Theorem ovolicc1 23284

Description: The measure of a closed interval is lower bounded by its length. (Contributed by Mario Carneiro, 13-Jun-2014.) (Proof shortened by Mario Carneiro, 25-Mar-2015.)

Hypotheses

Ref	Expression
ovolicc.1	|- ( ph -> A e. RR )
ovolicc.2	|- ( ph -> B e. RR )
ovolicc.3	|- ( ph -> A <_ B )
ovolicc1.4	|- G = ( n e. NN |-> if ( n = 1 , <. A , B >. , <. 0 , 0 >. ) )

Assertion

Ref	Expression
ovolicc1	|- ( ph -> ( vol* ` ( A [,] B ) ) <_ ( B - A ) )

Distinct variable groups:   A, n   B, n   n, G   ph, n

Proof of Theorem ovolicc1

Dummy variables k x z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovolicc.1	. . . 4 |- ( ph -> A e. RR )
2		ovolicc.2	. . . 4 |- ( ph -> B e. RR )
3		iccssre 12255	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
4	1, 2, 3	syl2anc 693	. . 3 |- ( ph -> ( A [,] B ) C_ RR )
5		ovolcl 23246	. . 3 |- ( ( A [,] B ) C_ RR -> ( vol* ` ( A [,] B ) ) e. RR* )
6	4, 5	syl 17	. 2 |- ( ph -> ( vol* ` ( A [,] B ) ) e. RR* )
7		ovolicc.3	. . . . . . . . . . 11 |- ( ph -> A <_ B )
8		df-br 4654	. . . . . . . . . . 11 |- ( A <_ B <-> <. A , B >. e. <_ )
9	7, 8	sylib 208	. . . . . . . . . 10 |- ( ph -> <. A , B >. e. <_ )
10		opelxpi 5148	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR ) -> <. A , B >. e. ( RR X. RR ) )
11	1, 2, 10	syl2anc 693	. . . . . . . . . 10 |- ( ph -> <. A , B >. e. ( RR X. RR ) )
12	9, 11	elind 3798	. . . . . . . . 9 |- ( ph -> <. A , B >. e. ( <_ i^i ( RR X. RR ) ) )
13	12	adantr 481	. . . . . . . 8 |- ( ( ph /\ n e. NN ) -> <. A , B >. e. ( <_ i^i ( RR X. RR ) ) )
14		0le0 11110	. . . . . . . . . 10 |- 0 <_ 0
15		df-br 4654	. . . . . . . . . 10 |- ( 0 <_ 0 <-> <. 0 , 0 >. e. <_ )
16	14, 15	mpbi 220	. . . . . . . . 9 |- <. 0 , 0 >. e. <_
17		0re 10040	. . . . . . . . . 10 |- 0 e. RR
18		opelxpi 5148	. . . . . . . . . 10 |- ( ( 0 e. RR /\ 0 e. RR ) -> <. 0 , 0 >. e. ( RR X. RR ) )
19	17, 17, 18	mp2an 708	. . . . . . . . 9 |- <. 0 , 0 >. e. ( RR X. RR )
20		elin 3796	. . . . . . . . 9 |- ( <. 0 , 0 >. e. ( <_ i^i ( RR X. RR ) ) <-> ( <. 0 , 0 >. e. <_ /\ <. 0 , 0 >. e. ( RR X. RR ) ) )
21	16, 19, 20	mpbir2an 955	. . . . . . . 8 |- <. 0 , 0 >. e. ( <_ i^i ( RR X. RR ) )
22		ifcl 4130	. . . . . . . 8 |- ( ( <. A , B >. e. ( <_ i^i ( RR X. RR ) ) /\ <. 0 , 0 >. e. ( <_ i^i ( RR X. RR ) ) ) -> if ( n = 1 , <. A , B >. , <. 0 , 0 >. ) e. ( <_ i^i ( RR X. RR ) ) )
23	13, 21, 22	sylancl 694	. . . . . . 7 |- ( ( ph /\ n e. NN ) -> if ( n = 1 , <. A , B >. , <. 0 , 0 >. ) e. ( <_ i^i ( RR X. RR ) ) )
24		ovolicc1.4	. . . . . . 7 |- G = ( n e. NN |-> if ( n = 1 , <. A , B >. , <. 0 , 0 >. ) )
25	23, 24	fmptd 6385	. . . . . 6 |- ( ph -> G : NN --> ( <_ i^i ( RR X. RR ) ) )
26		eqid 2622	. . . . . . 7 |- ( ( abs o. - ) o. G ) = ( ( abs o. - ) o. G )
27		eqid 2622	. . . . . . 7 |- seq 1 ( + , ( ( abs o. - ) o. G ) ) = seq 1 ( + , ( ( abs o. - ) o. G ) )
28	26, 27	ovolsf 23241	. . . . . 6 |- ( G : NN --> ( <_ i^i ( RR X. RR ) ) -> seq 1 ( + , ( ( abs o. - ) o. G ) ) : NN --> ( 0 [,) +oo ) )
29	25, 28	syl 17	. . . . 5 |- ( ph -> seq 1 ( + , ( ( abs o. - ) o. G ) ) : NN --> ( 0 [,) +oo ) )
30		frn 6053	. . . . 5 |- ( seq 1 ( + , ( ( abs o. - ) o. G ) ) : NN --> ( 0 [,) +oo ) -> ran seq 1 ( + , ( ( abs o. - ) o. G ) ) C_ ( 0 [,) +oo ) )
31	29, 30	syl 17	. . . 4 |- ( ph -> ran seq 1 ( + , ( ( abs o. - ) o. G ) ) C_ ( 0 [,) +oo ) )
32		icossxr 12258	. . . 4 |- ( 0 [,) +oo ) C_ RR*
33	31, 32	syl6ss 3615	. . 3 |- ( ph -> ran seq 1 ( + , ( ( abs o. - ) o. G ) ) C_ RR* )
34		supxrcl 12145	. . 3 |- ( ran seq 1 ( + , ( ( abs o. - ) o. G ) ) C_ RR* -> sup ( ran seq 1 ( + , ( ( abs o. - ) o. G ) ) , RR* , < ) e. RR* )
35	33, 34	syl 17	. 2 |- ( ph -> sup ( ran seq 1 ( + , ( ( abs o. - ) o. G ) ) , RR* , < ) e. RR* )
36	2, 1	resubcld 10458	. . 3 |- ( ph -> ( B - A ) e. RR )
37	36	rexrd 10089	. 2 |- ( ph -> ( B - A ) e. RR* )
38		1nn 11031	. . . . . . 7 |- 1 e. NN
39	38	a1i 11	. . . . . 6 |- ( ( ph /\ x e. ( A [,] B ) ) -> 1 e. NN )
40		op1stg 7180	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( 1st ` <. A , B >. ) = A )
41	1, 2, 40	syl2anc 693	. . . . . . . 8 |- ( ph -> ( 1st ` <. A , B >. ) = A )
42	41	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( 1st ` <. A , B >. ) = A )
43		elicc2 12238	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> ( x e. ( A [,] B ) <-> ( x e. RR /\ A <_ x /\ x <_ B ) ) )
44	1, 2, 43	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( x e. ( A [,] B ) <-> ( x e. RR /\ A <_ x /\ x <_ B ) ) )
45	44	biimpa 501	. . . . . . . 8 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( x e. RR /\ A <_ x /\ x <_ B ) )
46	45	simp2d 1074	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> A <_ x )
47	42, 46	eqbrtrd 4675	. . . . . 6 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( 1st ` <. A , B >. ) <_ x )
48	45	simp3d 1075	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> x <_ B )
49		op2ndg 7181	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( 2nd ` <. A , B >. ) = B )
50	1, 2, 49	syl2anc 693	. . . . . . . 8 |- ( ph -> ( 2nd ` <. A , B >. ) = B )
51	50	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( 2nd ` <. A , B >. ) = B )
52	48, 51	breqtrrd 4681	. . . . . 6 |- ( ( ph /\ x e. ( A [,] B ) ) -> x <_ ( 2nd ` <. A , B >. ) )
53		fveq2 6191	. . . . . . . . . . 11 |- ( n = 1 -> ( G ` n ) = ( G ` 1 ) )
54		iftrue 4092	. . . . . . . . . . . . 13 |- ( n = 1 -> if ( n = 1 , <. A , B >. , <. 0 , 0 >. ) = <. A , B >. )
55		opex 4932	. . . . . . . . . . . . 13 |- <. A , B >. e. _V
56	54, 24, 55	fvmpt 6282	. . . . . . . . . . . 12 |- ( 1 e. NN -> ( G ` 1 ) = <. A , B >. )
57	38, 56	ax-mp 5	. . . . . . . . . . 11 |- ( G ` 1 ) = <. A , B >.
58	53, 57	syl6eq 2672	. . . . . . . . . 10 |- ( n = 1 -> ( G ` n ) = <. A , B >. )
59	58	fveq2d 6195	. . . . . . . . 9 |- ( n = 1 -> ( 1st ` ( G ` n ) ) = ( 1st ` <. A , B >. ) )
60	59	breq1d 4663	. . . . . . . 8 |- ( n = 1 -> ( ( 1st ` ( G ` n ) ) <_ x <-> ( 1st ` <. A , B >. ) <_ x ) )
61	58	fveq2d 6195	. . . . . . . . 9 |- ( n = 1 -> ( 2nd ` ( G ` n ) ) = ( 2nd ` <. A , B >. ) )
62	61	breq2d 4665	. . . . . . . 8 |- ( n = 1 -> ( x <_ ( 2nd ` ( G ` n ) ) <-> x <_ ( 2nd ` <. A , B >. ) ) )
63	60, 62	anbi12d 747	. . . . . . 7 |- ( n = 1 -> ( ( ( 1st ` ( G ` n ) ) <_ x /\ x <_ ( 2nd ` ( G ` n ) ) ) <-> ( ( 1st ` <. A , B >. ) <_ x /\ x <_ ( 2nd ` <. A , B >. ) ) ) )
64	63	rspcev 3309	. . . . . 6 |- ( ( 1 e. NN /\ ( ( 1st ` <. A , B >. ) <_ x /\ x <_ ( 2nd ` <. A , B >. ) ) ) -> E. n e. NN ( ( 1st ` ( G ` n ) ) <_ x /\ x <_ ( 2nd ` ( G ` n ) ) ) )
65	39, 47, 52, 64	syl12anc 1324	. . . . 5 |- ( ( ph /\ x e. ( A [,] B ) ) -> E. n e. NN ( ( 1st ` ( G ` n ) ) <_ x /\ x <_ ( 2nd ` ( G ` n ) ) ) )
66	65	ralrimiva 2966	. . . 4 |- ( ph -> A. x e. ( A [,] B ) E. n e. NN ( ( 1st ` ( G ` n ) ) <_ x /\ x <_ ( 2nd ` ( G ` n ) ) ) )
67		ovolficc 23237	. . . . 5 |- ( ( ( A [,] B ) C_ RR /\ G : NN --> ( <_ i^i ( RR X. RR ) ) ) -> ( ( A [,] B ) C_ U. ran ( [,] o. G ) <-> A. x e. ( A [,] B ) E. n e. NN ( ( 1st ` ( G ` n ) ) <_ x /\ x <_ ( 2nd ` ( G ` n ) ) ) ) )
68	4, 25, 67	syl2anc 693	. . . 4 |- ( ph -> ( ( A [,] B ) C_ U. ran ( [,] o. G ) <-> A. x e. ( A [,] B ) E. n e. NN ( ( 1st ` ( G ` n ) ) <_ x /\ x <_ ( 2nd ` ( G ` n ) ) ) ) )
69	66, 68	mpbird 247	. . 3 |- ( ph -> ( A [,] B ) C_ U. ran ( [,] o. G ) )
70	27	ovollb2 23257	. . 3 |- ( ( G : NN --> ( <_ i^i ( RR X. RR ) ) /\ ( A [,] B ) C_ U. ran ( [,] o. G ) ) -> ( vol* ` ( A [,] B ) ) <_ sup ( ran seq 1 ( + , ( ( abs o. - ) o. G ) ) , RR* , < ) )
71	25, 69, 70	syl2anc 693	. 2 |- ( ph -> ( vol* ` ( A [,] B ) ) <_ sup ( ran seq 1 ( + , ( ( abs o. - ) o. G ) ) , RR* , < ) )
72		addid1 10216	. . . . . . . . 9 |- ( k e. CC -> ( k + 0 ) = k )
73	72	adantl 482	. . . . . . . 8 |- ( ( ( ph /\ x e. NN ) /\ k e. CC ) -> ( k + 0 ) = k )
74		nnuz 11723	. . . . . . . . . 10 |- NN = ( ZZ>= ` 1 )
75	38, 74	eleqtri 2699	. . . . . . . . 9 |- 1 e. ( ZZ>= ` 1 )
76	75	a1i 11	. . . . . . . 8 |- ( ( ph /\ x e. NN ) -> 1 e. ( ZZ>= ` 1 ) )
77		simpr 477	. . . . . . . . 9 |- ( ( ph /\ x e. NN ) -> x e. NN )
78	77, 74	syl6eleq 2711	. . . . . . . 8 |- ( ( ph /\ x e. NN ) -> x e. ( ZZ>= ` 1 ) )
79		rge0ssre 12280	. . . . . . . . . 10 |- ( 0 [,) +oo ) C_ RR
80	29	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ x e. NN ) -> seq 1 ( + , ( ( abs o. - ) o. G ) ) : NN --> ( 0 [,) +oo ) )
81		ffvelrn 6357	. . . . . . . . . . 11 |- ( ( seq 1 ( + , ( ( abs o. - ) o. G ) ) : NN --> ( 0 [,) +oo ) /\ 1 e. NN ) -> ( seq 1 ( + , ( ( abs o. - ) o. G ) ) ` 1 ) e. ( 0 [,) +oo ) )
82	80, 38, 81	sylancl 694	. . . . . . . . . 10 |- ( ( ph /\ x e. NN ) -> ( seq 1 ( + , ( ( abs o. - ) o. G ) ) ` 1 ) e. ( 0 [,) +oo ) )
83	79, 82	sseldi 3601	. . . . . . . . 9 |- ( ( ph /\ x e. NN ) -> ( seq 1 ( + , ( ( abs o. - ) o. G ) ) ` 1 ) e. RR )
84	83	recnd 10068	. . . . . . . 8 |- ( ( ph /\ x e. NN ) -> ( seq 1 ( + , ( ( abs o. - ) o. G ) ) ` 1 ) e. CC )
85	25	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ x e. NN ) /\ k e. ( ( 1 + 1 ) ... x ) ) -> G : NN --> ( <_ i^i ( RR X. RR ) ) )
86		elfzuz 12338	. . . . . . . . . . . . 13 |- ( k e. ( ( 1 + 1 ) ... x ) -> k e. ( ZZ>= ` ( 1 + 1 ) ) )
87	86	adantl 482	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. NN ) /\ k e. ( ( 1 + 1 ) ... x ) ) -> k e. ( ZZ>= ` ( 1 + 1 ) ) )
88		df-2 11079	. . . . . . . . . . . . 13 |- 2 = ( 1 + 1 )
89	88	fveq2i 6194	. . . . . . . . . . . 12 |- ( ZZ>= ` 2 ) = ( ZZ>= ` ( 1 + 1 ) )
90	87, 89	syl6eleqr 2712	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. NN ) /\ k e. ( ( 1 + 1 ) ... x ) ) -> k e. ( ZZ>= ` 2 ) )
91		eluz2nn 11726	. . . . . . . . . . 11 |- ( k e. ( ZZ>= ` 2 ) -> k e. NN )
92	90, 91	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ x e. NN ) /\ k e. ( ( 1 + 1 ) ... x ) ) -> k e. NN )
93	26	ovolfsval 23239	. . . . . . . . . 10 |- ( ( G : NN --> ( <_ i^i ( RR X. RR ) ) /\ k e. NN ) -> ( ( ( abs o. - ) o. G ) ` k ) = ( ( 2nd ` ( G ` k ) ) - ( 1st ` ( G ` k ) ) ) )
94	85, 92, 93	syl2anc 693	. . . . . . . . 9 |- ( ( ( ph /\ x e. NN ) /\ k e. ( ( 1 + 1 ) ... x ) ) -> ( ( ( abs o. - ) o. G ) ` k ) = ( ( 2nd ` ( G ` k ) ) - ( 1st ` ( G ` k ) ) ) )
95		eqeq1 2626	. . . . . . . . . . . . . . . . 17 |- ( n = k -> ( n = 1 <-> k = 1 ) )
96	95	ifbid 4108	. . . . . . . . . . . . . . . 16 |- ( n = k -> if ( n = 1 , <. A , B >. , <. 0 , 0 >. ) = if ( k = 1 , <. A , B >. , <. 0 , 0 >. ) )
97		opex 4932	. . . . . . . . . . . . . . . . 17 |- <. 0 , 0 >. e. _V
98	55, 97	ifex 4156	. . . . . . . . . . . . . . . 16 |- if ( k = 1 , <. A , B >. , <. 0 , 0 >. ) e. _V
99	96, 24, 98	fvmpt 6282	. . . . . . . . . . . . . . 15 |- ( k e. NN -> ( G ` k ) = if ( k = 1 , <. A , B >. , <. 0 , 0 >. ) )
100	92, 99	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. NN ) /\ k e. ( ( 1 + 1 ) ... x ) ) -> ( G ` k ) = if ( k = 1 , <. A , B >. , <. 0 , 0 >. ) )
101		eluz2b3 11762	. . . . . . . . . . . . . . . . . 18 |- ( k e. ( ZZ>= ` 2 ) <-> ( k e. NN /\ k =/= 1 ) )
102	101	simprbi 480	. . . . . . . . . . . . . . . . 17 |- ( k e. ( ZZ>= ` 2 ) -> k =/= 1 )
103	90, 102	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. NN ) /\ k e. ( ( 1 + 1 ) ... x ) ) -> k =/= 1 )
104	103	neneqd 2799	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. NN ) /\ k e. ( ( 1 + 1 ) ... x ) ) -> -. k = 1 )
105	104	iffalsed 4097	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. NN ) /\ k e. ( ( 1 + 1 ) ... x ) ) -> if ( k = 1 , <. A , B >. , <. 0 , 0 >. ) = <. 0 , 0 >. )
106	100, 105	eqtrd 2656	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. NN ) /\ k e. ( ( 1 + 1 ) ... x ) ) -> ( G ` k ) = <. 0 , 0 >. )
107	106	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. NN ) /\ k e. ( ( 1 + 1 ) ... x ) ) -> ( 2nd ` ( G ` k ) ) = ( 2nd ` <. 0 , 0 >. ) )
108		c0ex 10034	. . . . . . . . . . . . 13 |- 0 e. _V
109	108, 108	op2nd 7177	. . . . . . . . . . . 12 |- ( 2nd ` <. 0 , 0 >. ) = 0
110	107, 109	syl6eq 2672	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. NN ) /\ k e. ( ( 1 + 1 ) ... x ) ) -> ( 2nd ` ( G ` k ) ) = 0 )
111	106	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. NN ) /\ k e. ( ( 1 + 1 ) ... x ) ) -> ( 1st ` ( G ` k ) ) = ( 1st ` <. 0 , 0 >. ) )
112	108, 108	op1st 7176	. . . . . . . . . . . 12 |- ( 1st ` <. 0 , 0 >. ) = 0
113	111, 112	syl6eq 2672	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. NN ) /\ k e. ( ( 1 + 1 ) ... x ) ) -> ( 1st ` ( G ` k ) ) = 0 )
114	110, 113	oveq12d 6668	. . . . . . . . . 10 |- ( ( ( ph /\ x e. NN ) /\ k e. ( ( 1 + 1 ) ... x ) ) -> ( ( 2nd ` ( G ` k ) ) - ( 1st ` ( G ` k ) ) ) = ( 0 - 0 ) )
115		0m0e0 11130	. . . . . . . . . 10 |- ( 0 - 0 ) = 0
116	114, 115	syl6eq 2672	. . . . . . . . 9 |- ( ( ( ph /\ x e. NN ) /\ k e. ( ( 1 + 1 ) ... x ) ) -> ( ( 2nd ` ( G ` k ) ) - ( 1st ` ( G ` k ) ) ) = 0 )
117	94, 116	eqtrd 2656	. . . . . . . 8 |- ( ( ( ph /\ x e. NN ) /\ k e. ( ( 1 + 1 ) ... x ) ) -> ( ( ( abs o. - ) o. G ) ` k ) = 0 )
118	73, 76, 78, 84, 117	seqid2 12847	. . . . . . 7 |- ( ( ph /\ x e. NN ) -> ( seq 1 ( + , ( ( abs o. - ) o. G ) ) ` 1 ) = ( seq 1 ( + , ( ( abs o. - ) o. G ) ) ` x ) )
119		1z 11407	. . . . . . . 8 |- 1 e. ZZ
120	25	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. NN ) -> G : NN --> ( <_ i^i ( RR X. RR ) ) )
121	26	ovolfsval 23239	. . . . . . . . . 10 |- ( ( G : NN --> ( <_ i^i ( RR X. RR ) ) /\ 1 e. NN ) -> ( ( ( abs o. - ) o. G ) ` 1 ) = ( ( 2nd ` ( G ` 1 ) ) - ( 1st ` ( G ` 1 ) ) ) )
122	120, 38, 121	sylancl 694	. . . . . . . . 9 |- ( ( ph /\ x e. NN ) -> ( ( ( abs o. - ) o. G ) ` 1 ) = ( ( 2nd ` ( G ` 1 ) ) - ( 1st ` ( G ` 1 ) ) ) )
123	57	fveq2i 6194	. . . . . . . . . . 11 |- ( 2nd ` ( G ` 1 ) ) = ( 2nd ` <. A , B >. )
124	50	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ x e. NN ) -> ( 2nd ` <. A , B >. ) = B )
125	123, 124	syl5eq 2668	. . . . . . . . . 10 |- ( ( ph /\ x e. NN ) -> ( 2nd ` ( G ` 1 ) ) = B )
126	57	fveq2i 6194	. . . . . . . . . . 11 |- ( 1st ` ( G ` 1 ) ) = ( 1st ` <. A , B >. )
127	41	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ x e. NN ) -> ( 1st ` <. A , B >. ) = A )
128	126, 127	syl5eq 2668	. . . . . . . . . 10 |- ( ( ph /\ x e. NN ) -> ( 1st ` ( G ` 1 ) ) = A )
129	125, 128	oveq12d 6668	. . . . . . . . 9 |- ( ( ph /\ x e. NN ) -> ( ( 2nd ` ( G ` 1 ) ) - ( 1st ` ( G ` 1 ) ) ) = ( B - A ) )
130	122, 129	eqtrd 2656	. . . . . . . 8 |- ( ( ph /\ x e. NN ) -> ( ( ( abs o. - ) o. G ) ` 1 ) = ( B - A ) )
131	119, 130	seq1i 12815	. . . . . . 7 |- ( ( ph /\ x e. NN ) -> ( seq 1 ( + , ( ( abs o. - ) o. G ) ) ` 1 ) = ( B - A ) )
132	118, 131	eqtr3d 2658	. . . . . 6 |- ( ( ph /\ x e. NN ) -> ( seq 1 ( + , ( ( abs o. - ) o. G ) ) ` x ) = ( B - A ) )
133	36	leidd 10594	. . . . . . 7 |- ( ph -> ( B - A ) <_ ( B - A ) )
134	133	adantr 481	. . . . . 6 |- ( ( ph /\ x e. NN ) -> ( B - A ) <_ ( B - A ) )
135	132, 134	eqbrtrd 4675	. . . . 5 |- ( ( ph /\ x e. NN ) -> ( seq 1 ( + , ( ( abs o. - ) o. G ) ) ` x ) <_ ( B - A ) )
136	135	ralrimiva 2966	. . . 4 |- ( ph -> A. x e. NN ( seq 1 ( + , ( ( abs o. - ) o. G ) ) ` x ) <_ ( B - A ) )
137		ffn 6045	. . . . . 6 |- ( seq 1 ( + , ( ( abs o. - ) o. G ) ) : NN --> ( 0 [,) +oo ) -> seq 1 ( + , ( ( abs o. - ) o. G ) ) Fn NN )
138	29, 137	syl 17	. . . . 5 |- ( ph -> seq 1 ( + , ( ( abs o. - ) o. G ) ) Fn NN )
139		breq1 4656	. . . . . 6 |- ( z = ( seq 1 ( + , ( ( abs o. - ) o. G ) ) ` x ) -> ( z <_ ( B - A ) <-> ( seq 1 ( + , ( ( abs o. - ) o. G ) ) ` x ) <_ ( B - A ) ) )
140	139	ralrn 6362	. . . . 5 |- ( seq 1 ( + , ( ( abs o. - ) o. G ) ) Fn NN -> ( A. z e. ran seq 1 ( + , ( ( abs o. - ) o. G ) ) z <_ ( B - A ) <-> A. x e. NN ( seq 1 ( + , ( ( abs o. - ) o. G ) ) ` x ) <_ ( B - A ) ) )
141	138, 140	syl 17	. . . 4 |- ( ph -> ( A. z e. ran seq 1 ( + , ( ( abs o. - ) o. G ) ) z <_ ( B - A ) <-> A. x e. NN ( seq 1 ( + , ( ( abs o. - ) o. G ) ) ` x ) <_ ( B - A ) ) )
142	136, 141	mpbird 247	. . 3 |- ( ph -> A. z e. ran seq 1 ( + , ( ( abs o. - ) o. G ) ) z <_ ( B - A ) )
143		supxrleub 12156	. . . 4 |- ( ( ran seq 1 ( + , ( ( abs o. - ) o. G ) ) C_ RR* /\ ( B - A ) e. RR* ) -> ( sup ( ran seq 1 ( + , ( ( abs o. - ) o. G ) ) , RR* , < ) <_ ( B - A ) <-> A. z e. ran seq 1 ( + , ( ( abs o. - ) o. G ) ) z <_ ( B - A ) ) )
144	33, 37, 143	syl2anc 693	. . 3 |- ( ph -> ( sup ( ran seq 1 ( + , ( ( abs o. - ) o. G ) ) , RR* , < ) <_ ( B - A ) <-> A. z e. ran seq 1 ( + , ( ( abs o. - ) o. G ) ) z <_ ( B - A ) ) )
145	142, 144	mpbird 247	. 2 |- ( ph -> sup ( ran seq 1 ( + , ( ( abs o. - ) o. G ) ) , RR* , < ) <_ ( B - A ) )
146	6, 35, 37, 71, 145	xrletrd 11993	1 |- ( ph -> ( vol* ` ( A [,] B ) ) <_ ( B - A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   ifcif 4086   <.cop 4183   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   ran crn 5115   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   supcsup 8346   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   2c2 11070   ZZ>=cuz 11687   [,)cico 12177   [,]cicc 12178   ...cfz 12326   seqcseq 12801   abscabs 13974   vol*covol 23231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-ovol 23233

This theorem is referenced by:  ovolicc  23291