Theorem uzind4ALT 9989

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 9988 or uzind4ALT 9989 may be used; see comment for nnind 9320. (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

Step	Hyp	Ref	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 ) )
7	1, 2, 3, 4, 5, 6	uzind4 9988	1 |- ( N e. ( ZZ>= ` M ) -> ta )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   = wceq 1402   e. wcel 2209   ` cfv 5377  (class class class)co 6085   1c1 8180   + caddc 8182   ZZcz 9644   ZZ>=cuz 9921

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922

This theorem is used by: (None)