Theorem ovolshftlem1 22455

Description: Lemma for ovolshft 22457. (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 22412	. . . . . . . . . . . . . 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 474	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) e. RR /\ ( 2nd ` ( F ` n ) ) e. RR /\ ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) ) )
4	3	simp1d 1019	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( F ` n ) ) e. RR )
5	3	simp2d 1020	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( 2nd ` ( F ` n ) ) e. RR )
6		ovolshft.2	. . . . . . . . . . . . 13 |- ( ph -> C e. RR )
7	6	adantr 467	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> C e. RR )
8	3	simp3d 1021	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) )
9	4, 5, 7, 8	leadd1dd 10224	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) + C ) <_ ( ( 2nd ` ( F ` n ) ) + C ) )
10		df-br 4402	. . . . . . . . . . 11 |- ( ( ( 1st ` ( F ` n ) ) + C ) <_ ( ( 2nd ` ( F ` n ) ) + C ) <-> <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. e. <_ )
11	9, 10	sylib 200	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. e. <_ )
12	4, 7	readdcld 9667	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) + C ) e. RR )
13	5, 7	readdcld 9667	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( ( 2nd ` ( F ` n ) ) + C ) e. RR )
14		opelxp 4863	. . . . . . . . . . 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 669	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. e. ( RR X. RR ) )
16	11, 15	elind 3617	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. e. ( <_ i^i ( RR X. RR ) ) )
17		ovolshft.6	. . . . . . . . 9 |- G = ( n e. NN |-> <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. )
18	16, 17	fmptd 6044	. . . . . . . 8 |- ( ph -> G : NN --> ( <_ i^i ( RR X. RR ) ) )
19		eqid 2450	. . . . . . . . 9 |- ( ( abs o. - ) o. G ) = ( ( abs o. - ) o. G )
20	19	ovolfsf 22417	. . . . . . . 8 |- ( G : NN --> ( <_ i^i ( RR X. RR ) ) -> ( ( abs o. - ) o. G ) : NN --> ( 0 [,) +oo ) )
21		ffn 5726	. . . . . . . 8 |- ( ( ( abs o. - ) o. G ) : NN --> ( 0 [,) +oo ) -> ( ( abs o. - ) o. G ) Fn NN )
22	18, 20, 21	3syl 18	. . . . . . 7 |- ( ph -> ( ( abs o. - ) o. G ) Fn NN )
23		eqid 2450	. . . . . . . . 9 |- ( ( abs o. - ) o. F ) = ( ( abs o. - ) o. F )
24	23	ovolfsf 22417	. . . . . . . 8 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> ( ( abs o. - ) o. F ) : NN --> ( 0 [,) +oo ) )
25		ffn 5726	. . . . . . . 8 |- ( ( ( abs o. - ) o. F ) : NN --> ( 0 [,) +oo ) -> ( ( abs o. - ) o. F ) Fn NN )
26	1, 24, 25	3syl 18	. . . . . . 7 |- ( ph -> ( ( abs o. - ) o. F ) Fn NN )
27		opex 4663	. . . . . . . . . . . . . 14 |- <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. e. _V
28	17	fvmpt2 5955	. . . . . . . . . . . . . 14 |- ( ( n e. NN /\ <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. e. _V ) -> ( G ` n ) = <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. )
29	27, 28	mpan2 676	. . . . . . . . . . . . 13 |- ( n e. NN -> ( G ` n ) = <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. )
30	29	fveq2d 5867	. . . . . . . . . . . 12 |- ( n e. NN -> ( 2nd ` ( G ` n ) ) = ( 2nd ` <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. ) )
31		ovex 6316	. . . . . . . . . . . . 13 |- ( ( 1st ` ( F ` n ) ) + C ) e. _V
32		ovex 6316	. . . . . . . . . . . . 13 |- ( ( 2nd ` ( F ` n ) ) + C ) e. _V
33	31, 32	op2nd 6799	. . . . . . . . . . . 12 |- ( 2nd ` <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. ) = ( ( 2nd ` ( F ` n ) ) + C )
34	30, 33	syl6eq 2500	. . . . . . . . . . 11 |- ( n e. NN -> ( 2nd ` ( G ` n ) ) = ( ( 2nd ` ( F ` n ) ) + C ) )
35	29	fveq2d 5867	. . . . . . . . . . . 12 |- ( n e. NN -> ( 1st ` ( G ` n ) ) = ( 1st ` <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. ) )
36	31, 32	op1st 6798	. . . . . . . . . . . 12 |- ( 1st ` <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. ) = ( ( 1st ` ( F ` n ) ) + C )
37	35, 36	syl6eq 2500	. . . . . . . . . . 11 |- ( n e. NN -> ( 1st ` ( G ` n ) ) = ( ( 1st ` ( F ` n ) ) + C ) )
38	34, 37	oveq12d 6306	. . . . . . . . . 10 |- ( n e. NN -> ( ( 2nd ` ( G ` n ) ) - ( 1st ` ( G ` n ) ) ) = ( ( ( 2nd ` ( F ` n ) ) + C ) - ( ( 1st ` ( F ` n ) ) + C ) ) )
39	38	adantl 468	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( ( 2nd ` ( G ` n ) ) - ( 1st ` ( G ` n ) ) ) = ( ( ( 2nd ` ( F ` n ) ) + C ) - ( ( 1st ` ( F ` n ) ) + C ) ) )
40	5	recnd 9666	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( 2nd ` ( F ` n ) ) e. CC )
41	4	recnd 9666	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( F ` n ) ) e. CC )
42	7	recnd 9666	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> C e. CC )
43	40, 41, 42	pnpcan2d 10021	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( ( ( 2nd ` ( F ` n ) ) + C ) - ( ( 1st ` ( F ` n ) ) + C ) ) = ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) )
44	39, 43	eqtrd 2484	. . . . . . . 8 |- ( ( ph /\ n e. NN ) -> ( ( 2nd ` ( G ` n ) ) - ( 1st ` ( G ` n ) ) ) = ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) )
45	19	ovolfsval 22416	. . . . . . . . 9 |- ( ( G : NN --> ( <_ i^i ( RR X. RR ) ) /\ n e. NN ) -> ( ( ( abs o. - ) o. G ) ` n ) = ( ( 2nd ` ( G ` n ) ) - ( 1st ` ( G ` n ) ) ) )
46	18, 45	sylan 474	. . . . . . . 8 |- ( ( ph /\ n e. NN ) -> ( ( ( abs o. - ) o. G ) ` n ) = ( ( 2nd ` ( G ` n ) ) - ( 1st ` ( G ` n ) ) ) )
47	23	ovolfsval 22416	. . . . . . . . 9 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ n e. NN ) -> ( ( ( abs o. - ) o. F ) ` n ) = ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) )
48	1, 47	sylan 474	. . . . . . . 8 |- ( ( ph /\ n e. NN ) -> ( ( ( abs o. - ) o. F ) ` n ) = ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) )
49	44, 46, 48	3eqtr4d 2494	. . . . . . 7 |- ( ( ph /\ n e. NN ) -> ( ( ( abs o. - ) o. G ) ` n ) = ( ( ( abs o. - ) o. F ) ` n ) )
50	22, 26, 49	eqfnfvd 5977	. . . . . 6 |- ( ph -> ( ( abs o. - ) o. G ) = ( ( abs o. - ) o. F ) )
51	50	seqeq3d 12218	. . . . 5 |- ( ph -> seq 1 ( + , ( ( abs o. - ) o. G ) ) = seq 1 ( + , ( ( abs o. - ) o. F ) ) )
52		ovolshft.5	. . . . 5 |- S = seq 1 ( + , ( ( abs o. - ) o. F ) )
53	51, 52	syl6eqr 2502	. . . 4 |- ( ph -> seq 1 ( + , ( ( abs o. - ) o. G ) ) = S )
54	53	rneqd 5061	. . 3 |- ( ph -> ran seq 1 ( + , ( ( abs o. - ) o. G ) ) = ran S )
55	54	supeq1d 7957	. 2 |- ( ph -> sup ( ran seq 1 ( + , ( ( abs o. - ) o. G ) ) , RR* , < ) = sup ( ran S , RR* , < ) )
56		ovolshft.3	. . . . . . . . 9 |- ( ph -> B = { x e. RR | ( x - C ) e. A } )
57	56	eleq2d 2513	. . . . . . . 8 |- ( ph -> ( y e. B <-> y e. { x e. RR | ( x - C ) e. A } ) )
58		oveq1 6295	. . . . . . . . . 10 |- ( x = y -> ( x - C ) = ( y - C ) )
59	58	eleq1d 2512	. . . . . . . . 9 |- ( x = y -> ( ( x - C ) e. A <-> ( y - C ) e. A ) )
60	59	elrab 3195	. . . . . . . 8 |- ( y e. { x e. RR | ( x - C ) e. A } <-> ( y e. RR /\ ( y - C ) e. A ) )
61	57, 60	syl6bb 265	. . . . . . 7 |- ( ph -> ( y e. B <-> ( y e. RR /\ ( y - C ) e. A ) ) )
62	61	biimpa 487	. . . . . 6 |- ( ( ph /\ y e. B ) -> ( y e. RR /\ ( y - C ) e. A ) )
63		simprr 765	. . . . . . . 8 |- ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) -> ( y - C ) e. A )
64		ovolshft.8	. . . . . . . . . 10 |- ( ph -> A C_ U. ran ( (,) o. F ) )
65		ovolshft.1	. . . . . . . . . . 11 |- ( ph -> A C_ RR )
66		ovolfioo 22413	. . . . . . . . . . 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 ) ) ) ) )
67	65, 1, 66	syl2anc 666	. . . . . . . . . 10 |- ( ph -> ( A C_ U. ran ( (,) o. F ) <-> A. x e. A E. n e. NN ( ( 1st ` ( F ` n ) ) < x /\ x < ( 2nd ` ( F ` n ) ) ) ) )
68	64, 67	mpbid 214	. . . . . . . . 9 |- ( ph -> A. x e. A E. n e. NN ( ( 1st ` ( F ` n ) ) < x /\ x < ( 2nd ` ( F ` n ) ) ) )
69	68	adantr 467	. . . . . . . 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 ) ) ) )
70		breq2 4405	. . . . . . . . . . 11 |- ( x = ( y - C ) -> ( ( 1st ` ( F ` n ) ) < x <-> ( 1st ` ( F ` n ) ) < ( y - C ) ) )
71		breq1 4404	. . . . . . . . . . 11 |- ( x = ( y - C ) -> ( x < ( 2nd ` ( F ` n ) ) <-> ( y - C ) < ( 2nd ` ( F ` n ) ) ) )
72	70, 71	anbi12d 716	. . . . . . . . . 10 |- ( x = ( y - C ) -> ( ( ( 1st ` ( F ` n ) ) < x /\ x < ( 2nd ` ( F ` n ) ) ) <-> ( ( 1st ` ( F ` n ) ) < ( y - C ) /\ ( y - C ) < ( 2nd ` ( F ` n ) ) ) ) )
73	72	rexbidv 2900	. . . . . . . . 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 ) ) ) ) )
74	73	rspcv 3145	. . . . . . . 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 ) ) ) ) )
75	63, 69, 74	sylc 62	. . . . . . 7 |- ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) -> E. n e. NN ( ( 1st ` ( F ` n ) ) < ( y - C ) /\ ( y - C ) < ( 2nd ` ( F ` n ) ) ) )
76	37	adantl 468	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( 1st ` ( G ` n ) ) = ( ( 1st ` ( F ` n ) ) + C ) )
77	76	breq1d 4411	. . . . . . . . . 10 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( ( 1st ` ( G ` n ) ) < y <-> ( ( 1st ` ( F ` n ) ) + C ) < y ) )
78	4	adantlr 720	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( 1st ` ( F ` n ) ) e. RR )
79	6	ad2antrr 731	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> C e. RR )
80		simplrl 769	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> y e. RR )
81	78, 79, 80	ltaddsubd 10210	. . . . . . . . . 10 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( ( ( 1st ` ( F ` n ) ) + C ) < y <-> ( 1st ` ( F ` n ) ) < ( y - C ) ) )
82	77, 81	bitrd 257	. . . . . . . . 9 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( ( 1st ` ( G ` n ) ) < y <-> ( 1st ` ( F ` n ) ) < ( y - C ) ) )
83	34	adantl 468	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( 2nd ` ( G ` n ) ) = ( ( 2nd ` ( F ` n ) ) + C ) )
84	83	breq2d 4413	. . . . . . . . . 10 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( y < ( 2nd ` ( G ` n ) ) <-> y < ( ( 2nd ` ( F ` n ) ) + C ) ) )
85	5	adantlr 720	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( 2nd ` ( F ` n ) ) e. RR )
86	80, 79, 85	ltsubaddd 10206	. . . . . . . . . 10 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( ( y - C ) < ( 2nd ` ( F ` n ) ) <-> y < ( ( 2nd ` ( F ` n ) ) + C ) ) )
87	84, 86	bitr4d 260	. . . . . . . . 9 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( y < ( 2nd ` ( G ` n ) ) <-> ( y - C ) < ( 2nd ` ( F ` n ) ) ) )
88	82, 87	anbi12d 716	. . . . . . . 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 ) ) ) ) )
89	88	rexbidva 2897	. . . . . . 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 ) ) ) ) )
90	75, 89	mpbird 236	. . . . . 6 |- ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) -> E. n e. NN ( ( 1st ` ( G ` n ) ) < y /\ y < ( 2nd ` ( G ` n ) ) ) )
91	62, 90	syldan 473	. . . . 5 |- ( ( ph /\ y e. B ) -> E. n e. NN ( ( 1st ` ( G ` n ) ) < y /\ y < ( 2nd ` ( G ` n ) ) ) )
92	91	ralrimiva 2801	. . . 4 |- ( ph -> A. y e. B E. n e. NN ( ( 1st ` ( G ` n ) ) < y /\ y < ( 2nd ` ( G ` n ) ) ) )
93		ssrab2 3513	. . . . . 6 |- { x e. RR | ( x - C ) e. A } C_ RR
94	56, 93	syl6eqss 3481	. . . . 5 |- ( ph -> B C_ RR )
95		ovolfioo 22413	. . . . 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 ) ) ) ) )
96	94, 18, 95	syl2anc 666	. . . 4 |- ( ph -> ( B C_ U. ran ( (,) o. G ) <-> A. y e. B E. n e. NN ( ( 1st ` ( G ` n ) ) < y /\ y < ( 2nd ` ( G ` n ) ) ) ) )
97	92, 96	mpbird 236	. . 3 |- ( ph -> B C_ U. ran ( (,) o. G ) )
98		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* , < ) ) }
99		eqid 2450	. . . 4 |- seq 1 ( + , ( ( abs o. - ) o. G ) ) = seq 1 ( + , ( ( abs o. - ) o. G ) )
100	98, 99	elovolmr 22422	. . 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 )
101	18, 97, 100	syl2anc 666	. 2 |- ( ph -> sup ( ran seq 1 ( + , ( ( abs o. - ) o. G ) ) , RR* , < ) e. M )
102	55, 101	eqeltrrd 2529	1 |- ( ph -> sup ( ran S , RR* , < ) e. M )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 984   = wceq 1443   e. wcel 1886   A.wral 2736   E.wrex 2737   {crab 2740   _Vcvv 3044   i^i cin 3402   C_ wss 3403   <.cop 3973   U.cuni 4197   class class class wbr 4401   |-> cmpt 4460   X. cxp 4831   ran crn 4834   o. ccom 4837   Fn wfn 5576   -->wf 5577   ` cfv 5581  (class class class)co 6288   1stc1st 6788   2ndc2nd 6789   ^m cmap 7469   supcsup 7951   RRcr 9535   0cc0 9536   1c1 9537   + caddc 9539   +oocpnf 9669   RR*cxr 9671   < clt 9672   <_ cle 9673   - cmin 9857   NNcn 10606   (,)cioo 11632   [,)cico 11634   seqcseq 12210   abscabs 13290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-er 7360  df-map 7471  df-en 7567  df-dom 7568  df-sdom 7569  df-sup 7953  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-n0 10867  df-z 10935  df-uz 11157  df-rp 11300  df-ioo 11636  df-ico 11638  df-fz 11782  df-seq 12211  df-exp 12270  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292

This theorem is referenced by:  ovolshftlem2  22456