, 96sylan 283 . . . . . . . . 9  |-  ( ( z  e.  N.  /\  C  e.  Q. )  ->  ( [ <. (
z  +N  2o ) ,  1o >. ]  ~Q  .Q  C )  e.  Q. )
9897ancoms 268 . . . . . . . 8  |-  ( ( C  e.  Q.  /\  z  e.  N. )  ->  ( [ <. (
z  +N  2o ) ,  1o >. ]  ~Q  .Q  C )  e.  Q. )
99983ad2antl3 1163 . . . . . . 7  |-  ( ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  /\  z  e.  N. )  ->  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C )  e.  Q. )
100 simpl2 1003 . . . . . . 7  |-  ( ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  /\  z  e.  N. )  ->  B  e.  Q. )
101 ltaddnq 7437 . . . . . . 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 411 . . . . . 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 479 . . . . 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 5042 . . . . 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 411 . . . 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 7411 . . . . . . 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 411 . . . . . 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 4030 . . . . 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 479 . . . 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 147 . . 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 110 . . . . 5  |-  ( ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  /\  z  e.  N. )  ->  z  e.  N. )
112 breq2 4022 . . . . . . . 8  |-  ( x  =  ( z  +N  2o )  ->  ( 1o  <N  x  <->  1o  <N  ( z  +N  2o ) ) )
113 opeq1 3793 . . . . . . . . . . . 12  |-  ( x  =  ( z  +N  2o )  ->  <. x ,  1o >.  =  <. ( z  +N  2o ) ,  1o >. )
114113eceq1d 6596 . . . . . . . . . . 11  |-  ( x  =  ( z  +N  2o )  ->  [ <. x ,  1o >. ]  ~Q  =  [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  )
115114oveq1d 5912 . . . . . . . . . 10  |-  ( x  =  ( z  +N  2o )  ->  ( [ <. x ,  1o >. ]  ~Q  .Q  C
)  =  ( [
<. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C ) )
116115oveq2d 5913 . . . . . . . . 9  |-  ( x  =  ( z  +N  2o )  ->  ( B  +Q  ( [ <. x ,  1o >. ]  ~Q  .Q  C ) )  =  ( B  +Q  ( [ <. ( z  +N  2o ) ,  1o >. ]  ~Q  .Q  C
) ) )
117116breq2d 4030 . . . . . . . 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 473 . . . . . . 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 2856 . . . . . 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 115 . . . . 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 479 . . 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 433 . 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 2612 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 104    <-> wb 105    /\ w3a 980    = wceq 1364    e. wcel 2160   E.wrex 2469    C_ wss 3144   (/)c0 3437   <.cop 3610   class class class wbr 4018   Oncon0 4381   suc csuc 4383   omcom 4607    X. cxp 4642  (class class class)co 5897   1oc1o 6435   2oc2o 6436    +o coa 6439   [cec 6558   /.cqs 6559   N.cnpi 7302    +N cpli 7303    .N cmi 7304    <N clti 7305    ~Q ceq 7309   Q.cnq 7310    +Q cplq 7312    .Q cmq 7313    <Q cltq 7315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-1o 6442  df-2o 6443  df-oadd 6446  df-omul 6447  df-er 6560  df-ec 6562  df-qs 6566  df-ni 7334  df-pli 7335  df-mi 7336  df-lti 7337  df-plpq 7374  df-mpq 7375  df-enq 7377  df-nqqs 7378  df-plqqs 7379  df-mqqs 7380  df-1nqqs 7381  df-rq 7382  df-ltnqqs 7383
This theorem is referenced by:  prarloc  7533
  Copyright terms: Public domain W3C validator