Theorem ovolshftlem1 20951

Description: Lemma for ovolshft 20953. (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 20909	. . . . . . . . . . . . . 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 468	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) e. RR /\ ( 2nd ` ( F ` n ) ) e. RR /\ ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) ) )
4	3	simp1d 995	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( F ` n ) ) e. RR )
5	3	simp2d 996	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( 2nd ` ( F ` n ) ) e. RR )
6		ovolshft.2	. . . . . . . . . . . . 13 |- ( ph -> C e. RR )
7	6	adantr 462	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> C e. RR )
8	3	simp3d 997	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) )
9	4, 5, 7, 8	leadd1dd 9949	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) + C ) <_ ( ( 2nd ` ( F ` n ) ) + C ) )
10		df-br 4290	. . . . . . . . . . 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 9409	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) + C ) e. RR )
13	5, 7	readdcld 9409	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( ( 2nd ` ( F ` n ) ) + C ) e. RR )
14		opelxp 4865	. . . . . . . . . . 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 659	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. e. ( RR X. RR ) )
16	11, 15	elind 3537	. . . . . . . . 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 5864	. . . . . . . 8 |- ( ph -> G : NN --> ( <_ i^i ( RR X. RR ) ) )
19		eqid 2441	. . . . . . . . 9 |- ( ( abs o. - ) o. G ) = ( ( abs o. - ) o. G )
20	19	ovolfsf 20914	. . . . . . . 8 |- ( G : NN --> ( <_ i^i ( RR X. RR ) ) -> ( ( abs o. - ) o. G ) : NN --> ( 0 [,) +oo ) )
21		ffn 5556	. . . . . . . 8 |- ( ( ( abs o. - ) o. G ) : NN --> ( 0 [,) +oo ) -> ( ( abs o. - ) o. G ) Fn NN )
22	18, 20, 21	3syl 20	. . . . . . 7 |- ( ph -> ( ( abs o. - ) o. G ) Fn NN )
23		eqid 2441	. . . . . . . . 9 |- ( ( abs o. - ) o. F ) = ( ( abs o. - ) o. F )
24	23	ovolfsf 20914	. . . . . . . 8 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> ( ( abs o. - ) o. F ) : NN --> ( 0 [,) +oo ) )
25		ffn 5556	. . . . . . . 8 |- ( ( ( abs o. - ) o. F ) : NN --> ( 0 [,) +oo ) -> ( ( abs o. - ) o. F ) Fn NN )
26	1, 24, 25	3syl 20	. . . . . . 7 |- ( ph -> ( ( abs o. - ) o. F ) Fn NN )
27		opex 4553	. . . . . . . . . . . . . 14 |- <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. e. _V
28	17	fvmpt2 5778	. . . . . . . . . . . . . 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 666	. . . . . . . . . . . . 13 |- ( n e. NN -> ( G ` n ) = <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. )
30	29	fveq2d 5692	. . . . . . . . . . . 12 |- ( n e. NN -> ( 2nd ` ( G ` n ) ) = ( 2nd ` <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. ) )
31		ovex 6115	. . . . . . . . . . . . 13 |- ( ( 1st ` ( F ` n ) ) + C ) e. _V
32		ovex 6115	. . . . . . . . . . . . 13 |- ( ( 2nd ` ( F ` n ) ) + C ) e. _V
33	31, 32	op2nd 6585	. . . . . . . . . . . 12 |- ( 2nd ` <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. ) = ( ( 2nd ` ( F ` n ) ) + C )
34	30, 33	syl6eq 2489	. . . . . . . . . . 11 |- ( n e. NN -> ( 2nd ` ( G ` n ) ) = ( ( 2nd ` ( F ` n ) ) + C ) )
35	29	fveq2d 5692	. . . . . . . . . . . 12 |- ( n e. NN -> ( 1st ` ( G ` n ) ) = ( 1st ` <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. ) )
36	31, 32	op1st 6584	. . . . . . . . . . . 12 |- ( 1st ` <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. ) = ( ( 1st ` ( F ` n ) ) + C )
37	35, 36	syl6eq 2489	. . . . . . . . . . 11 |- ( n e. NN -> ( 1st ` ( G ` n ) ) = ( ( 1st ` ( F ` n ) ) + C ) )
38	34, 37	oveq12d 6108	. . . . . . . . . 10 |- ( n e. NN -> ( ( 2nd ` ( G ` n ) ) - ( 1st ` ( G ` n ) ) ) = ( ( ( 2nd ` ( F ` n ) ) + C ) - ( ( 1st ` ( F ` n ) ) + C ) ) )
39	38	adantl 463	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( ( 2nd ` ( G ` n ) ) - ( 1st ` ( G ` n ) ) ) = ( ( ( 2nd ` ( F ` n ) ) + C ) - ( ( 1st ` ( F ` n ) ) + C ) ) )
40	5	recnd 9408	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( 2nd ` ( F ` n ) ) e. CC )
41	4	recnd 9408	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( F ` n ) ) e. CC )
42	7	recnd 9408	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> C e. CC )
43	40, 41, 42	pnpcan2d 9753	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( ( ( 2nd ` ( F ` n ) ) + C ) - ( ( 1st ` ( F ` n ) ) + C ) ) = ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) )
44	39, 43	eqtrd 2473	. . . . . . . 8 |- ( ( ph /\ n e. NN ) -> ( ( 2nd ` ( G ` n ) ) - ( 1st ` ( G ` n ) ) ) = ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) )
45	19	ovolfsval 20913	. . . . . . . . 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 468	. . . . . . . 8 |- ( ( ph /\ n e. NN ) -> ( ( ( abs o. - ) o. G ) ` n ) = ( ( 2nd ` ( G ` n ) ) - ( 1st ` ( G ` n ) ) ) )
47	23	ovolfsval 20913	. . . . . . . . 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 468	. . . . . . . 8 |- ( ( ph /\ n e. NN ) -> ( ( ( abs o. - ) o. F ) ` n ) = ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) )
49	44, 46, 48	3eqtr4d 2483	. . . . . . 7 |- ( ( ph /\ n e. NN ) -> ( ( ( abs o. - ) o. G ) ` n ) = ( ( ( abs o. - ) o. F ) ` n ) )
50	22, 26, 49	eqfnfvd 5797	. . . . . 6 |- ( ph -> ( ( abs o. - ) o. G ) = ( ( abs o. - ) o. F ) )
51	50	seqeq3d 11810	. . . . 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 2491	. . . 4 |- ( ph -> seq 1 ( + , ( ( abs o. - ) o. G ) ) = S )
54	53	rneqd 5063	. . 3 |- ( ph -> ran seq 1 ( + , ( ( abs o. - ) o. G ) ) = ran S )
55	54	supeq1d 7692	. 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 2508	. . . . . . . 8 |- ( ph -> ( y e. B <-> y e. { x e. RR | ( x - C ) e. A } ) )
58		oveq1 6097	. . . . . . . . . 10 |- ( x = y -> ( x - C ) = ( y - C ) )
59	58	eleq1d 2507	. . . . . . . . 9 |- ( x = y -> ( ( x - C ) e. A <-> ( y - C ) e. A ) )
60	59	elrab 3114	. . . . . . . 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 481	. . . . . 6 |- ( ( ph /\ y e. B ) -> ( y e. RR /\ ( y - C ) e. A ) )
63		simprr 751	. . . . . . . 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 20910	. . . . . . . . . . 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 656	. . . . . . . . . 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 462	. . . . . . . 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 4293	. . . . . . . . . . 11 |- ( x = ( y - C ) -> ( ( 1st ` ( F ` n ) ) < x <-> ( 1st ` ( F ` n ) ) < ( y - C ) ) )
71		breq1 4292	. . . . . . . . . . 11 |- ( x = ( y - C ) -> ( x < ( 2nd ` ( F ` n ) ) <-> ( y - C ) < ( 2nd ` ( F ` n ) ) ) )
72	70, 71	anbi12d 705	. . . . . . . . . 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 2734	. . . . . . . . 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 3066	. . . . . . . 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 60	. . . . . . 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 463	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( 1st ` ( G ` n ) ) = ( ( 1st ` ( F ` n ) ) + C ) )
77	76	breq1d 4299	. . . . . . . . . 10 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( ( 1st ` ( G ` n ) ) < y <-> ( ( 1st ` ( F ` n ) ) + C ) < y ) )
78	4	adantlr 709	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( 1st ` ( F ` n ) ) e. RR )
79	6	ad2antrr 720	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> C e. RR )
80		simplrl 754	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> y e. RR )
81	78, 79, 80	ltaddsubd 9935	. . . . . . . . . 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 463	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( 2nd ` ( G ` n ) ) = ( ( 2nd ` ( F ` n ) ) + C ) )
84	83	breq2d 4301	. . . . . . . . . 10 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( y < ( 2nd ` ( G ` n ) ) <-> y < ( ( 2nd ` ( F ` n ) ) + C ) ) )
85	5	adantlr 709	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( 2nd ` ( F ` n ) ) e. RR )
86	80, 79, 85	ltsubaddd 9931	. . . . . . . . . 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 705	. . . . . . . 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 2730	. . . . . . 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 467	. . . . 5 |- ( ( ph /\ y e. B ) -> E. n e. NN ( ( 1st ` ( G ` n ) ) < y /\ y < ( 2nd ` ( G ` n ) ) ) )
92	91	ralrimiva 2797	. . . 4 |- ( ph -> A. y e. B E. n e. NN ( ( 1st ` ( G ` n ) ) < y /\ y < ( 2nd ` ( G ` n ) ) ) )
93		ssrab2 3434	. . . . . 6 |- { x e. RR | ( x - C ) e. A } C_ RR
94	56, 93	syl6eqss 3403	. . . . 5 |- ( ph -> B C_ RR )
95		ovolfioo 20910	. . . . 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 656	. . . 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 2441	. . . 4 |- seq 1 ( + , ( ( abs o. - ) o. G ) ) = seq 1 ( + , ( ( abs o. - ) o. G ) )
100	98, 99	elovolmr 20918	. . 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 656	. 2 |- ( ph -> sup ( ran seq 1 ( + , ( ( abs o. - ) o. G ) ) , RR* , < ) e. M )
102	55, 101	eqeltrrd 2516	1 |- ( ph -> sup ( ran S , RR* , < ) e. M )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 960   = wceq 1364   e. wcel 1761   A.wral 2713   E.wrex 2714   {crab 2717   _Vcvv 2970   i^i cin 3324   C_ wss 3325   <.cop 3880   U.cuni 4088   class class class wbr 4289   e. cmpt 4347   X. cxp 4834   ran crn 4837   o. ccom 4840   Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575   ^m cmap 7210   supcsup 7686   RRcr 9277   0cc0 9278   1c1 9279   + caddc 9281   +oocpnf 9411   RR*cxr 9413   < clt 9414   <_ cle 9415   - cmin 9591   NNcn 10318   (,)cioo 11296   [,)cico 11298   seqcseq 11802   abscabs 12719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-ioo 11300  df-ico 11302  df-fz 11434  df-seq 11803  df-exp 11862  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721

This theorem is referenced by:  ovolshftlem2  20952