Theorem uzind4ALT 9607

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 9606 or uzind4ALT 9607 may be used; see comment for nnind 8953. (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 9606	1 |- ( N e. ( ZZ>= ` M ) -> ta )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2160   ` cfv 5231  (class class class)co 5891   1c1 7830   + caddc 7832   ZZcz 9271   ZZ>=cuz 9546

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7920  ax-resscn 7921  ax-1cn 7922  ax-1re 7923  ax-icn 7924  ax-addcl 7925  ax-addrcl 7926  ax-mulcl 7927  ax-addcom 7929  ax-addass 7931  ax-distr 7933  ax-i2m1 7934  ax-0lt1 7935  ax-0id 7937  ax-rnegex 7938  ax-cnre 7940  ax-pre-ltirr 7941  ax-pre-ltwlin 7942  ax-pre-lttrn 7943  ax-pre-ltadd 7945

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-pnf 8012  df-mnf 8013  df-xr 8014  df-ltxr 8015  df-le 8016  df-sub 8148  df-neg 8149  df-inn 8938  df-n0 9195  df-z 9272  df-uz 9547

This theorem is referenced by: (None)