Theorem ovolshftlem1 22212

Description: Lemma for ovolshft 22214. (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 22170	. . . . . . . . . . . . . 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 469	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) e. RR /\ ( 2nd ` ( F ` n ) ) e. RR /\ ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) ) )
4	3	simp1d 1009	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( F ` n ) ) e. RR )
5	3	simp2d 1010	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( 2nd ` ( F ` n ) ) e. RR )
6		ovolshft.2	. . . . . . . . . . . . 13 |- ( ph -> C e. RR )
7	6	adantr 463	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> C e. RR )
8	3	simp3d 1011	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) )
9	4, 5, 7, 8	leadd1dd 10206	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) + C ) <_ ( ( 2nd ` ( F ` n ) ) + C ) )
10		df-br 4396	. . . . . . . . . . 11 |- ( ( ( 1st ` ( F ` n ) ) + C ) <_ ( ( 2nd ` ( F ` n ) ) + C ) <-> <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. e. <_ )
11	9, 10	sylib 196	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. e. <_ )
12	4, 7	readdcld 9653	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) + C ) e. RR )
13	5, 7	readdcld 9653	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( ( 2nd ` ( F ` n ) ) + C ) e. RR )
14		opelxp 4853	. . . . . . . . . . 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 662	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. e. ( RR X. RR ) )
16	11, 15	elind 3627	. . . . . . . . 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 6033	. . . . . . . 8 |- ( ph -> G : NN --> ( <_ i^i ( RR X. RR ) ) )
19		eqid 2402	. . . . . . . . 9 |- ( ( abs o. - ) o. G ) = ( ( abs o. - ) o. G )
20	19	ovolfsf 22175	. . . . . . . 8 |- ( G : NN --> ( <_ i^i ( RR X. RR ) ) -> ( ( abs o. - ) o. G ) : NN --> ( 0 [,) +oo ) )
21		ffn 5714	. . . . . . . 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 2402	. . . . . . . . 9 |- ( ( abs o. - ) o. F ) = ( ( abs o. - ) o. F )
24	23	ovolfsf 22175	. . . . . . . 8 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> ( ( abs o. - ) o. F ) : NN --> ( 0 [,) +oo ) )
25		ffn 5714	. . . . . . . 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 4655	. . . . . . . . . . . . . 14 |- <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. e. _V
28	17	fvmpt2 5941	. . . . . . . . . . . . . 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 669	. . . . . . . . . . . . 13 |- ( n e. NN -> ( G ` n ) = <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. )
30	29	fveq2d 5853	. . . . . . . . . . . 12 |- ( n e. NN -> ( 2nd ` ( G ` n ) ) = ( 2nd ` <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. ) )
31		ovex 6306	. . . . . . . . . . . . 13 |- ( ( 1st ` ( F ` n ) ) + C ) e. _V
32		ovex 6306	. . . . . . . . . . . . 13 |- ( ( 2nd ` ( F ` n ) ) + C ) e. _V
33	31, 32	op2nd 6793	. . . . . . . . . . . 12 |- ( 2nd ` <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. ) = ( ( 2nd ` ( F ` n ) ) + C )
34	30, 33	syl6eq 2459	. . . . . . . . . . 11 |- ( n e. NN -> ( 2nd ` ( G ` n ) ) = ( ( 2nd ` ( F ` n ) ) + C ) )
35	29	fveq2d 5853	. . . . . . . . . . . 12 |- ( n e. NN -> ( 1st ` ( G ` n ) ) = ( 1st ` <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. ) )
36	31, 32	op1st 6792	. . . . . . . . . . . 12 |- ( 1st ` <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. ) = ( ( 1st ` ( F ` n ) ) + C )
37	35, 36	syl6eq 2459	. . . . . . . . . . 11 |- ( n e. NN -> ( 1st ` ( G ` n ) ) = ( ( 1st ` ( F ` n ) ) + C ) )
38	34, 37	oveq12d 6296	. . . . . . . . . 10 |- ( n e. NN -> ( ( 2nd ` ( G ` n ) ) - ( 1st ` ( G ` n ) ) ) = ( ( ( 2nd ` ( F ` n ) ) + C ) - ( ( 1st ` ( F ` n ) ) + C ) ) )
39	38	adantl 464	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( ( 2nd ` ( G ` n ) ) - ( 1st ` ( G ` n ) ) ) = ( ( ( 2nd ` ( F ` n ) ) + C ) - ( ( 1st ` ( F ` n ) ) + C ) ) )
40	5	recnd 9652	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( 2nd ` ( F ` n ) ) e. CC )
41	4	recnd 9652	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( F ` n ) ) e. CC )
42	7	recnd 9652	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> C e. CC )
43	40, 41, 42	pnpcan2d 10005	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( ( ( 2nd ` ( F ` n ) ) + C ) - ( ( 1st ` ( F ` n ) ) + C ) ) = ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) )
44	39, 43	eqtrd 2443	. . . . . . . 8 |- ( ( ph /\ n e. NN ) -> ( ( 2nd ` ( G ` n ) ) - ( 1st ` ( G ` n ) ) ) = ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) )
45	19	ovolfsval 22174	. . . . . . . . 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 469	. . . . . . . 8 |- ( ( ph /\ n e. NN ) -> ( ( ( abs o. - ) o. G ) ` n ) = ( ( 2nd ` ( G ` n ) ) - ( 1st ` ( G ` n ) ) ) )
47	23	ovolfsval 22174	. . . . . . . . 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 469	. . . . . . . 8 |- ( ( ph /\ n e. NN ) -> ( ( ( abs o. - ) o. F ) ` n ) = ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) )
49	44, 46, 48	3eqtr4d 2453	. . . . . . 7 |- ( ( ph /\ n e. NN ) -> ( ( ( abs o. - ) o. G ) ` n ) = ( ( ( abs o. - ) o. F ) ` n ) )
50	22, 26, 49	eqfnfvd 5962	. . . . . 6 |- ( ph -> ( ( abs o. - ) o. G ) = ( ( abs o. - ) o. F ) )
51	50	seqeq3d 12159	. . . . 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 2461	. . . 4 |- ( ph -> seq 1 ( + , ( ( abs o. - ) o. G ) ) = S )
54	53	rneqd 5051	. . 3 |- ( ph -> ran seq 1 ( + , ( ( abs o. - ) o. G ) ) = ran S )
55	54	supeq1d 7939	. 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 2472	. . . . . . . 8 |- ( ph -> ( y e. B <-> y e. { x e. RR | ( x - C ) e. A } ) )
58		oveq1 6285	. . . . . . . . . 10 |- ( x = y -> ( x - C ) = ( y - C ) )
59	58	eleq1d 2471	. . . . . . . . 9 |- ( x = y -> ( ( x - C ) e. A <-> ( y - C ) e. A ) )
60	59	elrab 3207	. . . . . . . 8 |- ( y e. { x e. RR | ( x - C ) e. A } <-> ( y e. RR /\ ( y - C ) e. A ) )
61	57, 60	syl6bb 261	. . . . . . 7 |- ( ph -> ( y e. B <-> ( y e. RR /\ ( y - C ) e. A ) ) )
62	61	biimpa 482	. . . . . 6 |- ( ( ph /\ y e. B ) -> ( y e. RR /\ ( y - C ) e. A ) )
63		simprr 758	. . . . . . . 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 22171	. . . . . . . . . . 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 659	. . . . . . . . . 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 210	. . . . . . . . 9 |- ( ph -> A. x e. A E. n e. NN ( ( 1st ` ( F ` n ) ) < x /\ x < ( 2nd ` ( F ` n ) ) ) )
69	68	adantr 463	. . . . . . . 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 4399	. . . . . . . . . . 11 |- ( x = ( y - C ) -> ( ( 1st ` ( F ` n ) ) < x <-> ( 1st ` ( F ` n ) ) < ( y - C ) ) )
71		breq1 4398	. . . . . . . . . . 11 |- ( x = ( y - C ) -> ( x < ( 2nd ` ( F ` n ) ) <-> ( y - C ) < ( 2nd ` ( F ` n ) ) ) )
72	70, 71	anbi12d 709	. . . . . . . . . 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 2918	. . . . . . . . 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 3156	. . . . . . . 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 59	. . . . . . 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 464	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( 1st ` ( G ` n ) ) = ( ( 1st ` ( F ` n ) ) + C ) )
77	76	breq1d 4405	. . . . . . . . . 10 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( ( 1st ` ( G ` n ) ) < y <-> ( ( 1st ` ( F ` n ) ) + C ) < y ) )
78	4	adantlr 713	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( 1st ` ( F ` n ) ) e. RR )
79	6	ad2antrr 724	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> C e. RR )
80		simplrl 762	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> y e. RR )
81	78, 79, 80	ltaddsubd 10192	. . . . . . . . . 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 253	. . . . . . . . 9 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( ( 1st ` ( G ` n ) ) < y <-> ( 1st ` ( F ` n ) ) < ( y - C ) ) )
83	34	adantl 464	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( 2nd ` ( G ` n ) ) = ( ( 2nd ` ( F ` n ) ) + C ) )
84	83	breq2d 4407	. . . . . . . . . 10 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( y < ( 2nd ` ( G ` n ) ) <-> y < ( ( 2nd ` ( F ` n ) ) + C ) ) )
85	5	adantlr 713	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( 2nd ` ( F ` n ) ) e. RR )
86	80, 79, 85	ltsubaddd 10188	. . . . . . . . . 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 256	. . . . . . . . 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 709	. . . . . . . 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 2915	. . . . . . 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 232	. . . . . 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 468	. . . . 5 |- ( ( ph /\ y e. B ) -> E. n e. NN ( ( 1st ` ( G ` n ) ) < y /\ y < ( 2nd ` ( G ` n ) ) ) )
92	91	ralrimiva 2818	. . . 4 |- ( ph -> A. y e. B E. n e. NN ( ( 1st ` ( G ` n ) ) < y /\ y < ( 2nd ` ( G ` n ) ) ) )
93		ssrab2 3524	. . . . . 6 |- { x e. RR | ( x - C ) e. A } C_ RR
94	56, 93	syl6eqss 3492	. . . . 5 |- ( ph -> B C_ RR )
95		ovolfioo 22171	. . . . 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 659	. . . 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 232	. . 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 2402	. . . 4 |- seq 1 ( + , ( ( abs o. - ) o. G ) ) = seq 1 ( + , ( ( abs o. - ) o. G ) )
100	98, 99	elovolmr 22179	. . 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 659	. 2 |- ( ph -> sup ( ran seq 1 ( + , ( ( abs o. - ) o. G ) ) , RR* , < ) e. M )
102	55, 101	eqeltrrd 2491	1 |- ( ph -> sup ( ran S , RR* , < ) e. M )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 974   = wceq 1405   e. wcel 1842   A.wral 2754   E.wrex 2755   {crab 2758   _Vcvv 3059   i^i cin 3413   C_ wss 3414   <.cop 3978   U.cuni 4191   class class class wbr 4395   |-> cmpt 4453   X. cxp 4821   ran crn 4824   o. ccom 4827   Fn wfn 5564   -->wf 5565   ` cfv 5569  (class class class)co 6278   1stc1st 6782   2ndc2nd 6783   ^m cmap 7457   supcsup 7934   RRcr 9521   0cc0 9522   1c1 9523   + caddc 9525   +oocpnf 9655   RR*cxr 9657   < clt 9658   <_ cle 9659   - cmin 9841   NNcn 10576   (,)cioo 11582   [,)cico 11584   seqcseq 12151   abscabs 13216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-sup 7935  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-ioo 11586  df-ico 11588  df-fz 11727  df-seq 12152  df-exp 12211  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218

This theorem is referenced by:  ovolshftlem2  22213