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Theorem nn0min 28391
Description: Extracting the minimum positive integer for which a property  ch does not hold. This uses substitutions similar to nn0ind 11037. (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 466 . . . 4  |-  ( (
ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta ) )  ->  E. n  e.  NN  ps )
3 nfv 1755 . . . . . . . . . 10  |-  F/ m ph
4 nfra1 2803 . . . . . . . . . 10  |-  F/ m A. m  e.  NN0  ( -.  th  ->  -. 
ta )
53, 4nfan 1988 . . . . . . . . 9  |-  F/ m
( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta ) )
6 nfv 1755 . . . . . . . . 9  |-  F/ m  -.  [ k  /  n ] ps
75, 6nfim 1980 . . . . . . . 8  |-  F/ m
( ( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta )
)  ->  -.  [ k  /  n ] ps )
8 dfsbcq2 3302 . . . . . . . . . 10  |-  ( k  =  1  ->  ( [ k  /  n ] ps  <->  [. 1  /  n ]. ps ) )
98notbid 295 . . . . . . . . 9  |-  ( k  =  1  ->  ( -.  [ k  /  n ] ps  <->  -.  [. 1  /  n ]. ps )
)
109imbi2d 317 . . . . . . . 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 1755 . . . . . . . . . . 11  |-  F/ n th
12 nn0min.1 . . . . . . . . . . 11  |-  ( n  =  m  ->  ( ps 
<->  th ) )
1311, 12sbhypf 3128 . . . . . . . . . 10  |-  ( k  =  m  ->  ( [ k  /  n ] ps  <->  th ) )
1413notbid 295 . . . . . . . . 9  |-  ( k  =  m  ->  ( -.  [ k  /  n ] ps  <->  -.  th )
)
1514imbi2d 317 . . . . . . . 8  |-  ( k  =  m  ->  (
( ( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta )
)  ->  -.  [ k  /  n ] ps ) 
<->  ( ( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta )
)  ->  -.  th )
) )
16 nfv 1755 . . . . . . . . . . 11  |-  F/ n ta
17 nn0min.2 . . . . . . . . . . 11  |-  ( n  =  ( m  + 
1 )  ->  ( ps 
<->  ta ) )
1816, 17sbhypf 3128 . . . . . . . . . 10  |-  ( k  =  ( m  + 
1 )  ->  ( [ k  /  n ] ps  <->  ta ) )
1918notbid 295 . . . . . . . . 9  |-  ( k  =  ( m  + 
1 )  ->  ( -.  [ k  /  n ] ps  <->  -.  ta )
)
2019imbi2d 317 . . . . . . . 8  |-  ( k  =  ( m  + 
1 )  ->  (
( ( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta )
)  ->  -.  [ k  /  n ] ps ) 
<->  ( ( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta )
)  ->  -.  ta )
) )
21 sbequ12r 2052 . . . . . . . . . 10  |-  ( k  =  n  ->  ( [ k  /  n ] ps  <->  ps ) )
2221notbid 295 . . . . . . . . 9  |-  ( k  =  n  ->  ( -.  [ k  /  n ] ps  <->  -.  ps )
)
2322imbi2d 317 . . . . . . . 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 10891 . . . . . . . . . 10  |-  0  e.  NN0
2611, 12sbie 2206 . . . . . . . . . . . . . 14  |-  ( [ m  /  n ] ps 
<->  th )
27 nfv 1755 . . . . . . . . . . . . . . 15  |-  F/ n ch
28 nn0min.0 . . . . . . . . . . . . . . 15  |-  ( n  =  0  ->  ( ps 
<->  ch ) )
2927, 28sbhypf 3128 . . . . . . . . . . . . . 14  |-  ( m  =  0  ->  ( [ m  /  n ] ps  <->  ch ) )
3026, 29syl5bbr 262 . . . . . . . . . . . . 13  |-  ( m  =  0  ->  ( th 
<->  ch ) )
3130notbid 295 . . . . . . . . . . . 12  |-  ( m  =  0  ->  ( -.  th  <->  -.  ch )
)
32 oveq1 6312 . . . . . . . . . . . . . . . 16  |-  ( m  =  0  ->  (
m  +  1 )  =  ( 0  +  1 ) )
33 0p1e1 10728 . . . . . . . . . . . . . . . 16  |-  ( 0  +  1 )  =  1
3432, 33syl6eq 2479 . . . . . . . . . . . . . . 15  |-  ( m  =  0  ->  (
m  +  1 )  =  1 )
35 1nn 10627 . . . . . . . . . . . . . . . 16  |-  1  e.  NN
36 eleq1 2495 . . . . . . . . . . . . . . . 16  |-  ( ( m  +  1 )  =  1  ->  (
( m  +  1 )  e.  NN  <->  1  e.  NN ) )
3735, 36mpbiri 236 . . . . . . . . . . . . . . 15  |-  ( ( m  +  1 )  =  1  ->  (
m  +  1 )  e.  NN )
3817sbcieg 3332 . . . . . . . . . . . . . . 15  |-  ( ( m  +  1 )  e.  NN  ->  ( [. ( m  +  1 )  /  n ]. ps 
<->  ta ) )
3934, 37, 383syl 18 . . . . . . . . . . . . . 14  |-  ( m  =  0  ->  ( [. ( m  +  1 )  /  n ]. ps 
<->  ta ) )
4034sbceq1d 3304 . . . . . . . . . . . . . 14  |-  ( m  =  0  ->  ( [. ( m  +  1 )  /  n ]. ps 
<-> 
[. 1  /  n ]. ps ) )
4139, 40bitr3d 258 . . . . . . . . . . . . 13  |-  ( m  =  0  ->  ( ta 
<-> 
[. 1  /  n ]. ps ) )
4241notbid 295 . . . . . . . . . . . 12  |-  ( m  =  0  ->  ( -.  ta  <->  -.  [. 1  /  n ]. ps )
)
4331, 42imbi12d 321 . . . . . . . . . . 11  |-  ( m  =  0  ->  (
( -.  th  ->  -. 
ta )  <->  ( -.  ch  ->  -.  [. 1  /  n ]. ps )
) )
4443rspcv 3178 . . . . . . . . . 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 471 . . . . . . . 8  |-  ( (
ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta ) )  ->  -.  [. 1  /  n ]. ps )
47 cbvralsv 3065 . . . . . . . . . . 11  |-  ( A. m  e.  NN0  ( -. 
th  ->  -.  ta )  <->  A. k  e.  NN0  [ k  /  m ] ( -.  th  ->  -.  ta ) )
48 nnnn0 10883 . . . . . . . . . . . 12  |-  ( m  e.  NN  ->  m  e.  NN0 )
49 sbequ12r 2052 . . . . . . . . . . . . 13  |-  ( k  =  m  ->  ( [ k  /  m ] ( -.  th  ->  -.  ta )  <->  ( -.  th 
->  -.  ta ) ) )
5049rspcv 3178 . . . . . . . . . . . 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 220 . . . . . . . . . 10  |-  ( m  e.  NN  ->  ( A. m  e.  NN0  ( -.  th  ->  -. 
ta )  ->  ( -.  th  ->  -.  ta )
) )
5352adantld 468 . . . . . . . . 9  |-  ( m  e.  NN  ->  (
( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta ) )  ->  ( -.  th  ->  -.  ta ) ) )
5453a2d 29 . . . . . . . 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 28389 . . . . . . 7  |-  ( n  e.  NN  ->  (
( ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta ) )  ->  -.  ps )
)
5655rgen 2781 . . . . . 6  |-  A. n  e.  NN  ( ( ph  /\ 
A. m  e.  NN0  ( -.  th  ->  -. 
ta ) )  ->  -.  ps )
57 r19.21v 2827 . . . . . 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 211 . . . . 5  |-  ( (
ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta ) )  ->  A. n  e.  NN  -.  ps )
59 ralnex 2868 . . . . 5  |-  ( A. n  e.  NN  -.  ps 
<->  -.  E. n  e.  NN  ps )
6058, 59sylib 199 . . . 4  |-  ( (
ph  /\  A. m  e.  NN0  ( -.  th  ->  -.  ta ) )  ->  -.  E. n  e.  NN  ps )
612, 60pm2.65da 578 . . 3  |-  ( ph  ->  -.  A. m  e. 
NN0  ( -.  th  ->  -.  ta ) )
62 imnan 423 . . . 4  |-  ( ( -.  th  ->  -.  ta )  <->  -.  ( -.  th 
/\  ta ) )
6362ralbii 2853 . . 3  |-  ( A. m  e.  NN0  ( -. 
th  ->  -.  ta )  <->  A. m  e.  NN0  -.  ( -.  th  /\  ta ) )
6461, 63sylnib 305 . 2  |-  ( ph  ->  -.  A. m  e. 
NN0  -.  ( -.  th 
/\  ta ) )
65 dfrex2 2873 . 2  |-  ( E. m  e.  NN0  ( -.  th  /\  ta )  <->  -. 
A. m  e.  NN0  -.  ( -.  th  /\  ta ) )
6664, 65sylibr 215 1  |-  ( ph  ->  E. m  e.  NN0  ( -.  th  /\  ta ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   [wsb 1790    e. wcel 1872   A.wral 2771   E.wrex 2772   [.wsbc 3299  (class class class)co 6305   0cc0 9546   1c1 9547    + caddc 9549   NNcn 10616   NN0cn0 10876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-om 6707  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-pnf 9684  df-mnf 9685  df-ltxr 9687  df-nn 10617  df-n0 10877
This theorem is referenced by:  archirng  28512
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