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Theorem uzind2 8540
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 8486 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  N  <->  ( M  +  1 )  <_  N ) )
2 peano2z 8468 . . . . . . 7  |-  ( M  e.  ZZ  ->  ( M  +  1 )  e.  ZZ )
3 uzind2.1 . . . . . . . . . 10  |-  ( j  =  ( M  + 
1 )  ->  ( ph 
<->  ps ) )
43imbi2d 228 . . . . . . . . 9  |-  ( j  =  ( M  + 
1 )  ->  (
( M  e.  ZZ  ->  ph )  <->  ( M  e.  ZZ  ->  ps )
) )
5 uzind2.2 . . . . . . . . . 10  |-  ( j  =  k  ->  ( ph 
<->  ch ) )
65imbi2d 228 . . . . . . . . 9  |-  ( j  =  k  ->  (
( M  e.  ZZ  ->  ph )  <->  ( M  e.  ZZ  ->  ch )
) )
7 uzind2.3 . . . . . . . . . 10  |-  ( j  =  ( k  +  1 )  ->  ( ph 
<->  th ) )
87imbi2d 228 . . . . . . . . 9  |-  ( j  =  ( k  +  1 )  ->  (
( M  e.  ZZ  ->  ph )  <->  ( M  e.  ZZ  ->  th )
) )
9 uzind2.4 . . . . . . . . . 10  |-  ( j  =  N  ->  ( ph 
<->  ta ) )
109imbi2d 228 . . . . . . . . 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 8486 . . . . . . . . . . . . . . 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 1141 . . . . . . . . . . . . . . 15  |-  ( ( M  e.  ZZ  /\  k  e.  ZZ )  ->  ( M  <  k  ->  ( ch  ->  th )
) )
1613, 15sylbird 168 . . . . . . . . . . . . . 14  |-  ( ( M  e.  ZZ  /\  k  e.  ZZ )  ->  ( ( M  + 
1 )  <_  k  ->  ( ch  ->  th )
) )
1716ex 113 . . . . . . . . . . . . 13  |-  ( M  e.  ZZ  ->  (
k  e.  ZZ  ->  ( ( M  +  1 )  <_  k  ->  ( ch  ->  th )
) ) )
1817com3l 80 . . . . . . . . . . . 12  |-  ( k  e.  ZZ  ->  (
( M  +  1 )  <_  k  ->  ( M  e.  ZZ  ->  ( ch  ->  th )
) ) )
1918imp 122 . . . . . . . . . . 11  |-  ( ( k  e.  ZZ  /\  ( M  +  1
)  <_  k )  ->  ( M  e.  ZZ  ->  ( ch  ->  th )
) )
20193adant1 957 . . . . . . . . . 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 8539 . . . . . . . 8  |-  ( ( ( M  +  1 )  e.  ZZ  /\  N  e.  ZZ  /\  ( M  +  1 )  <_  N )  -> 
( M  e.  ZZ  ->  ta ) )
23223exp 1138 . . . . . . 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 82 . . . . 5  |-  ( M  e.  ZZ  ->  ( N  e.  ZZ  ->  ( M  e.  ZZ  ->  ( ( M  +  1 )  <_  N  ->  ta ) ) ) )
2625pm2.43a 50 . . . 4  |-  ( M  e.  ZZ  ->  ( N  e.  ZZ  ->  ( ( M  +  1 )  <_  N  ->  ta ) ) )
2726imp 122 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  + 
1 )  <_  N  ->  ta ) )
281, 27sylbid 148 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  N  ->  ta ) )
29283impia 1136 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <  N )  ->  ta )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434   class class class wbr 3793  (class class class)co 5543   1c1 7044    + caddc 7046    < clt 7215    <_ cle 7216   ZZcz 8432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-ltadd 7154
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-inn 8107  df-n0 8356  df-z 8433
This theorem is referenced by: (None)
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