Theorem uzind4s 9385

Description: Induction on the upper set of integers that starts at an integer M, using explicit substitution. The hypotheses are the basis and the induction step. (Contributed by NM, 4-Nov-2005.)

Hypotheses

Ref	Expression
uzind4s.1	|- ( M e. ZZ -> [. M / k ]. ph )
uzind4s.2	|- ( k e. ( ZZ>= ` M ) -> ( ph -> [. ( k + 1 ) / k ]. ph ) )

Assertion

Ref	Expression
uzind4s	|- ( N e. ( ZZ>= ` M ) -> [. N / k ]. ph )

Distinct variable group:   k, M

Allowed substitution hints:   ph( k)   N( k)

Proof of Theorem uzind4s

Dummy variables m j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 2912	. 2 |- ( j = M -> ( [ j / k ] ph <-> [. M / k ]. ph ) )
2		sbequ 1812	. 2 |- ( j = m -> ( [ j / k ] ph <-> [ m / k ] ph ) )
3		dfsbcq2 2912	. 2 |- ( j = ( m + 1 ) -> ( [ j / k ] ph <-> [. ( m + 1 ) / k ]. ph ) )
4		dfsbcq2 2912	. 2 |- ( j = N -> ( [ j / k ] ph <-> [. N / k ]. ph ) )
5		uzind4s.1	. 2 |- ( M e. ZZ -> [. M / k ]. ph )
6		nfv 1508	. . . 4 |- F/ k m e. ( ZZ>= ` M )
7		nfs1v 1912	. . . . 5 |- F/ k [ m / k ] ph
8		nfsbc1v 2927	. . . . 5 |- F/ k [. ( m + 1 ) / k ]. ph
9	7, 8	nfim 1551	. . . 4 |- F/ k ( [ m / k ] ph -> [. ( m + 1 ) / k ]. ph )
10	6, 9	nfim 1551	. . 3 |- F/ k ( m e. ( ZZ>= ` M ) -> ( [ m / k ] ph -> [. ( m + 1 ) / k ]. ph ) )
11		eleq1 2202	. . . 4 |- ( k = m -> ( k e. ( ZZ>= ` M ) <-> m e. ( ZZ>= ` M ) ) )
12		sbequ12 1744	. . . . 5 |- ( k = m -> ( ph <-> [ m / k ] ph ) )
13		oveq1 5781	. . . . . 6 |- ( k = m -> ( k + 1 ) = ( m + 1 ) )
14	13	sbceq1d 2914	. . . . 5 |- ( k = m -> ( [. ( k + 1 ) / k ]. ph <-> [. ( m + 1 ) / k ]. ph ) )
15	12, 14	imbi12d 233	. . . 4 |- ( k = m -> ( ( ph -> [. ( k + 1 ) / k ]. ph ) <-> ( [ m / k ] ph -> [. ( m + 1 ) / k ]. ph ) ) )
16	11, 15	imbi12d 233	. . 3 |- ( k = m -> ( ( k e. ( ZZ>= ` M ) -> ( ph -> [. ( k + 1 ) / k ]. ph ) ) <-> ( m e. ( ZZ>= ` M ) -> ( [ m / k ] ph -> [. ( m + 1 ) / k ]. ph ) ) ) )
17		uzind4s.2	. . 3 |- ( k e. ( ZZ>= ` M ) -> ( ph -> [. ( k + 1 ) / k ]. ph ) )
18	10, 16, 17	chvar 1730	. 2 |- ( m e. ( ZZ>= ` M ) -> ( [ m / k ] ph -> [. ( m + 1 ) / k ]. ph ) )
19	1, 2, 3, 4, 5, 18	uzind4 9383	1 |- ( N e. ( ZZ>= ` M ) -> [. N / k ]. ph )

Syntax hints:   -> wi 4   e. wcel 1480   [wsb 1735   [.wsbc 2909   ` cfv 5123  (class class class)co 5774   1c1 7621   + caddc 7623   ZZcz 9054   ZZ>=cuz 9326

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-ltadd 7736

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-inn 8721  df-n0 8978  df-z 9055  df-uz 9327

This theorem is referenced by: (None)