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Theorem uzind4ALT 11218
Description: Induction on the upper set of integers that starts at an integer  M. The last four hypotheses give us the substitution instances we need; the first two are the basis and the induction step. Either uzind4 11217 or uzind4ALT 11218 may be used; see comment for nnind 10627. (Contributed by NM, 7-Sep-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
uzind4ALT.5  |-  ( M  e.  ZZ  ->  ps )
uzind4ALT.6  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( ch  ->  th ) )
uzind4ALT.1  |-  ( j  =  M  ->  ( ph 
<->  ps ) )
uzind4ALT.2  |-  ( j  =  k  ->  ( ph 
<->  ch ) )
uzind4ALT.3  |-  ( j  =  ( k  +  1 )  ->  ( ph 
<->  th ) )
uzind4ALT.4  |-  ( j  =  N  ->  ( ph 
<->  ta ) )
Assertion
Ref Expression
uzind4ALT  |-  ( N  e.  ( ZZ>= `  M
)  ->  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 uzind4ALT
StepHypRef Expression
1 uzind4ALT.1 . 2  |-  ( j  =  M  ->  ( ph 
<->  ps ) )
2 uzind4ALT.2 . 2  |-  ( j  =  k  ->  ( ph 
<->  ch ) )
3 uzind4ALT.3 . 2  |-  ( j  =  ( k  +  1 )  ->  ( ph 
<->  th ) )
4 uzind4ALT.4 . 2  |-  ( j  =  N  ->  ( ph 
<->  ta ) )
5 uzind4ALT.5 . 2  |-  ( M  e.  ZZ  ->  ps )
6 uzind4ALT.6 . 2  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( ch  ->  th ) )
71, 2, 3, 4, 5, 6uzind4 11217 1  |-  ( N  e.  ( ZZ>= `  M
)  ->  ta )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    = wceq 1437    e. wcel 1870   ` cfv 5601  (class class class)co 6305   1c1 9539    + caddc 9541   ZZcz 10937   ZZ>=cuz 11159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
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 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  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-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160
This theorem is referenced by: (None)
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