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Theorem uzind2 8828
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 8774 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  N  <->  ( M  +  1 )  <_  N ) )
2 peano2z 8756 . . . . . . 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 8774 . . . . . . . . . . . . . . 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 1145 . . . . . . . . . . . . . . 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 961 . . . . . . . . . 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 8827 . . . . . . . 8  |-  ( ( ( M  +  1 )  e.  ZZ  /\  N  e.  ZZ  /\  ( M  +  1 )  <_  N )  -> 
( M  e.  ZZ  ->  ta ) )
23223exp 1142 . . . . . . 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 1140 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 924    = wceq 1289    e. wcel 1438   class class class wbr 3837  (class class class)co 5634   1c1 7330    + caddc 7332    < clt 7501    <_ cle 7502   ZZcz 8720
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-ltadd 7440
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-inn 8395  df-n0 8644  df-z 8721
This theorem is referenced by: (None)
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