Theorem uzind4ALT 9523

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 9522 or uzind4ALT 9523 may be used; see comment for nnind 8869. (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 9522	1 |- ( N e. ( ZZ>= ` M ) -> ta )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   e. wcel 2136   ` cfv 5187  (class class class)co 5841   1c1 7750   + caddc 7752   ZZcz 9187   ZZ>=cuz 9462

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4099  ax-pow 4152  ax-pr 4186  ax-un 4410  ax-setind 4513  ax-cnex 7840  ax-resscn 7841  ax-1cn 7842  ax-1re 7843  ax-icn 7844  ax-addcl 7845  ax-addrcl 7846  ax-mulcl 7847  ax-addcom 7849  ax-addass 7851  ax-distr 7853  ax-i2m1 7854  ax-0lt1 7855  ax-0id 7857  ax-rnegex 7858  ax-cnre 7860  ax-pre-ltirr 7861  ax-pre-ltwlin 7862  ax-pre-lttrn 7863  ax-pre-ltadd 7865

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-nel 2431  df-ral 2448  df-rex 2449  df-reu 2450  df-rab 2452  df-v 2727  df-sbc 2951  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-int 3824  df-br 3982  df-opab 4043  df-mpt 4044  df-id 4270  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-rn 4614  df-res 4615  df-ima 4616  df-iota 5152  df-fun 5189  df-fn 5190  df-f 5191  df-fv 5195  df-riota 5797  df-ov 5844  df-oprab 5845  df-mpo 5846  df-pnf 7931  df-mnf 7932  df-xr 7933  df-ltxr 7934  df-le 7935  df-sub 8067  df-neg 8068  df-inn 8854  df-n0 9111  df-z 9188  df-uz 9463

This theorem is referenced by: (None)