Theorem uzind4s2 9481

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. Use this instead of uzind4s 9480 when j and k must be distinct in [. ( k + 1 ) / j ]. ph. (Contributed by NM, 16-Nov-2005.)

Hypotheses

Ref	Expression
uzind4s2.1	|- ( M e. ZZ -> [. M / j ]. ph )
uzind4s2.2	|- ( k e. ( ZZ>= ` M ) -> ( [. k / j ]. ph -> [. ( k + 1 ) / j ]. ph ) )

Assertion

Ref	Expression
uzind4s2	|- ( N e. ( ZZ>= ` M ) -> [. N / j ]. ph )

Distinct variable groups:   k, M   ph, k   j, k

Allowed substitution hints:   ph( j)   M( j)   N( j, k)

Proof of Theorem uzind4s2

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

Step	Hyp	Ref	Expression
1		dfsbcq 2935	. 2 |- ( m = M -> ( [. m / j ]. ph <-> [. M / j ]. ph ) )
2		dfsbcq 2935	. 2 |- ( m = n -> ( [. m / j ]. ph <-> [. n / j ]. ph ) )
3		dfsbcq 2935	. 2 |- ( m = ( n + 1 ) -> ( [. m / j ]. ph <-> [. ( n + 1 ) / j ]. ph ) )
4		dfsbcq 2935	. 2 |- ( m = N -> ( [. m / j ]. ph <-> [. N / j ]. ph ) )
5		uzind4s2.1	. 2 |- ( M e. ZZ -> [. M / j ]. ph )
6		dfsbcq 2935	. . . 4 |- ( k = n -> ( [. k / j ]. ph <-> [. n / j ]. ph ) )
7		oveq1 5821	. . . . 5 |- ( k = n -> ( k + 1 ) = ( n + 1 ) )
8	7	sbceq1d 2938	. . . 4 |- ( k = n -> ( [. ( k + 1 ) / j ]. ph <-> [. ( n + 1 ) / j ]. ph ) )
9	6, 8	imbi12d 233	. . 3 |- ( k = n -> ( ( [. k / j ]. ph -> [. ( k + 1 ) / j ]. ph ) <-> ( [. n / j ]. ph -> [. ( n + 1 ) / j ]. ph ) ) )
10		uzind4s2.2	. . 3 |- ( k e. ( ZZ>= ` M ) -> ( [. k / j ]. ph -> [. ( k + 1 ) / j ]. ph ) )
11	9, 10	vtoclga 2775	. 2 |- ( n e. ( ZZ>= ` M ) -> ( [. n / j ]. ph -> [. ( n + 1 ) / j ]. ph ) )
12	1, 2, 3, 4, 5, 11	uzind4 9478	1 |- ( N e. ( ZZ>= ` M ) -> [. N / j ]. ph )

Syntax hints:   -> wi 4   e. wcel 2125   [.wsbc 2933   ` cfv 5163  (class class class)co 5814   1c1 7712   + caddc 7714   ZZcz 9146   ZZ>=cuz 9418

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 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490  ax-cnex 7802  ax-resscn 7803  ax-1cn 7804  ax-1re 7805  ax-icn 7806  ax-addcl 7807  ax-addrcl 7808  ax-mulcl 7809  ax-addcom 7811  ax-addass 7813  ax-distr 7815  ax-i2m1 7816  ax-0lt1 7817  ax-0id 7819  ax-rnegex 7820  ax-cnre 7822  ax-pre-ltirr 7823  ax-pre-ltwlin 7824  ax-pre-lttrn 7825  ax-pre-ltadd 7827

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 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-nel 2420  df-ral 2437  df-rex 2438  df-reu 2439  df-rab 2441  df-v 2711  df-sbc 2934  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-int 3804  df-br 3962  df-opab 4022  df-mpt 4023  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-fv 5171  df-riota 5770  df-ov 5817  df-oprab 5818  df-mpo 5819  df-pnf 7893  df-mnf 7894  df-xr 7895  df-ltxr 7896  df-le 7897  df-sub 8027  df-neg 8028  df-inn 8813  df-n0 9070  df-z 9147  df-uz 9419

This theorem is referenced by: (None)