Theorem uzind4ALT 9680

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 9679 or uzind4ALT 9680 may be used; see comment for nnind 9023. (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 9679	1 |- ( N e. ( ZZ>= ` M ) -> ta )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2167   ` cfv 5259  (class class class)co 5925   1c1 7897   + caddc 7899   ZZcz 9343   ZZ>=cuz 9618

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-inn 9008  df-n0 9267  df-z 9344  df-uz 9619

This theorem is referenced by: (None)