Theorem uzind4s 9412

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 2916	. 2 |- ( j = M -> ( [ j / k ] ph <-> [. M / k ]. ph ) )
2		sbequ 1813	. 2 |- ( j = m -> ( [ j / k ] ph <-> [ m / k ] ph ) )
3		dfsbcq2 2916	. 2 |- ( j = ( m + 1 ) -> ( [ j / k ] ph <-> [. ( m + 1 ) / k ]. ph ) )
4		dfsbcq2 2916	. 2 |- ( j = N -> ( [ j / k ] ph <-> [. N / k ]. ph ) )
5		uzind4s.1	. 2 |- ( M e. ZZ -> [. M / k ]. ph )
6		nfv 1509	. . . 4 |- F/ k m e. ( ZZ>= ` M )
7		nfs1v 1913	. . . . 5 |- F/ k [ m / k ] ph
8		nfsbc1v 2931	. . . . 5 |- F/ k [. ( m + 1 ) / k ]. ph
9	7, 8	nfim 1552	. . . 4 |- F/ k ( [ m / k ] ph -> [. ( m + 1 ) / k ]. ph )
10	6, 9	nfim 1552	. . 3 |- F/ k ( m e. ( ZZ>= ` M ) -> ( [ m / k ] ph -> [. ( m + 1 ) / k ]. ph ) )
11		eleq1 2203	. . . 4 |- ( k = m -> ( k e. ( ZZ>= ` M ) <-> m e. ( ZZ>= ` M ) ) )
12		sbequ12 1745	. . . . 5 |- ( k = m -> ( ph <-> [ m / k ] ph ) )
13		oveq1 5789	. . . . . 6 |- ( k = m -> ( k + 1 ) = ( m + 1 ) )
14	13	sbceq1d 2918	. . . . 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 1731	. 2 |- ( m e. ( ZZ>= ` M ) -> ( [ m / k ] ph -> [. ( m + 1 ) / k ]. ph ) )
19	1, 2, 3, 4, 5, 18	uzind4 9410	1 |- ( N e. ( ZZ>= ` M ) -> [. N / k ]. ph )

Syntax hints:   -> wi 4   e. wcel 1481   [wsb 1736   [.wsbc 2913   ` cfv 5131  (class class class)co 5782   1c1 7645   + caddc 7647   ZZcz 9078   ZZ>=cuz 9350

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-addcom 7744  ax-addass 7746  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-0id 7752  ax-rnegex 7753  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-ltadd 7760

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-inn 8745  df-n0 9002  df-z 9079  df-uz 9351

This theorem is referenced by: (None)