Theorem uzind4s 9746

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 3008	. 2 |- ( j = M -> ( [ j / k ] ph <-> [. M / k ]. ph ) )
2		sbequ 1864	. 2 |- ( j = m -> ( [ j / k ] ph <-> [ m / k ] ph ) )
3		dfsbcq2 3008	. 2 |- ( j = ( m + 1 ) -> ( [ j / k ] ph <-> [. ( m + 1 ) / k ]. ph ) )
4		dfsbcq2 3008	. 2 |- ( j = N -> ( [ j / k ] ph <-> [. N / k ]. ph ) )
5		uzind4s.1	. 2 |- ( M e. ZZ -> [. M / k ]. ph )
6		nfv 1552	. . . 4 |- F/ k m e. ( ZZ>= ` M )
7		nfs1v 1968	. . . . 5 |- F/ k [ m / k ] ph
8		nfsbc1v 3024	. . . . 5 |- F/ k [. ( m + 1 ) / k ]. ph
9	7, 8	nfim 1596	. . . 4 |- F/ k ( [ m / k ] ph -> [. ( m + 1 ) / k ]. ph )
10	6, 9	nfim 1596	. . 3 |- F/ k ( m e. ( ZZ>= ` M ) -> ( [ m / k ] ph -> [. ( m + 1 ) / k ]. ph ) )
11		eleq1 2270	. . . 4 |- ( k = m -> ( k e. ( ZZ>= ` M ) <-> m e. ( ZZ>= ` M ) ) )
12		sbequ12 1795	. . . . 5 |- ( k = m -> ( ph <-> [ m / k ] ph ) )
13		oveq1 5974	. . . . . 6 |- ( k = m -> ( k + 1 ) = ( m + 1 ) )
14	13	sbceq1d 3010	. . . . 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 1781	. 2 |- ( m e. ( ZZ>= ` M ) -> ( [ m / k ] ph -> [. ( m + 1 ) / k ]. ph ) )
19	1, 2, 3, 4, 5, 18	uzind4 9744	1 |- ( N e. ( ZZ>= ` M ) -> [. N / k ]. ph )

Syntax hints:   -> wi 4   [wsb 1786   e. wcel 2178   [.wsbc 3005   ` cfv 5290  (class class class)co 5967   1c1 7961   + caddc 7963   ZZcz 9407   ZZ>=cuz 9683

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-inn 9072  df-n0 9331  df-z 9408  df-uz 9684

This theorem is referenced by: (None)