Theorem uzind4s 9785

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 3031	. 2 |- ( j = M -> ( [ j / k ] ph <-> [. M / k ]. ph ) )
2		sbequ 1886	. 2 |- ( j = m -> ( [ j / k ] ph <-> [ m / k ] ph ) )
3		dfsbcq2 3031	. 2 |- ( j = ( m + 1 ) -> ( [ j / k ] ph <-> [. ( m + 1 ) / k ]. ph ) )
4		dfsbcq2 3031	. 2 |- ( j = N -> ( [ j / k ] ph <-> [. N / k ]. ph ) )
5		uzind4s.1	. 2 |- ( M e. ZZ -> [. M / k ]. ph )
6		nfv 1574	. . . 4 |- F/ k m e. ( ZZ>= ` M )
7		nfs1v 1990	. . . . 5 |- F/ k [ m / k ] ph
8		nfsbc1v 3047	. . . . 5 |- F/ k [. ( m + 1 ) / k ]. ph
9	7, 8	nfim 1618	. . . 4 |- F/ k ( [ m / k ] ph -> [. ( m + 1 ) / k ]. ph )
10	6, 9	nfim 1618	. . 3 |- F/ k ( m e. ( ZZ>= ` M ) -> ( [ m / k ] ph -> [. ( m + 1 ) / k ]. ph ) )
11		eleq1 2292	. . . 4 |- ( k = m -> ( k e. ( ZZ>= ` M ) <-> m e. ( ZZ>= ` M ) ) )
12		sbequ12 1817	. . . . 5 |- ( k = m -> ( ph <-> [ m / k ] ph ) )
13		oveq1 6008	. . . . . 6 |- ( k = m -> ( k + 1 ) = ( m + 1 ) )
14	13	sbceq1d 3033	. . . . 5 |- ( k = m -> ( [. ( k + 1 ) / k ]. ph <-> [. ( m + 1 ) / k ]. ph ) )
15	12, 14	imbi12d 234	. . . 4 |- ( k = m -> ( ( ph -> [. ( k + 1 ) / k ]. ph ) <-> ( [ m / k ] ph -> [. ( m + 1 ) / k ]. ph ) ) )
16	11, 15	imbi12d 234	. . 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 1803	. 2 |- ( m e. ( ZZ>= ` M ) -> ( [ m / k ] ph -> [. ( m + 1 ) / k ]. ph ) )
19	1, 2, 3, 4, 5, 18	uzind4 9783	1 |- ( N e. ( ZZ>= ` M ) -> [. N / k ]. ph )

Syntax hints:   -> wi 4   [wsb 1808   e. wcel 2200   [.wsbc 3028   ` cfv 5318  (class class class)co 6001   1c1 8000   + caddc 8002   ZZcz 9446   ZZ>=cuz 9722

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-n0 9370  df-z 9447  df-uz 9723

This theorem is referenced by: (None)