Theorem ovolshftlem1 21007

Description: Lemma for ovolshft 21009. (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 20965	. . . . . . . . . . . . . 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 471	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) e. RR /\ ( 2nd ` ( F ` n ) ) e. RR /\ ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) ) )
4	3	simp1d 1000	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( F ` n ) ) e. RR )
5	3	simp2d 1001	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( 2nd ` ( F ` n ) ) e. RR )
6		ovolshft.2	. . . . . . . . . . . . 13 |- ( ph -> C e. RR )
7	6	adantr 465	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> C e. RR )
8	3	simp3d 1002	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) )
9	4, 5, 7, 8	leadd1dd 9968	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) + C ) <_ ( ( 2nd ` ( F ` n ) ) + C ) )
10		df-br 4308	. . . . . . . . . . 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 9428	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) + C ) e. RR )
13	5, 7	readdcld 9428	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( ( 2nd ` ( F ` n ) ) + C ) e. RR )
14		opelxp 4884	. . . . . . . . . . 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 664	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. e. ( RR X. RR ) )
16	11, 15	elind 3555	. . . . . . . . 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 5882	. . . . . . . 8 |- ( ph -> G : NN --> ( <_ i^i ( RR X. RR ) ) )
19		eqid 2443	. . . . . . . . 9 |- ( ( abs o. - ) o. G ) = ( ( abs o. - ) o. G )
20	19	ovolfsf 20970	. . . . . . . 8 |- ( G : NN --> ( <_ i^i ( RR X. RR ) ) -> ( ( abs o. - ) o. G ) : NN --> ( 0 [,) +oo ) )
21		ffn 5574	. . . . . . . 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 2443	. . . . . . . . 9 |- ( ( abs o. - ) o. F ) = ( ( abs o. - ) o. F )
24	23	ovolfsf 20970	. . . . . . . 8 |- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> ( ( abs o. - ) o. F ) : NN --> ( 0 [,) +oo ) )
25		ffn 5574	. . . . . . . 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 4571	. . . . . . . . . . . . . 14 |- <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. e. _V
28	17	fvmpt2 5796	. . . . . . . . . . . . . 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 671	. . . . . . . . . . . . 13 |- ( n e. NN -> ( G ` n ) = <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. )
30	29	fveq2d 5710	. . . . . . . . . . . 12 |- ( n e. NN -> ( 2nd ` ( G ` n ) ) = ( 2nd ` <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. ) )
31		ovex 6131	. . . . . . . . . . . . 13 |- ( ( 1st ` ( F ` n ) ) + C ) e. _V
32		ovex 6131	. . . . . . . . . . . . 13 |- ( ( 2nd ` ( F ` n ) ) + C ) e. _V
33	31, 32	op2nd 6601	. . . . . . . . . . . 12 |- ( 2nd ` <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. ) = ( ( 2nd ` ( F ` n ) ) + C )
34	30, 33	syl6eq 2491	. . . . . . . . . . 11 |- ( n e. NN -> ( 2nd ` ( G ` n ) ) = ( ( 2nd ` ( F ` n ) ) + C ) )
35	29	fveq2d 5710	. . . . . . . . . . . 12 |- ( n e. NN -> ( 1st ` ( G ` n ) ) = ( 1st ` <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. ) )
36	31, 32	op1st 6600	. . . . . . . . . . . 12 |- ( 1st ` <. ( ( 1st ` ( F ` n ) ) + C ) , ( ( 2nd ` ( F ` n ) ) + C ) >. ) = ( ( 1st ` ( F ` n ) ) + C )
37	35, 36	syl6eq 2491	. . . . . . . . . . 11 |- ( n e. NN -> ( 1st ` ( G ` n ) ) = ( ( 1st ` ( F ` n ) ) + C ) )
38	34, 37	oveq12d 6124	. . . . . . . . . 10 |- ( n e. NN -> ( ( 2nd ` ( G ` n ) ) - ( 1st ` ( G ` n ) ) ) = ( ( ( 2nd ` ( F ` n ) ) + C ) - ( ( 1st ` ( F ` n ) ) + C ) ) )
39	38	adantl 466	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( ( 2nd ` ( G ` n ) ) - ( 1st ` ( G ` n ) ) ) = ( ( ( 2nd ` ( F ` n ) ) + C ) - ( ( 1st ` ( F ` n ) ) + C ) ) )
40	5	recnd 9427	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( 2nd ` ( F ` n ) ) e. CC )
41	4	recnd 9427	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( F ` n ) ) e. CC )
42	7	recnd 9427	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> C e. CC )
43	40, 41, 42	pnpcan2d 9772	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( ( ( 2nd ` ( F ` n ) ) + C ) - ( ( 1st ` ( F ` n ) ) + C ) ) = ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) )
44	39, 43	eqtrd 2475	. . . . . . . 8 |- ( ( ph /\ n e. NN ) -> ( ( 2nd ` ( G ` n ) ) - ( 1st ` ( G ` n ) ) ) = ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) )
45	19	ovolfsval 20969	. . . . . . . . 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 471	. . . . . . . 8 |- ( ( ph /\ n e. NN ) -> ( ( ( abs o. - ) o. G ) ` n ) = ( ( 2nd ` ( G ` n ) ) - ( 1st ` ( G ` n ) ) ) )
47	23	ovolfsval 20969	. . . . . . . . 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 471	. . . . . . . 8 |- ( ( ph /\ n e. NN ) -> ( ( ( abs o. - ) o. F ) ` n ) = ( ( 2nd ` ( F ` n ) ) - ( 1st ` ( F ` n ) ) ) )
49	44, 46, 48	3eqtr4d 2485	. . . . . . 7 |- ( ( ph /\ n e. NN ) -> ( ( ( abs o. - ) o. G ) ` n ) = ( ( ( abs o. - ) o. F ) ` n ) )
50	22, 26, 49	eqfnfvd 5815	. . . . . 6 |- ( ph -> ( ( abs o. - ) o. G ) = ( ( abs o. - ) o. F ) )
51	50	seqeq3d 11829	. . . . 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 2493	. . . 4 |- ( ph -> seq 1 ( + , ( ( abs o. - ) o. G ) ) = S )
54	53	rneqd 5082	. . 3 |- ( ph -> ran seq 1 ( + , ( ( abs o. - ) o. G ) ) = ran S )
55	54	supeq1d 7711	. 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 2510	. . . . . . . 8 |- ( ph -> ( y e. B <-> y e. { x e. RR | ( x - C ) e. A } ) )
58		oveq1 6113	. . . . . . . . . 10 |- ( x = y -> ( x - C ) = ( y - C ) )
59	58	eleq1d 2509	. . . . . . . . 9 |- ( x = y -> ( ( x - C ) e. A <-> ( y - C ) e. A ) )
60	59	elrab 3132	. . . . . . . 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 484	. . . . . 6 |- ( ( ph /\ y e. B ) -> ( y e. RR /\ ( y - C ) e. A ) )
63		simprr 756	. . . . . . . 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 20966	. . . . . . . . . . 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 661	. . . . . . . . . 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 465	. . . . . . . 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 4311	. . . . . . . . . . 11 |- ( x = ( y - C ) -> ( ( 1st ` ( F ` n ) ) < x <-> ( 1st ` ( F ` n ) ) < ( y - C ) ) )
71		breq1 4310	. . . . . . . . . . 11 |- ( x = ( y - C ) -> ( x < ( 2nd ` ( F ` n ) ) <-> ( y - C ) < ( 2nd ` ( F ` n ) ) ) )
72	70, 71	anbi12d 710	. . . . . . . . . 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 2751	. . . . . . . . 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 3084	. . . . . . . 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 466	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( 1st ` ( G ` n ) ) = ( ( 1st ` ( F ` n ) ) + C ) )
77	76	breq1d 4317	. . . . . . . . . 10 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( ( 1st ` ( G ` n ) ) < y <-> ( ( 1st ` ( F ` n ) ) + C ) < y ) )
78	4	adantlr 714	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( 1st ` ( F ` n ) ) e. RR )
79	6	ad2antrr 725	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> C e. RR )
80		simplrl 759	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> y e. RR )
81	78, 79, 80	ltaddsubd 9954	. . . . . . . . . 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 466	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( 2nd ` ( G ` n ) ) = ( ( 2nd ` ( F ` n ) ) + C ) )
84	83	breq2d 4319	. . . . . . . . . 10 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( y < ( 2nd ` ( G ` n ) ) <-> y < ( ( 2nd ` ( F ` n ) ) + C ) ) )
85	5	adantlr 714	. . . . . . . . . . 11 |- ( ( ( ph /\ ( y e. RR /\ ( y - C ) e. A ) ) /\ n e. NN ) -> ( 2nd ` ( F ` n ) ) e. RR )
86	80, 79, 85	ltsubaddd 9950	. . . . . . . . . 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 710	. . . . . . . 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 2747	. . . . . . 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 470	. . . . 5 |- ( ( ph /\ y e. B ) -> E. n e. NN ( ( 1st ` ( G ` n ) ) < y /\ y < ( 2nd ` ( G ` n ) ) ) )
92	91	ralrimiva 2814	. . . 4 |- ( ph -> A. y e. B E. n e. NN ( ( 1st ` ( G ` n ) ) < y /\ y < ( 2nd ` ( G ` n ) ) ) )
93		ssrab2 3452	. . . . . 6 |- { x e. RR | ( x - C ) e. A } C_ RR
94	56, 93	syl6eqss 3421	. . . . 5 |- ( ph -> B C_ RR )
95		ovolfioo 20966	. . . . 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 661	. . . 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 2443	. . . 4 |- seq 1 ( + , ( ( abs o. - ) o. G ) ) = seq 1 ( + , ( ( abs o. - ) o. G ) )
100	98, 99	elovolmr 20974	. . 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 661	. 2 |- ( ph -> sup ( ran seq 1 ( + , ( ( abs o. - ) o. G ) ) , RR* , < ) e. M )
102	55, 101	eqeltrrd 2518	1 |- ( ph -> sup ( ran S , RR* , < ) e. M )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   A.wral 2730   E.wrex 2731   {crab 2734   _Vcvv 2987   i^i cin 3342   C_ wss 3343   <.cop 3898   U.cuni 4106   class class class wbr 4307   e. cmpt 4365   X. cxp 4853   ran crn 4856   o. ccom 4859   Fn wfn 5428   -->wf 5429   ` cfv 5433  (class class class)co 6106   1stc1st 6590   2ndc2nd 6591   ^m cmap 7229   supcsup 7705   RRcr 9296   0cc0 9297   1c1 9298   + caddc 9300   +oocpnf 9430   RR*cxr 9432   < clt 9433   <_ cle 9434   - cmin 9610   NNcn 10337   (,)cioo 11315   [,)cico 11317   seqcseq 11821   abscabs 12738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374  ax-pre-sup 9375

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-1st 6592  df-2nd 6593  df-recs 6847  df-rdg 6881  df-er 7116  df-map 7231  df-en 7326  df-dom 7327  df-sdom 7328  df-sup 7706  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-div 10009  df-nn 10338  df-2 10395  df-3 10396  df-n0 10595  df-z 10662  df-uz 10877  df-rp 11007  df-ioo 11319  df-ico 11321  df-fz 11453  df-seq 11822  df-exp 11881  df-cj 12603  df-re 12604  df-im 12605  df-sqr 12739  df-abs 12740

This theorem is referenced by:  ovolshftlem2  21008