Theorem ovolshftlem1 19358

Description: Lemma for ovolshft 19360. (Contributed by Mario Carneiro, 22-Mar-2014.)

Hypotheses

Ref	Expression
ovolshft.1	|- ( ph -> A C_ RR )
ovolshft.2	|- ( ph -> C e. RR )
ovolshft.3	|- ( ph -> B = { x e. RR | ( x - C ) e. A } )
ovolshft.4	|- M = { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( B C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) }
ovolshft.5	|- S = seq 1 ( + , ( ( abs o. - ) o. F ) )
ovolshft.6	|- G = ( n e. NN |-> <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. )
ovolshft.7	|- ( ph -> F : NN --> ( <_ i^i ( RR X. RR ) ) )
ovolshft.8	|- ( ph -> A C_ U. ran ( (,) o. F ) )

Assertion

Ref	Expression
ovolshftlem1	|- ( ph -> sup ( ran S , RR* , < ) e. M )

Distinct variable groups:   f, n, x, y, A   C, f, n, x, y   n, F, x   f, G, n, y   B, f, n, y   ph, f, n, y

Allowed substitution hints:   ph( x)   B( x)   S( x, y, f, n)   F( y, f)   G( x)   M( x, y, f, n)

Proof of Theorem ovolshftlem1

Step	Hyp	Ref	Expression
1		ovolshft.7	. . . . . . . . . . . . . 14 |- ( ph -> F : NN --> ( <_ i^i ( RR X. RR ) ) )
2		ovolfcl 19316	. . . . . . . . . . . . . 14 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) e. RR /\ ( 2nd ` ( F ` n ) ) e. RR /\ ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) ) )
3	1, 2	sylan 458	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) e. RR /\ ( 2nd ` ( F ` n ) ) e. RR /\ ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) ) )
4	3	simp1d 969	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( F ` n ) ) e. RR )
5	3	simp2d 970	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( 2nd ` ( F ` n ) ) e. RR )
6		ovolshft.2	. . . . . . . . . . . . 13 |- ( ph -> C e. RR )
7	6	adantr 452	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> C e. RR )
8	3	simp3d 971	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) )
9	4, 5, 7, 8	leadd1dd 9596	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) + C ) <_ ( ( 2nd ` ( F ` n ) ) + C ) )
10		df-br 4173	. . . . . . . . . . 11 |- ( ( ( 1st ` ( F ` n ) ) + C ) <_ ( ( 2nd ` ( F ` n ) ) + C ) <-> <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. e. <_ )
11	9, 10	sylib 189	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. e. <_ )
12	4, 7	readdcld 9071	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) + C ) e. RR )
13	5, 7	readdcld 9071	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( ( 2nd ` ( F ` n ) ) + C ) e. RR )
14		opelxp 4867	. . . . . . . . . . 11 |- ( <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. e. ( RR X. RR ) <-> ( ( ( 1st ` ( F ` n ) ) + C ) e. RR /\ ( ( 2nd ` ( F ` n ) ) + C ) e. RR ) )
15	12, 13, 14	sylanbrc 646	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. e. ( RR X. RR ) )
16		elin 3490	. . . . . . . . . 10 |- ( <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. e. ( <_ i^i ( RR X. RR ) ) <-> ( <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. e. <_ /\ <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. e. ( RR X. RR ) ) )
17	11, 15, 16	sylanbrc 646	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. e. ( <_ i^i ( RR X. RR ) ) )
18		ovolshft.6	. . . . . . . . 9 |- G = ( n e. NN |-> <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. )
19	17, 18	fmptd 5852	. . . . . . . 8 |- ( ph -> G : NN --> ( <_ i^i ( RR X. RR ) ) )
20		eqid 2404	. . . . . . . . 9 |- ( ( abs o. - ) o. G ) = ( ( abs o. - ) o. G )
21	20	ovolfsf 19321	. . . . . . . 8 |- ( G : NN --> ( <_ i^i ( RR X. RR ) ) -> ( ( abs o. - ) o. G ) : NN --> ( 0 [,) +oo ) )
22		ffn 5550	. . . . . . . 8 |- ( ( ( abs o. - ) o. G ) : NN --> ( 0 [,) +oo ) -> ( ( abs o. - ) o. G ) Fn NN )
23	19, 21, 22	3syl 19	. . . . . . 7 |- ( ph -> ( ( abs o. - ) o. G ) Fn NN )
24		eqid 2404	. . . . . . . . 9 |- ( ( abs o. - ) o. F ) = ( ( abs o. - ) o. F )
25	24	ovolfsf 19321	. . . . . . . 8 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> ( ( abs o. - ) o. F ) : NN --> ( 0 [,) +oo ) )
26		ffn 5550	. . . . . . . 8 |- ( ( ( abs o. - ) o. F ) : NN --> ( 0 [,) +oo ) -> ( ( abs o. - ) o. F ) Fn NN )
27	1, 25, 26	3syl 19	. . . . . . 7 |- ( ph -> ( ( abs o. - ) o. F ) Fn NN )
28		opex 4387	. . . . . . . . . . . . . 14 |- <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. e. _V
29	18	fvmpt2 5771	. . . . . . . . . . . . . 14 |- ( ( n e. NN /\ <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. e. _V ) -> ( G ` n ) = <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. )
30	28, 29	mpan2 653	. . . . . . . . . . . . 13 |- ( n e. NN -> ( G ` n ) = <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. )
31	30	fveq2d 5691	. . . . . . . . . . . 12 |- ( n e. NN -> ( 2nd ` ( G ` n ) ) = ( 2nd ` <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. ) )
32		ovex 6065	. . . . . . . . . . . . 13 |- ( ( 1st ` ( F ` n ) ) + C ) e. _V
33		ovex 6065	. . . . . . . . . . . . 13 |- ( ( 2nd ` ( F ` n ) ) + C ) e. _V
34	32, 33	op2nd 6315	. . . . . . . . . . . 12 |- ( 2nd ` <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. ) = ( ( 2nd ` ( F ` n ) ) + C )
35	31, 34	syl6eq 2452	. . . . . . . . . . 11 |- ( n e. NN -> ( 2nd ` ( G ` n ) ) = ( ( 2nd ` ( F ` n ) ) + C ) )
36	30	fveq2d 5691	. . . . . . . . . . . 12 |- ( n e. NN -> ( 1st ` ( G ` n ) ) = ( 1st ` <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. ) )
37	32, 33	op1st 6314	. . . . . . . . . . . 12 |- ( 1st ` <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. ) = ( ( 1st ` ( F ` n ) ) + C )
38	36, 37	syl6eq 2452	. . . . . . . . . . 11 |- ( n e. NN -> ( 1st ` ( G ` n ) ) = ( ( 1st ` ( F ` n ) ) + C ) )
39	35, 38	oveq12d 6058	. . . . . . . . . 10 |- ( n e. NN -> ( ( 2nd ` ( G ` n ) ) - ( 1st ` ( G ` n ) ) ) = ( ( ( 2nd ` ( F ` n ) ) + C ) - ( ( 1st ` ( F ` n ) ) + C ) ) )
40	39	adantl 453	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( ( 2nd ` ( G ` n ) ) - ( 1st ` ( G ` n ) ) ) = ( ( ( 2nd ` ( F ` n ) ) + C ) - ( ( 1st ` ( F ` n ) ) + C ) ) )
41	5	recnd 9070	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( 2nd ` ( F ` n ) ) e. CC )
42	4	recnd 9070	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( F ` n ) ) e. CC )
43	7	recnd 9070	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> C e. CC )
44	41, 42, 43	pnpcan2d 9405	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( ( ( 2nd ` ( F ` n ) ) + C ) - ( ( 1st ` ( F ` n ) ) + C ) ) = ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) )
45	40, 44	eqtrd 2436	. . . . . . . 8 |- ( ( ph /\ n e. NN ) -> ( ( 2nd ` ( G ` n ) ) - ( 1st ` ( G ` n ) ) ) = ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) )
46	20	ovolfsval 19320	. . . . . . . . 9 |- ( ( G : NN --> ( <_ i^i ( RR X. RR ) ) /\ n e. NN ) -> ( ( ( abs o. - ) o. G ) ` n ) = ( ( 2nd ` ( G ` n ) ) - ( 1st ` ( G ` n ) ) ) )
47	19, 46	sylan 458	. . . . . . . 8 |- ( ( ph /\ n e. NN ) -> ( ( ( abs o. - ) o. G ) ` n ) = ( ( 2nd ` ( G ` n ) ) - ( 1st ` ( G ` n ) ) ) )
48	24	ovolfsval 19320	. . . . . . . . 9 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ n e. NN ) -> ( ( ( abs o. - ) o. F ) ` n ) = ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) )
49	1, 48	sylan 458	. . . . . . . 8 |- ( ( ph /\ n e. NN ) -> ( ( ( abs o. - ) o. F ) ` n ) = ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) )
50	45, 47, 49	3eqtr4d 2446	. . . . . . 7 |- ( ( ph /\ n e. NN ) -> ( ( ( abs o. - ) o. G ) ` n ) = ( ( ( abs o. - ) o. F ) ` n ) )
51	23, 27, 50	eqfnfvd 5789	. . . . . 6 |- ( ph -> ( ( abs o. - ) o. G ) = ( ( abs o. - ) o. F ) )
52	51	seqeq3d 11286	. . . . 5 |- ( ph -> seq 1 ( + , ( ( abs o. - ) o. G ) ) = seq 1 ( + , ( ( abs o. - ) o. F ) ) )
53		ovolshft.5	. . . . 5 |- S = seq 1 ( + , ( ( abs o. - ) o. F ) )
54	52, 53	syl6eqr 2454	. . . 4 |- ( ph -> seq 1 ( + , ( ( abs o. - ) o. G ) ) = S )
55	54	rneqd 5056	. . 3 |- ( ph -> ran seq 1 ( + , ( ( abs o. - ) o. G ) ) = ran S )
56	55	supeq1d 7409	. 2 |- ( ph -> sup ( ran seq 1 ( + , ( ( abs o. - ) o. G ) ) , RR* , < ) = sup ( ran S , RR* , < ) )
57		ovolshft.3	. . . . . . . . 9 |- ( ph -> B = { x e. RR | ( x - C ) e. A } )
58	57	eleq2d 2471	. . . . . . . 8 |- ( ph -> ( y e. B <-> y e. { x e. RR | ( x - C ) e. A } ) )
59		oveq1 6047	. . . . . . . . . 10 |- ( x = y -> ( x - C ) = ( y - C ) )
60	59	eleq1d 2470	. . . . . . . . 9 |- ( x = y -> ( ( x - C ) e. A <-> ( y - C ) e. A ) )
61	60	elrab 3052	. . . . . . . 8 |- ( y e. { x e. RR | ( x - C ) e. A } <-> ( y e. RR /\ ( y - C ) e. A ) )
62	58, 61	syl6bb 253	. . . . . . 7 |- ( ph -> ( y e. B <-> ( y e. RR /\ ( y - C ) e. A ) ) )
63	62	biimpa 471	. . . . . 6 |- ( ( ph /\ y e. B ) -> ( y e. RR /\ ( y - C ) e. A ) )
64		simprr 734	. . . . . . . 8 |- ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) -> ( y - C ) e. A )
65		ovolshft.8	. . . . . . . . . 10 |- ( ph -> A C_ U. ran ( (,) o. F ) )
66		ovolshft.1	. . . . . . . . . . 11 |- ( ph -> A C_ RR )
67		ovolfioo 19317	. . . . . . . . . . 11 |- ( ( A C_ RR /\ F : NN --> ( <_ i^i ( RR X. RR ) ) ) -> ( A C_ U. ran ( (,) o. F ) <-> A. x e. A E. n e. NN ( ( 1st ` ( F ` n ) ) < x /\ x < ( 2nd ` ( F ` n ) ) ) ) )
68	66, 1, 67	syl2anc 643	. . . . . . . . . 10 |- ( ph -> ( A C_ U. ran ( (,) o. F ) <-> A. x e. A E. n e. NN ( ( 1st ` ( F ` n ) ) < x /\ x < ( 2nd ` ( F ` n ) ) ) ) )
69	65, 68	mpbid 202	. . . . . . . . 9 |- ( ph -> A. x e. A E. n e. NN ( ( 1st ` ( F ` n ) ) < x /\ x < ( 2nd ` ( F ` n ) ) ) )
70	69	adantr 452	. . . . . . . 8 |- ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) -> A. x e. A E. n e. NN ( ( 1st ` ( F ` n ) ) < x /\ x < ( 2nd ` ( F ` n ) ) ) )
71		breq2 4176	. . . . . . . . . . 11 |- ( x = ( y - C ) -> ( ( 1st ` ( F ` n ) ) < x <-> ( 1st ` ( F ` n ) ) < ( y - C ) ) )
72		breq1 4175	. . . . . . . . . . 11 |- ( x = ( y - C ) -> ( x < ( 2nd ` ( F ` n ) ) <-> ( y - C ) < ( 2nd ` ( F ` n ) ) ) )
73	71, 72	anbi12d 692	. . . . . . . . . 10 |- ( x = ( y - C ) -> ( ( ( 1st ` ( F ` n ) ) < x /\ x < ( 2nd ` ( F ` n ) ) ) <-> ( ( 1st ` ( F ` n ) ) < ( y - C ) /\ ( y - C ) < ( 2nd ` ( F ` n ) ) ) ) )
74	73	rexbidv 2687	. . . . . . . . 9 |- ( x = ( y - C ) -> ( E. n e. NN ( ( 1st ` ( F ` n ) ) < x /\ x < ( 2nd ` ( F ` n ) ) ) <-> E. n e. NN ( ( 1st ` ( F ` n ) ) < ( y - C ) /\ ( y - C ) < ( 2nd ` ( F ` n ) ) ) ) )
75	74	rspcv 3008	. . . . . . . 8 |- ( ( y - C ) e. A -> ( A. x e. A E. n e. NN ( ( 1st ` ( F ` n ) ) < x /\ x < ( 2nd ` ( F ` n ) ) ) -> E. n e. NN ( ( 1st ` ( F ` n ) ) < ( y - C ) /\ ( y - C ) < ( 2nd ` ( F ` n ) ) ) ) )
76	64, 70, 75	sylc 58	. . . . . . 7 |- ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) -> E. n e. NN ( ( 1st ` ( F ` n ) ) < ( y - C ) /\ ( y - C ) < ( 2nd ` ( F ` n ) ) ) )
77	38	adantl 453	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( 1st ` ( G ` n ) ) = ( ( 1st ` ( F ` n ) ) + C ) )
78	77	breq1d 4182	. . . . . . . . . 10 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( ( 1st ` ( G ` n ) ) < y <-> ( ( 1st ` ( F ` n ) ) + C ) < y ) )
79	4	adantlr 696	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( 1st ` ( F ` n ) ) e. RR )
80	6	ad2antrr 707	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> C e. RR )
81		simplrl 737	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> y e. RR )
82	79, 80, 81	ltaddsubd 9582	. . . . . . . . . 10 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( ( ( 1st ` ( F ` n ) ) + C ) < y <-> ( 1st ` ( F ` n ) ) < ( y - C ) ) )
83	78, 82	bitrd 245	. . . . . . . . 9 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( ( 1st ` ( G ` n ) ) < y <-> ( 1st ` ( F ` n ) ) < ( y - C ) ) )
84	35	adantl 453	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( 2nd ` ( G ` n ) ) = ( ( 2nd ` ( F ` n ) ) + C ) )
85	84	breq2d 4184	. . . . . . . . . 10 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( y < ( 2nd ` ( G ` n ) ) <-> y < ( ( 2nd ` ( F ` n ) ) + C ) ) )
86	5	adantlr 696	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( 2nd ` ( F ` n ) ) e. RR )
87	81, 80, 86	ltsubaddd 9578	. . . . . . . . . 10 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( ( y - C ) < ( 2nd ` ( F ` n ) ) <-> y < ( ( 2nd ` ( F ` n ) ) + C ) ) )
88	85, 87	bitr4d 248	. . . . . . . . 9 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( y < ( 2nd ` ( G ` n ) ) <-> ( y - C ) < ( 2nd ` ( F ` n ) ) ) )
89	83, 88	anbi12d 692	. . . . . . . 8 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( ( ( 1st ` ( G ` n ) ) < y /\ y < ( 2nd ` ( G ` n ) ) ) <-> ( ( 1st ` ( F ` n ) ) < ( y - C ) /\ ( y - C ) < ( 2nd ` ( F ` n ) ) ) ) )
90	89	rexbidva 2683	. . . . . . 7 |- ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) -> ( E. n e. NN ( ( 1st ` ( G ` n ) ) < y /\ y < ( 2nd ` ( G ` n ) ) ) <-> E. n e. NN ( ( 1st ` ( F ` n ) ) < ( y - C ) /\ ( y - C ) < ( 2nd ` ( F ` n ) ) ) ) )
91	76, 90	mpbird 224	. . . . . 6 |- ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) -> E. n e. NN ( ( 1st ` ( G ` n ) ) < y /\ y < ( 2nd ` ( G ` n ) ) ) )
92	63, 91	syldan 457	. . . . 5 |- ( ( ph /\ y e. B ) -> E. n e. NN ( ( 1st ` ( G ` n ) ) < y /\ y < ( 2nd ` ( G ` n ) ) ) )
93	92	ralrimiva 2749	. . . 4 |- ( ph -> A. y e. B E. n e. NN ( ( 1st ` ( G ` n ) ) < y /\ y < ( 2nd ` ( G ` n ) ) ) )
94		ssrab2 3388	. . . . . 6 |- { x e. RR | ( x - C ) e. A } C_ RR
95	57, 94	syl6eqss 3358	. . . . 5 |- ( ph -> B C_ RR )
96		ovolfioo 19317	. . . . 5 |- ( ( B C_ RR /\ G : NN --> ( <_ i^i ( RR X. RR ) ) ) -> ( B C_ U. ran ( (,) o. G ) <-> A. y e. B E. n e. NN ( ( 1st ` ( G ` n ) ) < y /\ y < ( 2nd ` ( G ` n ) ) ) ) )
97	95, 19, 96	syl2anc 643	. . . 4 |- ( ph -> ( B C_ U. ran ( (,) o. G ) <-> A. y e. B E. n e. NN ( ( 1st ` ( G ` n ) ) < y /\ y < ( 2nd ` ( G ` n ) ) ) ) )
98	93, 97	mpbird 224	. . 3 |- ( ph -> B C_ U. ran ( (,) o. G ) )
99		ovolshft.4	. . . 4 |- M = { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( B C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) }
100		eqid 2404	. . . 4 |- seq 1 ( + , ( ( abs o. - ) o. G ) ) = seq 1 ( + , ( ( abs o. - ) o. G ) )
101	99, 100	elovolmr 19325	. . 3 |- ( ( G : NN --> ( <_ i^i ( RR X. RR ) ) /\ B C_ U. ran ( (,) o. G ) ) -> sup ( ran seq 1 ( + , ( ( abs o. - ) o. G ) ) , RR* , < ) e. M )
102	19, 98, 101	syl2anc 643	. 2 |- ( ph -> sup ( ran seq 1 ( + , ( ( abs o. - ) o. G ) ) , RR* , < ) e. M )
103	56, 102	eqeltrrd 2479	1 |- ( ph -> sup ( ran S , RR* , < ) e. M )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   A.wral 2666   E.wrex 2667   {crab 2670   _Vcvv 2916   i^i cin 3279   C_ wss 3280   <.cop 3777   U.cuni 3975   class class class wbr 4172   e. cmpt 4226   X. cxp 4835   ran crn 4838   o. ccom 4841   Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307   ^m cmap 6977   supcsup 7403   RRcr 8945   0cc0 8946   1c1 8947   + caddc 8949   +oocpnf 9073   RR*cxr 9075   < clt 9076   <_ cle 9077   - cmin 9247   NNcn 9956   (,)cioo 10872   [,)cico 10874   seq cseq 11278   abscabs 11994

This theorem is referenced by:  ovolshftlem2  19359

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-ioo 10876  df-ico 10878  df-fz 11000  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996