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Theorem uzind2 9364
Description: Induction on the upper integers that start after an integer  M. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 25-Jul-2005.)
Hypotheses
Ref Expression
uzind2.1  |-  ( j  =  ( M  + 
1 )  ->  ( ph 
<->  ps ) )
uzind2.2  |-  ( j  =  k  ->  ( ph 
<->  ch ) )
uzind2.3  |-  ( j  =  ( k  +  1 )  ->  ( ph 
<->  th ) )
uzind2.4  |-  ( j  =  N  ->  ( ph 
<->  ta ) )
uzind2.5  |-  ( M  e.  ZZ  ->  ps )
uzind2.6  |-  ( ( M  e.  ZZ  /\  k  e.  ZZ  /\  M  <  k )  ->  ( ch  ->  th ) )
Assertion
Ref Expression
uzind2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <  N )  ->  ta )
Distinct variable groups:    j, N    ps, j    ch, j    th, j    ta, j    ph, k    j, k, M
Allowed substitution hints:    ph( j)    ps( k)    ch( k)    th( k)    ta( k)    N( k)

Proof of Theorem uzind2
StepHypRef Expression
1 zltp1le 9306 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  N  <->  ( M  +  1 )  <_  N ) )
2 peano2z 9288 . . . . . . 7  |-  ( M  e.  ZZ  ->  ( M  +  1 )  e.  ZZ )
3 uzind2.1 . . . . . . . . . 10  |-  ( j  =  ( M  + 
1 )  ->  ( ph 
<->  ps ) )
43imbi2d 230 . . . . . . . . 9  |-  ( j  =  ( M  + 
1 )  ->  (
( M  e.  ZZ  ->  ph )  <->  ( M  e.  ZZ  ->  ps )
) )
5 uzind2.2 . . . . . . . . . 10  |-  ( j  =  k  ->  ( ph 
<->  ch ) )
65imbi2d 230 . . . . . . . . 9  |-  ( j  =  k  ->  (
( M  e.  ZZ  ->  ph )  <->  ( M  e.  ZZ  ->  ch )
) )
7 uzind2.3 . . . . . . . . . 10  |-  ( j  =  ( k  +  1 )  ->  ( ph 
<->  th ) )
87imbi2d 230 . . . . . . . . 9  |-  ( j  =  ( k  +  1 )  ->  (
( M  e.  ZZ  ->  ph )  <->  ( M  e.  ZZ  ->  th )
) )
9 uzind2.4 . . . . . . . . . 10  |-  ( j  =  N  ->  ( ph 
<->  ta ) )
109imbi2d 230 . . . . . . . . 9  |-  ( j  =  N  ->  (
( M  e.  ZZ  ->  ph )  <->  ( M  e.  ZZ  ->  ta )
) )
11 uzind2.5 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  ps )
1211a1i 9 . . . . . . . . 9  |-  ( ( M  +  1 )  e.  ZZ  ->  ( M  e.  ZZ  ->  ps ) )
13 zltp1le 9306 . . . . . . . . . . . . . . 15  |-  ( ( M  e.  ZZ  /\  k  e.  ZZ )  ->  ( M  <  k  <->  ( M  +  1 )  <_  k ) )
14 uzind2.6 . . . . . . . . . . . . . . . 16  |-  ( ( M  e.  ZZ  /\  k  e.  ZZ  /\  M  <  k )  ->  ( ch  ->  th ) )
15143expia 1205 . . . . . . . . . . . . . . 15  |-  ( ( M  e.  ZZ  /\  k  e.  ZZ )  ->  ( M  <  k  ->  ( ch  ->  th )
) )
1613, 15sylbird 170 . . . . . . . . . . . . . 14  |-  ( ( M  e.  ZZ  /\  k  e.  ZZ )  ->  ( ( M  + 
1 )  <_  k  ->  ( ch  ->  th )
) )
1716ex 115 . . . . . . . . . . . . 13  |-  ( M  e.  ZZ  ->  (
k  e.  ZZ  ->  ( ( M  +  1 )  <_  k  ->  ( ch  ->  th )
) ) )
1817com3l 81 . . . . . . . . . . . 12  |-  ( k  e.  ZZ  ->  (
( M  +  1 )  <_  k  ->  ( M  e.  ZZ  ->  ( ch  ->  th )
) ) )
1918imp 124 . . . . . . . . . . 11  |-  ( ( k  e.  ZZ  /\  ( M  +  1
)  <_  k )  ->  ( M  e.  ZZ  ->  ( ch  ->  th )
) )
20193adant1 1015 . . . . . . . . . 10  |-  ( ( ( M  +  1 )  e.  ZZ  /\  k  e.  ZZ  /\  ( M  +  1 )  <_  k )  -> 
( M  e.  ZZ  ->  ( ch  ->  th )
) )
2120a2d 26 . . . . . . . . 9  |-  ( ( ( M  +  1 )  e.  ZZ  /\  k  e.  ZZ  /\  ( M  +  1 )  <_  k )  -> 
( ( M  e.  ZZ  ->  ch )  ->  ( M  e.  ZZ  ->  th ) ) )
224, 6, 8, 10, 12, 21uzind 9363 . . . . . . . 8  |-  ( ( ( M  +  1 )  e.  ZZ  /\  N  e.  ZZ  /\  ( M  +  1 )  <_  N )  -> 
( M  e.  ZZ  ->  ta ) )
23223exp 1202 . . . . . . 7  |-  ( ( M  +  1 )  e.  ZZ  ->  ( N  e.  ZZ  ->  ( ( M  +  1 )  <_  N  ->  ( M  e.  ZZ  ->  ta ) ) ) )
242, 23syl 14 . . . . . 6  |-  ( M  e.  ZZ  ->  ( N  e.  ZZ  ->  ( ( M  +  1 )  <_  N  ->  ( M  e.  ZZ  ->  ta ) ) ) )
2524com34 83 . . . . 5  |-  ( M  e.  ZZ  ->  ( N  e.  ZZ  ->  ( M  e.  ZZ  ->  ( ( M  +  1 )  <_  N  ->  ta ) ) ) )
2625pm2.43a 51 . . . 4  |-  ( M  e.  ZZ  ->  ( N  e.  ZZ  ->  ( ( M  +  1 )  <_  N  ->  ta ) ) )
2726imp 124 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  + 
1 )  <_  N  ->  ta ) )
281, 27sylbid 150 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  N  ->  ta ) )
29283impia 1200 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <  N )  ->  ta )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   class class class wbr 4003  (class class class)co 5874   1c1 7811    + caddc 7813    < clt 7991    <_ cle 7992   ZZcz 9252
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-ltadd 7926
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-inn 8919  df-n0 9176  df-z 9253
This theorem is referenced by: (None)
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