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 ∈ ℤ → [ 𝑀 / 𝑗 ] 𝜑 )
	uzind4s2.2	⊢ ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) → ( [ 𝑘 / 𝑗 ] 𝜑 → [ ( 𝑘 + 1 ) / 𝑗 ] 𝜑 ) )
Assertion	uzind4s2	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → [ 𝑁 / 𝑗 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		uzind4s2.1	⊢ ( 𝑀 ∈ ℤ → [ 𝑀 / 𝑗 ] 𝜑 )
2		uzind4s2.2	⊢ ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) → ( [ 𝑘 / 𝑗 ] 𝜑 → [ ( 𝑘 + 1 ) / 𝑗 ] 𝜑 ) )
3		dfsbcq	⊢ ( 𝑚 = 𝑀 → ( [ 𝑚 / 𝑗 ] 𝜑 ↔ [ 𝑀 / 𝑗 ] 𝜑 ) )
4		dfsbcq	⊢ ( 𝑚 = 𝑛 → ( [ 𝑚 / 𝑗 ] 𝜑 ↔ [ 𝑛 / 𝑗 ] 𝜑 ) )
5		dfsbcq	⊢ ( 𝑚 = ( 𝑛 + 1 ) → ( [ 𝑚 / 𝑗 ] 𝜑 ↔ [ ( 𝑛 + 1 ) / 𝑗 ] 𝜑 ) )
6		dfsbcq	⊢ ( 𝑚 = 𝑁 → ( [ 𝑚 / 𝑗 ] 𝜑 ↔ [ 𝑁 / 𝑗 ] 𝜑 ) )
7		dfsbcq	⊢ ( 𝑘 = 𝑛 → ( [ 𝑘 / 𝑗 ] 𝜑 ↔ [ 𝑛 / 𝑗 ] 𝜑 ) )
8		oveq1	⊢ ( 𝑘 = 𝑛 → ( 𝑘 + 1 ) = ( 𝑛 + 1 ) )
9	8	sbceq1d	⊢ ( 𝑘 = 𝑛 → ( [ ( 𝑘 + 1 ) / 𝑗 ] 𝜑 ↔ [ ( 𝑛 + 1 ) / 𝑗 ] 𝜑 ) )
10	7 9	imbi12d	⊢ ( 𝑘 = 𝑛 → ( ( [ 𝑘 / 𝑗 ] 𝜑 → [ ( 𝑘 + 1 ) / 𝑗 ] 𝜑 ) ↔ ( [ 𝑛 / 𝑗 ] 𝜑 → [ ( 𝑛 + 1 ) / 𝑗 ] 𝜑 ) ) )
11	10 2	vtoclga	⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) → ( [ 𝑛 / 𝑗 ] 𝜑 → [ ( 𝑛 + 1 ) / 𝑗 ] 𝜑 ) )
12	3 4 5 6 1 11	uzind4	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → [ 𝑁 / 𝑗 ] 𝜑 )