Theorem uzind4s2 9747

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 9746 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 3007	. 2 |- ( m = M -> ( [. m / j ]. ph <-> [. M / j ]. ph ) )
2		dfsbcq 3007	. 2 |- ( m = n -> ( [. m / j ]. ph <-> [. n / j ]. ph ) )
3		dfsbcq 3007	. 2 |- ( m = ( n + 1 ) -> ( [. m / j ]. ph <-> [. ( n + 1 ) / j ]. ph ) )
4		dfsbcq 3007	. 2 |- ( m = N -> ( [. m / j ]. ph <-> [. N / j ]. ph ) )
5		uzind4s2.1	. 2 |- ( M e. ZZ -> [. M / j ]. ph )
6		dfsbcq 3007	. . . 4 |- ( k = n -> ( [. k / j ]. ph <-> [. n / j ]. ph ) )
7		oveq1 5974	. . . . 5 |- ( k = n -> ( k + 1 ) = ( n + 1 ) )
8	7	sbceq1d 3010	. . . 4 |- ( k = n -> ( [. ( k + 1 ) / j ]. ph <-> [. ( n + 1 ) / j ]. ph ) )
9	6, 8	imbi12d 234	. . 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 2844	. 2 |- ( n e. ( ZZ>= ` M ) -> ( [. n / j ]. ph -> [. ( n + 1 ) / j ]. ph ) )
12	1, 2, 3, 4, 5, 11	uzind4 9744	1 |- ( N e. ( ZZ>= ` M ) -> [. N / j ]. ph )

Syntax hints:   -> wi 4   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)