Theorem uzind4s 9549

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 2958	. 2 |- ( j = M -> ( [ j / k ] ph <-> [. M / k ]. ph ) )
2		sbequ 1833	. 2 |- ( j = m -> ( [ j / k ] ph <-> [ m / k ] ph ) )
3		dfsbcq2 2958	. 2 |- ( j = ( m + 1 ) -> ( [ j / k ] ph <-> [. ( m + 1 ) / k ]. ph ) )
4		dfsbcq2 2958	. 2 |- ( j = N -> ( [ j / k ] ph <-> [. N / k ]. ph ) )
5		uzind4s.1	. 2 |- ( M e. ZZ -> [. M / k ]. ph )
6		nfv 1521	. . . 4 |- F/ k m e. ( ZZ>= ` M )
7		nfs1v 1932	. . . . 5 |- F/ k [ m / k ] ph
8		nfsbc1v 2973	. . . . 5 |- F/ k [. ( m + 1 ) / k ]. ph
9	7, 8	nfim 1565	. . . 4 |- F/ k ( [ m / k ] ph -> [. ( m + 1 ) / k ]. ph )
10	6, 9	nfim 1565	. . 3 |- F/ k ( m e. ( ZZ>= ` M ) -> ( [ m / k ] ph -> [. ( m + 1 ) / k ]. ph ) )
11		eleq1 2233	. . . 4 |- ( k = m -> ( k e. ( ZZ>= ` M ) <-> m e. ( ZZ>= ` M ) ) )
12		sbequ12 1764	. . . . 5 |- ( k = m -> ( ph <-> [ m / k ] ph ) )
13		oveq1 5860	. . . . . 6 |- ( k = m -> ( k + 1 ) = ( m + 1 ) )
14	13	sbceq1d 2960	. . . . 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 1750	. 2 |- ( m e. ( ZZ>= ` M ) -> ( [ m / k ] ph -> [. ( m + 1 ) / k ]. ph ) )
19	1, 2, 3, 4, 5, 18	uzind4 9547	1 |- ( N e. ( ZZ>= ` M ) -> [. N / k ]. ph )

Syntax hints:   -> wi 4   [wsb 1755   e. wcel 2141   [.wsbc 2955   ` cfv 5198  (class class class)co 5853   1c1 7775   + caddc 7777   ZZcz 9212   ZZ>=cuz 9487

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488

This theorem is referenced by: (None)