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Theorem nn0min 26095
Description: Extracting the minimum positive integer for which a property  ch does not hold. This uses substitutions similar to nn0ind 10743. (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 1673 . . . . . . . . . 10  |-  F/ m ph
4 nfra1 2771 . . . . . . . . . 10  |-  F/ m A. m  e.  NN0  ( -.  th  ->  -. 
ta )
53, 4nfan 1861 . . . . . . . . 9  |-  F/ m
( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta ) )
6 nfv 1673 . . . . . . . . 9  |-  F/ m  -.  [ k  /  n ] ps
75, 6nfim 1853 . . . . . . . 8  |-  F/ m
( ( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta )
)  ->  -.  [ k  /  n ] ps )
8 dfsbcq2 3194 . . . . . . . . . 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 1673 . . . . . . . . . . 11  |-  F/ n th
12 nn0min.1 . . . . . . . . . . 11  |-  ( n  =  m  ->  ( ps 
<->  th ) )
1311, 12sbhypf 3024 . . . . . . . . . 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 1673 . . . . . . . . . . 11  |-  F/ n ta
17 nn0min.2 . . . . . . . . . . 11  |-  ( n  =  ( m  + 
1 )  ->  ( ps 
<->  ta ) )
1816, 17sbhypf 3024 . . . . . . . . . 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 1937 . . . . . . . . . 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 10599 . . . . . . . . . 10  |-  0  e.  NN0
2611, 12sbie 2100 . . . . . . . . . . . . . 14  |-  ( [ m  /  n ] ps 
<->  th )
27 nfv 1673 . . . . . . . . . . . . . . 15  |-  F/ n ch
28 nn0min.0 . . . . . . . . . . . . . . 15  |-  ( n  =  0  ->  ( ps 
<->  ch ) )
2927, 28sbhypf 3024 . . . . . . . . . . . . . 14  |-  ( m  =  0  ->  ( [ m  /  n ] ps  <->  ch ) )
3026, 29syl5bbr 259 . . . . . . . . . . . . 13  |-  ( m  =  0  ->  ( th 
<->  ch ) )
3130notbid 294 . . . . . . . . . . . 12  |-  ( m  =  0  ->  ( -.  th  <->  -.  ch )
)
32 oveq1 6103 . . . . . . . . . . . . . . . 16  |-  ( m  =  0  ->  (
m  +  1 )  =  ( 0  +  1 ) )
33 0p1e1 10438 . . . . . . . . . . . . . . . 16  |-  ( 0  +  1 )  =  1
3432, 33syl6eq 2491 . . . . . . . . . . . . . . 15  |-  ( m  =  0  ->  (
m  +  1 )  =  1 )
35 1nn 10338 . . . . . . . . . . . . . . . 16  |-  1  e.  NN
36 eleq1 2503 . . . . . . . . . . . . . . . 16  |-  ( ( m  +  1 )  =  1  ->  (
( m  +  1 )  e.  NN  <->  1  e.  NN ) )
3735, 36mpbiri 233 . . . . . . . . . . . . . . 15  |-  ( ( m  +  1 )  =  1  ->  (
m  +  1 )  e.  NN )
3817sbcieg 3224 . . . . . . . . . . . . . . 15  |-  ( ( m  +  1 )  e.  NN  ->  ( [. ( m  +  1 )  /  n ]. ps 
<->  ta ) )
3934, 37, 383syl 20 . . . . . . . . . . . . . 14  |-  ( m  =  0  ->  ( [. ( m  +  1 )  /  n ]. ps 
<->  ta ) )
40 dfsbcq 3193 . . . . . . . . . . . . . . 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 3074 . . . . . . . . . 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 1673 . . . . . . . . . . . 12  |-  F/ k ( -.  th  ->  -. 
ta )
4948nfs1 2054 . . . . . . . . . . . 12  |-  F/ m [ k  /  m ] ( -.  th  ->  -.  ta )
50 sbequ12 1936 . . . . . . . . . . . 12  |-  ( m  =  k  ->  (
( -.  th  ->  -. 
ta )  <->  [ k  /  m ] ( -. 
th  ->  -.  ta )
) )
5148, 49, 50cbvral 2948 . . . . . . . . . . 11  |-  ( A. m  e.  NN0  ( -. 
th  ->  -.  ta )  <->  A. k  e.  NN0  [ k  /  m ] ( -.  th  ->  -.  ta ) )
52 nnnn0 10591 . . . . . . . . . . . 12  |-  ( m  e.  NN  ->  m  e.  NN0 )
53 sbequ12r 1937 . . . . . . . . . . . . 13  |-  ( k  =  m  ->  ( [ k  /  m ] ( -.  th  ->  -.  ta )  <->  ( -.  th 
->  -.  ta ) ) )
5448, 53rspc 3072 . . . . . . . . . . . 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 26094 . . . . . . 7  |-  ( n  e.  NN  ->  (
( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta ) )  ->  -.  ps )
)
6059rgen 2786 . . . . . 6  |-  A. n  e.  NN  ( ( ph  /\ 
A. m  e.  NN0  ( -.  th  ->  -. 
ta ) )  ->  -.  ps )
61 r19.21v 2808 . . . . . 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 2730 . . . . 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 2744 . . . 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 2733 . 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 1369   [wsb 1700    e. wcel 1756   A.wral 2720   E.wrex 2721   [.wsbc 3191  (class class class)co 6096   0cc0 9287   1c1 9288    + caddc 9290   NNcn 10327   NN0cn0 10584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-om 6482  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-ltxr 9428  df-nn 10328  df-n0 10585
This theorem is referenced by:  archirng  26210
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