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Theorem nn0min 27435
Description: Extracting the minimum positive integer for which a property  ch does not hold. This uses substitutions similar to nn0ind 10969. (Contributed by Thierry Arnoux, 6-May-2018.)
Hypotheses
Ref Expression
nn0min.0  |-  ( n  =  0  ->  ( ps 
<->  ch ) )
nn0min.1  |-  ( n  =  m  ->  ( ps 
<->  th ) )
nn0min.2  |-  ( n  =  ( m  + 
1 )  ->  ( ps 
<->  ta ) )
nn0min.3  |-  ( ph  ->  -.  ch )
nn0min.4  |-  ( ph  ->  E. n  e.  NN  ps )
Assertion
Ref Expression
nn0min  |-  ( ph  ->  E. m  e.  NN0  ( -.  th  /\  ta ) )
Distinct variable groups:    m, n, ph    ps, m    ta, n    th, n    ch, m, n
Allowed substitution hints:    ps( n)    th( m)    ta( m)

Proof of Theorem nn0min
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 nn0min.4 . . . . 5  |-  ( ph  ->  E. n  e.  NN  ps )
21adantr 465 . . . 4  |-  ( (
ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta ) )  ->  E. n  e.  NN  ps )
3 nfv 1683 . . . . . . . . . 10  |-  F/ m ph
4 nfra1 2848 . . . . . . . . . 10  |-  F/ m A. m  e.  NN0  ( -.  th  ->  -. 
ta )
53, 4nfan 1875 . . . . . . . . 9  |-  F/ m
( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta ) )
6 nfv 1683 . . . . . . . . 9  |-  F/ m  -.  [ k  /  n ] ps
75, 6nfim 1867 . . . . . . . 8  |-  F/ m
( ( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta )
)  ->  -.  [ k  /  n ] ps )
8 dfsbcq2 3339 . . . . . . . . . 10  |-  ( k  =  1  ->  ( [ k  /  n ] ps  <->  [. 1  /  n ]. ps ) )
98notbid 294 . . . . . . . . 9  |-  ( k  =  1  ->  ( -.  [ k  /  n ] ps  <->  -.  [. 1  /  n ]. ps )
)
109imbi2d 316 . . . . . . . 8  |-  ( k  =  1  ->  (
( ( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta )
)  ->  -.  [ k  /  n ] ps ) 
<->  ( ( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta )
)  ->  -.  [. 1  /  n ]. ps )
) )
11 nfv 1683 . . . . . . . . . . 11  |-  F/ n th
12 nn0min.1 . . . . . . . . . . 11  |-  ( n  =  m  ->  ( ps 
<->  th ) )
1311, 12sbhypf 3165 . . . . . . . . . 10  |-  ( k  =  m  ->  ( [ k  /  n ] ps  <->  th ) )
1413notbid 294 . . . . . . . . 9  |-  ( k  =  m  ->  ( -.  [ k  /  n ] ps  <->  -.  th )
)
1514imbi2d 316 . . . . . . . 8  |-  ( k  =  m  ->  (
( ( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta )
)  ->  -.  [ k  /  n ] ps ) 
<->  ( ( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta )
)  ->  -.  th )
) )
16 nfv 1683 . . . . . . . . . . 11  |-  F/ n ta
17 nn0min.2 . . . . . . . . . . 11  |-  ( n  =  ( m  + 
1 )  ->  ( ps 
<->  ta ) )
1816, 17sbhypf 3165 . . . . . . . . . 10  |-  ( k  =  ( m  + 
1 )  ->  ( [ k  /  n ] ps  <->  ta ) )
1918notbid 294 . . . . . . . . 9  |-  ( k  =  ( m  + 
1 )  ->  ( -.  [ k  /  n ] ps  <->  -.  ta )
)
2019imbi2d 316 . . . . . . . 8  |-  ( k  =  ( m  + 
1 )  ->  (
( ( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta )
)  ->  -.  [ k  /  n ] ps ) 
<->  ( ( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta )
)  ->  -.  ta )
) )
21 sbequ12r 1962 . . . . . . . . . 10  |-  ( k  =  n  ->  ( [ k  /  n ] ps  <->  ps ) )
2221notbid 294 . . . . . . . . 9  |-  ( k  =  n  ->  ( -.  [ k  /  n ] ps  <->  -.  ps )
)
2322imbi2d 316 . . . . . . . 8  |-  ( k  =  n  ->  (
( ( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta )
)  ->  -.  [ k  /  n ] ps ) 
<->  ( ( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta )
)  ->  -.  ps )
) )
24 nn0min.3 . . . . . . . . 9  |-  ( ph  ->  -.  ch )
25 0nn0 10822 . . . . . . . . . 10  |-  0  e.  NN0
2611, 12sbie 2123 . . . . . . . . . . . . . 14  |-  ( [ m  /  n ] ps 
<->  th )
27 nfv 1683 . . . . . . . . . . . . . . 15  |-  F/ n ch
28 nn0min.0 . . . . . . . . . . . . . . 15  |-  ( n  =  0  ->  ( ps 
<->  ch ) )
2927, 28sbhypf 3165 . . . . . . . . . . . . . 14  |-  ( m  =  0  ->  ( [ m  /  n ] ps  <->  ch ) )
3026, 29syl5bbr 259 . . . . . . . . . . . . 13  |-  ( m  =  0  ->  ( th 
<->  ch ) )
3130notbid 294 . . . . . . . . . . . 12  |-  ( m  =  0  ->  ( -.  th  <->  -.  ch )
)
32 oveq1 6302 . . . . . . . . . . . . . . . 16  |-  ( m  =  0  ->  (
m  +  1 )  =  ( 0  +  1 ) )
33 0p1e1 10659 . . . . . . . . . . . . . . . 16  |-  ( 0  +  1 )  =  1
3432, 33syl6eq 2524 . . . . . . . . . . . . . . 15  |-  ( m  =  0  ->  (
m  +  1 )  =  1 )
35 1nn 10559 . . . . . . . . . . . . . . . 16  |-  1  e.  NN
36 eleq1 2539 . . . . . . . . . . . . . . . 16  |-  ( ( m  +  1 )  =  1  ->  (
( m  +  1 )  e.  NN  <->  1  e.  NN ) )
3735, 36mpbiri 233 . . . . . . . . . . . . . . 15  |-  ( ( m  +  1 )  =  1  ->  (
m  +  1 )  e.  NN )
3817sbcieg 3369 . . . . . . . . . . . . . . 15  |-  ( ( m  +  1 )  e.  NN  ->  ( [. ( m  +  1 )  /  n ]. ps 
<->  ta ) )
3934, 37, 383syl 20 . . . . . . . . . . . . . 14  |-  ( m  =  0  ->  ( [. ( m  +  1 )  /  n ]. ps 
<->  ta ) )
40 dfsbcq 3338 . . . . . . . . . . . . . . 15  |-  ( ( m  +  1 )  =  1  ->  ( [. ( m  +  1 )  /  n ]. ps 
<-> 
[. 1  /  n ]. ps ) )
4134, 40syl 16 . . . . . . . . . . . . . 14  |-  ( m  =  0  ->  ( [. ( m  +  1 )  /  n ]. ps 
<-> 
[. 1  /  n ]. ps ) )
4239, 41bitr3d 255 . . . . . . . . . . . . 13  |-  ( m  =  0  ->  ( ta 
<-> 
[. 1  /  n ]. ps ) )
4342notbid 294 . . . . . . . . . . . 12  |-  ( m  =  0  ->  ( -.  ta  <->  -.  [. 1  /  n ]. ps )
)
4431, 43imbi12d 320 . . . . . . . . . . 11  |-  ( m  =  0  ->  (
( -.  th  ->  -. 
ta )  <->  ( -.  ch  ->  -.  [. 1  /  n ]. ps )
) )
4544rspcv 3215 . . . . . . . . . 10  |-  ( 0  e.  NN0  ->  ( A. m  e.  NN0  ( -. 
th  ->  -.  ta )  ->  ( -.  ch  ->  -. 
[. 1  /  n ]. ps ) ) )
4625, 45ax-mp 5 . . . . . . . . 9  |-  ( A. m  e.  NN0  ( -. 
th  ->  -.  ta )  ->  ( -.  ch  ->  -. 
[. 1  /  n ]. ps ) )
4724, 46mpan9 469 . . . . . . . 8  |-  ( (
ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta ) )  ->  -.  [. 1  /  n ]. ps )
48 nfv 1683 . . . . . . . . . . . 12  |-  F/ k ( -.  th  ->  -. 
ta )
4948nfs1 2077 . . . . . . . . . . . 12  |-  F/ m [ k  /  m ] ( -.  th  ->  -.  ta )
50 sbequ12 1961 . . . . . . . . . . . 12  |-  ( m  =  k  ->  (
( -.  th  ->  -. 
ta )  <->  [ k  /  m ] ( -. 
th  ->  -.  ta )
) )
5148, 49, 50cbvral 3089 . . . . . . . . . . 11  |-  ( A. m  e.  NN0  ( -. 
th  ->  -.  ta )  <->  A. k  e.  NN0  [ k  /  m ] ( -.  th  ->  -.  ta ) )
52 nnnn0 10814 . . . . . . . . . . . 12  |-  ( m  e.  NN  ->  m  e.  NN0 )
53 sbequ12r 1962 . . . . . . . . . . . . 13  |-  ( k  =  m  ->  ( [ k  /  m ] ( -.  th  ->  -.  ta )  <->  ( -.  th 
->  -.  ta ) ) )
5448, 53rspc 3213 . . . . . . . . . . . 12  |-  ( m  e.  NN0  ->  ( A. k  e.  NN0  [ k  /  m ] ( -.  th  ->  -.  ta )  ->  ( -. 
th  ->  -.  ta )
) )
5552, 54syl 16 . . . . . . . . . . 11  |-  ( m  e.  NN  ->  ( A. k  e.  NN0  [ k  /  m ]
( -.  th  ->  -. 
ta )  ->  ( -.  th  ->  -.  ta )
) )
5651, 55syl5bi 217 . . . . . . . . . 10  |-  ( m  e.  NN  ->  ( A. m  e.  NN0  ( -.  th  ->  -. 
ta )  ->  ( -.  th  ->  -.  ta )
) )
5756adantld 467 . . . . . . . . 9  |-  ( m  e.  NN  ->  (
( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta ) )  ->  ( -.  th  ->  -.  ta ) ) )
5857a2d 26 . . . . . . . 8  |-  ( m  e.  NN  ->  (
( ( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta )
)  ->  -.  th )  ->  ( ( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta )
)  ->  -.  ta )
) )
597, 10, 15, 20, 23, 47, 58nnindf 27434 . . . . . . 7  |-  ( n  e.  NN  ->  (
( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta ) )  ->  -.  ps )
)
6059rgen 2827 . . . . . 6  |-  A. n  e.  NN  ( ( ph  /\ 
A. m  e.  NN0  ( -.  th  ->  -. 
ta ) )  ->  -.  ps )
61 r19.21v 2872 . . . . . 6  |-  ( A. n  e.  NN  (
( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta ) )  ->  -.  ps )  <->  ( ( ph  /\  A. m  e.  NN0  ( -. 
th  ->  -.  ta )
)  ->  A. n  e.  NN  -.  ps )
)
6260, 61mpbi 208 . . . . 5  |-  ( (
ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta ) )  ->  A. n  e.  NN  -.  ps )
63 ralnex 2913 . . . . 5  |-  ( A. n  e.  NN  -.  ps 
<->  -.  E. n  e.  NN  ps )
6462, 63sylib 196 . . . 4  |-  ( (
ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta ) )  ->  -.  E. n  e.  NN  ps )
652, 64pm2.65da 576 . . 3  |-  ( ph  ->  -.  A. m  e. 
NN0  ( -.  th  ->  -.  ta ) )
66 imnan 422 . . . . 5  |-  ( ( -.  th  ->  -.  ta )  <->  -.  ( -.  th 
/\  ta ) )
6766ralbii 2898 . . . 4  |-  ( A. m  e.  NN0  ( -. 
th  ->  -.  ta )  <->  A. m  e.  NN0  -.  ( -.  th  /\  ta ) )
6867notbii 296 . . 3  |-  ( -. 
A. m  e.  NN0  ( -.  th  ->  -. 
ta )  <->  -.  A. m  e.  NN0  -.  ( -. 
th  /\  ta )
)
6965, 68sylib 196 . 2  |-  ( ph  ->  -.  A. m  e. 
NN0  -.  ( -.  th 
/\  ta ) )
70 dfrex2 2918 . 2  |-  ( E. m  e.  NN0  ( -.  th  /\  ta )  <->  -. 
A. m  e.  NN0  -.  ( -.  th  /\  ta ) )
7169, 70sylibr 212 1  |-  ( ph  ->  E. m  e.  NN0  ( -.  th  /\  ta ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   [wsb 1711    e. wcel 1767   A.wral 2817   E.wrex 2818   [.wsbc 3336  (class class class)co 6295   0cc0 9504   1c1 9505    + caddc 9507   NNcn 10548   NN0cn0 10807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-om 6696  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-ltxr 9645  df-nn 10549  df-n0 10808
This theorem is referenced by:  archirng  27556
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