Theorem fzofzp1

Description: If a point is in a half-open range, the next point is in the closed range. (Contributed by Stefan O'Rear, 23-Aug-2015)

	Ref	Expression
Assertion	fzofzp1	|- ( C e. ( A ..^ B ) -> ( C + 1 ) e. ( A ... B ) )

Proof

Step	Hyp	Ref	Expression
1		elfzoel1	|- ( C e. ( A ..^ B ) -> A e. ZZ )
2		uzid	|- ( A e. ZZ -> A e. ( ZZ>= ` A ) )
3		peano2uz	|- ( A e. ( ZZ>= ` A ) -> ( A + 1 ) e. ( ZZ>= ` A ) )
4		fzoss1	|- ( ( A + 1 ) e. ( ZZ>= ` A ) -> ( ( A + 1 ) ..^ ( B + 1 ) ) C_ ( A ..^ ( B + 1 ) ) )
5	1 2 3 4	4syl	|- ( C e. ( A ..^ B ) -> ( ( A + 1 ) ..^ ( B + 1 ) ) C_ ( A ..^ ( B + 1 ) ) )
6		1z	|- 1 e. ZZ
7		fzoaddel	|- ( ( C e. ( A ..^ B ) /\ 1 e. ZZ ) -> ( C + 1 ) e. ( ( A + 1 ) ..^ ( B + 1 ) ) )
8	6 7	mpan2	|- ( C e. ( A ..^ B ) -> ( C + 1 ) e. ( ( A + 1 ) ..^ ( B + 1 ) ) )
9	5 8	sseldd	|- ( C e. ( A ..^ B ) -> ( C + 1 ) e. ( A ..^ ( B + 1 ) ) )
10		elfzoel2	|- ( C e. ( A ..^ B ) -> B e. ZZ )
11		fzval3	|- ( B e. ZZ -> ( A ... B ) = ( A ..^ ( B + 1 ) ) )
12	10 11	syl	|- ( C e. ( A ..^ B ) -> ( A ... B ) = ( A ..^ ( B + 1 ) ) )
13	9 12	eleqtrrd	|- ( C e. ( A ..^ B ) -> ( C + 1 ) e. ( A ... B ) )