Theorem uzind4s2

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

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

Proof

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