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Theorem uzind2 9117
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 9062 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  N  <->  ( M  +  1 )  <_  N ) )
2 peano2z 9044 . . . . . . 7  |-  ( M  e.  ZZ  ->  ( M  +  1 )  e.  ZZ )
3 uzind2.1 . . . . . . . . . 10  |-  ( j  =  ( M  + 
1 )  ->  ( ph 
<->  ps ) )
43imbi2d 229 . . . . . . . . 9  |-  ( j  =  ( M  + 
1 )  ->  (
( M  e.  ZZ  ->  ph )  <->  ( M  e.  ZZ  ->  ps )
) )
5 uzind2.2 . . . . . . . . . 10  |-  ( j  =  k  ->  ( ph 
<->  ch ) )
65imbi2d 229 . . . . . . . . 9  |-  ( j  =  k  ->  (
( M  e.  ZZ  ->  ph )  <->  ( M  e.  ZZ  ->  ch )
) )
7 uzind2.3 . . . . . . . . . 10  |-  ( j  =  ( k  +  1 )  ->  ( ph 
<->  th ) )
87imbi2d 229 . . . . . . . . 9  |-  ( j  =  ( k  +  1 )  ->  (
( M  e.  ZZ  ->  ph )  <->  ( M  e.  ZZ  ->  th )
) )
9 uzind2.4 . . . . . . . . . 10  |-  ( j  =  N  ->  ( ph 
<->  ta ) )
109imbi2d 229 . . . . . . . . 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 9062 . . . . . . . . . . . . . . 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 1166 . . . . . . . . . . . . . . 15  |-  ( ( M  e.  ZZ  /\  k  e.  ZZ )  ->  ( M  <  k  ->  ( ch  ->  th )
) )
1613, 15sylbird 169 . . . . . . . . . . . . . 14  |-  ( ( M  e.  ZZ  /\  k  e.  ZZ )  ->  ( ( M  + 
1 )  <_  k  ->  ( ch  ->  th )
) )
1716ex 114 . . . . . . . . . . . . 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 123 . . . . . . . . . . 11  |-  ( ( k  e.  ZZ  /\  ( M  +  1
)  <_  k )  ->  ( M  e.  ZZ  ->  ( ch  ->  th )
) )
20193adant1 982 . . . . . . . . . 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 9116 . . . . . . . 8  |-  ( ( ( M  +  1 )  e.  ZZ  /\  N  e.  ZZ  /\  ( M  +  1 )  <_  N )  -> 
( M  e.  ZZ  ->  ta ) )
23223exp 1163 . . . . . . 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 123 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  + 
1 )  <_  N  ->  ta ) )
281, 27sylbid 149 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  N  ->  ta ) )
29283impia 1161 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <  N )  ->  ta )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 945    = wceq 1314    e. wcel 1463   class class class wbr 3897  (class class class)co 5740   1c1 7585    + caddc 7587    < clt 7764    <_ cle 7765   ZZcz 9008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-addcom 7684  ax-addass 7686  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-0id 7692  ax-rnegex 7693  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-ltadd 7700
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-inn 8681  df-n0 8932  df-z 9009
This theorem is referenced by: (None)
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