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Theorem nn0min 28076
Description: Extracting the minimum positive integer for which a property  ch does not hold. This uses substitutions similar to nn0ind 11000. (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 1730 . . . . . . . . . 10  |-  F/ m ph
4 nfra1 2787 . . . . . . . . . 10  |-  F/ m A. m  e.  NN0  ( -.  th  ->  -. 
ta )
53, 4nfan 1958 . . . . . . . . 9  |-  F/ m
( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta ) )
6 nfv 1730 . . . . . . . . 9  |-  F/ m  -.  [ k  /  n ] ps
75, 6nfim 1950 . . . . . . . 8  |-  F/ m
( ( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta )
)  ->  -.  [ k  /  n ] ps )
8 dfsbcq2 3282 . . . . . . . . . 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 1730 . . . . . . . . . . 11  |-  F/ n th
12 nn0min.1 . . . . . . . . . . 11  |-  ( n  =  m  ->  ( ps 
<->  th ) )
1311, 12sbhypf 3108 . . . . . . . . . 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 1730 . . . . . . . . . . 11  |-  F/ n ta
17 nn0min.2 . . . . . . . . . . 11  |-  ( n  =  ( m  + 
1 )  ->  ( ps 
<->  ta ) )
1816, 17sbhypf 3108 . . . . . . . . . 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 2023 . . . . . . . . . 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 10853 . . . . . . . . . 10  |-  0  e.  NN0
2611, 12sbie 2175 . . . . . . . . . . . . . 14  |-  ( [ m  /  n ] ps 
<->  th )
27 nfv 1730 . . . . . . . . . . . . . . 15  |-  F/ n ch
28 nn0min.0 . . . . . . . . . . . . . . 15  |-  ( n  =  0  ->  ( ps 
<->  ch ) )
2927, 28sbhypf 3108 . . . . . . . . . . . . . 14  |-  ( m  =  0  ->  ( [ m  /  n ] ps  <->  ch ) )
3026, 29syl5bbr 261 . . . . . . . . . . . . 13  |-  ( m  =  0  ->  ( th 
<->  ch ) )
3130notbid 294 . . . . . . . . . . . 12  |-  ( m  =  0  ->  ( -.  th  <->  -.  ch )
)
32 oveq1 6287 . . . . . . . . . . . . . . . 16  |-  ( m  =  0  ->  (
m  +  1 )  =  ( 0  +  1 ) )
33 0p1e1 10690 . . . . . . . . . . . . . . . 16  |-  ( 0  +  1 )  =  1
3432, 33syl6eq 2461 . . . . . . . . . . . . . . 15  |-  ( m  =  0  ->  (
m  +  1 )  =  1 )
35 1nn 10589 . . . . . . . . . . . . . . . 16  |-  1  e.  NN
36 eleq1 2476 . . . . . . . . . . . . . . . 16  |-  ( ( m  +  1 )  =  1  ->  (
( m  +  1 )  e.  NN  <->  1  e.  NN ) )
3735, 36mpbiri 235 . . . . . . . . . . . . . . 15  |-  ( ( m  +  1 )  =  1  ->  (
m  +  1 )  e.  NN )
3817sbcieg 3312 . . . . . . . . . . . . . . 15  |-  ( ( m  +  1 )  e.  NN  ->  ( [. ( m  +  1 )  /  n ]. ps 
<->  ta ) )
3934, 37, 383syl 18 . . . . . . . . . . . . . 14  |-  ( m  =  0  ->  ( [. ( m  +  1 )  /  n ]. ps 
<->  ta ) )
4034sbceq1d 3284 . . . . . . . . . . . . . 14  |-  ( m  =  0  ->  ( [. ( m  +  1 )  /  n ]. ps 
<-> 
[. 1  /  n ]. ps ) )
4139, 40bitr3d 257 . . . . . . . . . . . . 13  |-  ( m  =  0  ->  ( ta 
<-> 
[. 1  /  n ]. ps ) )
4241notbid 294 . . . . . . . . . . . 12  |-  ( m  =  0  ->  ( -.  ta  <->  -.  [. 1  /  n ]. ps )
)
4331, 42imbi12d 320 . . . . . . . . . . 11  |-  ( m  =  0  ->  (
( -.  th  ->  -. 
ta )  <->  ( -.  ch  ->  -.  [. 1  /  n ]. ps )
) )
4443rspcv 3158 . . . . . . . . . 10  |-  ( 0  e.  NN0  ->  ( A. m  e.  NN0  ( -. 
th  ->  -.  ta )  ->  ( -.  ch  ->  -. 
[. 1  /  n ]. ps ) ) )
4525, 44ax-mp 5 . . . . . . . . 9  |-  ( A. m  e.  NN0  ( -. 
th  ->  -.  ta )  ->  ( -.  ch  ->  -. 
[. 1  /  n ]. ps ) )
4624, 45mpan9 469 . . . . . . . 8  |-  ( (
ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta ) )  ->  -.  [. 1  /  n ]. ps )
47 cbvralsv 3047 . . . . . . . . . . 11  |-  ( A. m  e.  NN0  ( -. 
th  ->  -.  ta )  <->  A. k  e.  NN0  [ k  /  m ] ( -.  th  ->  -.  ta ) )
48 nnnn0 10845 . . . . . . . . . . . 12  |-  ( m  e.  NN  ->  m  e.  NN0 )
49 sbequ12r 2023 . . . . . . . . . . . . 13  |-  ( k  =  m  ->  ( [ k  /  m ] ( -.  th  ->  -.  ta )  <->  ( -.  th 
->  -.  ta ) ) )
5049rspcv 3158 . . . . . . . . . . . 12  |-  ( m  e.  NN0  ->  ( A. k  e.  NN0  [ k  /  m ] ( -.  th  ->  -.  ta )  ->  ( -. 
th  ->  -.  ta )
) )
5148, 50syl 17 . . . . . . . . . . 11  |-  ( m  e.  NN  ->  ( A. k  e.  NN0  [ k  /  m ]
( -.  th  ->  -. 
ta )  ->  ( -.  th  ->  -.  ta )
) )
5247, 51syl5bi 219 . . . . . . . . . 10  |-  ( m  e.  NN  ->  ( A. m  e.  NN0  ( -.  th  ->  -. 
ta )  ->  ( -.  th  ->  -.  ta )
) )
5352adantld 467 . . . . . . . . 9  |-  ( m  e.  NN  ->  (
( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta ) )  ->  ( -.  th  ->  -.  ta ) ) )
5453a2d 28 . . . . . . . 8  |-  ( m  e.  NN  ->  (
( ( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta )
)  ->  -.  th )  ->  ( ( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta )
)  ->  -.  ta )
) )
557, 10, 15, 20, 23, 46, 54nnindf 28074 . . . . . . 7  |-  ( n  e.  NN  ->  (
( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta ) )  ->  -.  ps )
)
5655rgen 2766 . . . . . 6  |-  A. n  e.  NN  ( ( ph  /\ 
A. m  e.  NN0  ( -.  th  ->  -. 
ta ) )  ->  -.  ps )
57 r19.21v 2811 . . . . . 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 )
)
5856, 57mpbi 210 . . . . 5  |-  ( (
ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta ) )  ->  A. n  e.  NN  -.  ps )
59 ralnex 2852 . . . . 5  |-  ( A. n  e.  NN  -.  ps 
<->  -.  E. n  e.  NN  ps )
6058, 59sylib 198 . . . 4  |-  ( (
ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta ) )  ->  -.  E. n  e.  NN  ps )
612, 60pm2.65da 576 . . 3  |-  ( ph  ->  -.  A. m  e. 
NN0  ( -.  th  ->  -.  ta ) )
62 imnan 422 . . . 4  |-  ( ( -.  th  ->  -.  ta )  <->  -.  ( -.  th 
/\  ta ) )
6362ralbii 2837 . . 3  |-  ( A. m  e.  NN0  ( -. 
th  ->  -.  ta )  <->  A. m  e.  NN0  -.  ( -.  th  /\  ta ) )
6461, 63sylnib 304 . 2  |-  ( ph  ->  -.  A. m  e. 
NN0  -.  ( -.  th 
/\  ta ) )
65 dfrex2 2857 . 2  |-  ( E. m  e.  NN0  ( -.  th  /\  ta )  <->  -. 
A. m  e.  NN0  -.  ( -.  th  /\  ta ) )
6664, 65sylibr 214 1  |-  ( ph  ->  E. m  e.  NN0  ( -.  th  /\  ta ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 186    /\ wa 369    = wceq 1407   [wsb 1765    e. wcel 1844   A.wral 2756   E.wrex 2757   [.wsbc 3279  (class class class)co 6280   0cc0 9524   1c1 9525    + caddc 9527   NNcn 10578   NN0cn0 10838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-om 6686  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-pnf 9662  df-mnf 9663  df-ltxr 9665  df-nn 10579  df-n0 10839
This theorem is referenced by:  archirng  28197
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