Theorem fourierdlem8

Description: A partition interval is a subset of the partitioned interval. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	fourierdlem8.a	|- ( ph -> A e. RR* )
	fourierdlem8.b	|- ( ph -> B e. RR* )
	fourierdlem8.q	|- ( ph -> Q : ( 0 ... M ) --> ( A [,] B ) )
	fourierdlem8.i	|- ( ph -> I e. ( 0 ..^ M ) )
Assertion	fourierdlem8	|- ( ph -> ( ( Q ` I ) [,] ( Q ` ( I + 1 ) ) ) C_ ( A [,] B ) )

Proof

Step	Hyp	Ref	Expression
1		fourierdlem8.a	|- ( ph -> A e. RR* )
2		fourierdlem8.b	|- ( ph -> B e. RR* )
3		fourierdlem8.q	|- ( ph -> Q : ( 0 ... M ) --> ( A [,] B ) )
4		fourierdlem8.i	|- ( ph -> I e. ( 0 ..^ M ) )
5	1	adantr	|- ( ( ph /\ x e. ( ( Q ` I ) [,] ( Q ` ( I + 1 ) ) ) ) -> A e. RR* )
6	2	adantr	|- ( ( ph /\ x e. ( ( Q ` I ) [,] ( Q ` ( I + 1 ) ) ) ) -> B e. RR* )
7	3	adantr	|- ( ( ph /\ x e. ( ( Q ` I ) [,] ( Q ` ( I + 1 ) ) ) ) -> Q : ( 0 ... M ) --> ( A [,] B ) )
8	4	adantr	|- ( ( ph /\ x e. ( ( Q ` I ) [,] ( Q ` ( I + 1 ) ) ) ) -> I e. ( 0 ..^ M ) )
9		simpr	|- ( ( ph /\ x e. ( ( Q ` I ) [,] ( Q ` ( I + 1 ) ) ) ) -> x e. ( ( Q ` I ) [,] ( Q ` ( I + 1 ) ) ) )
10	5 6 7 8 9	fourierdlem1	|- ( ( ph /\ x e. ( ( Q ` I ) [,] ( Q ` ( I + 1 ) ) ) ) -> x e. ( A [,] B ) )
11	10	ralrimiva	|- ( ph -> A. x e. ( ( Q ` I ) [,] ( Q ` ( I + 1 ) ) ) x e. ( A [,] B ) )
12		dfss3	|- ( ( ( Q ` I ) [,] ( Q ` ( I + 1 ) ) ) C_ ( A [,] B ) <-> A. x e. ( ( Q ` I ) [,] ( Q ` ( I + 1 ) ) ) x e. ( A [,] B ) )
13	11 12	sylibr	|- ( ph -> ( ( Q ` I ) [,] ( Q ` ( I + 1 ) ) ) C_ ( A [,] B ) )