Theorem ovolval5lem2 40867

Description: |- ( ( ph /\ n e. NN ) -> <. ( ( 1st ( F n ) ) - ( W / ( 2 ^ n ) ) ) , ( 2nd ( F n ) ) >. e. ( RR X. RR ) ) (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
ovolval5lem2.q	|- Q = { z e. RR* | E. f e. ( ( RR X. RR ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ z = (Σ^ ` ( ( vol o. (,) ) o. f ) ) ) }
ovolval5lem2.y	|- ( ph -> Y = (Σ^ ` ( ( vol o. [,) ) o. F ) ) )
ovolval5lem2.z	|- Z = (Σ^ ` ( ( vol o. (,) ) o. G ) )
ovolval5lem2.f	|- ( ph -> F : NN --> ( RR X. RR ) )
ovolval5lem2.s	|- ( ph -> A C_ U. ran ( [,) o. F ) )
ovolval5lem2.w	|- ( ph -> W e. RR+ )
ovolval5lem2.g	|- G = ( n e. NN |-> <. ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) , ( 2nd ` ( F ` n ) ) >. )

Assertion

Ref	Expression
ovolval5lem2	|- ( ph -> E. z e. Q z <_ ( Y +e W ) )

Distinct variable groups:   A, f, z   n, F   f, G   n, G   z, Q   n, W   z, W   z, Y   f, Z, z   ph, n

Allowed substitution hints:   ph( z, f)   A( n)   Q( f, n)   F( z, f)   G( z)   W( f)   Y( f, n)   Z( n)

Proof of Theorem ovolval5lem2

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovolval5lem2.z	. . . . . 6 |- Z = (Σ^ ` ( ( vol o. (,) ) o. G ) )
2	1	a1i 11	. . . . 5 |- ( ph -> Z = (Σ^ ` ( ( vol o. (,) ) o. G ) ) )
3		nnex 11026	. . . . . . 7 |- NN e. _V
4	3	a1i 11	. . . . . 6 |- ( ph -> NN e. _V )
5		volioof 40204	. . . . . . . 8 |- ( vol o. (,) ) : ( RR* X. RR* ) --> ( 0 [,] +oo )
6	5	a1i 11	. . . . . . 7 |- ( ph -> ( vol o. (,) ) : ( RR* X. RR* ) --> ( 0 [,] +oo ) )
7		rexpssxrxp 10084	. . . . . . . 8 |- ( RR X. RR ) C_ ( RR* X. RR* )
8	7	a1i 11	. . . . . . 7 |- ( ph -> ( RR X. RR ) C_ ( RR* X. RR* ) )
9		ovolval5lem2.f	. . . . . . . . . . . 12 |- ( ph -> F : NN --> ( RR X. RR ) )
10	9	ffvelrnda 6359	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( F ` n ) e. ( RR X. RR ) )
11		xp1st 7198	. . . . . . . . . . 11 |- ( ( F ` n ) e. ( RR X. RR ) -> ( 1st ` ( F ` n ) ) e. RR )
12	10, 11	syl 17	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( F ` n ) ) e. RR )
13		ovolval5lem2.w	. . . . . . . . . . . . 13 |- ( ph -> W e. RR+ )
14	13	rpred 11872	. . . . . . . . . . . 12 |- ( ph -> W e. RR )
15	14	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> W e. RR )
16		2nn 11185	. . . . . . . . . . . . . . 15 |- 2 e. NN
17	16	a1i 11	. . . . . . . . . . . . . 14 |- ( n e. NN -> 2 e. NN )
18		nnnn0 11299	. . . . . . . . . . . . . 14 |- ( n e. NN -> n e. NN0 )
19	17, 18	nnexpcld 13030	. . . . . . . . . . . . 13 |- ( n e. NN -> ( 2 ^ n ) e. NN )
20	19	nnred 11035	. . . . . . . . . . . 12 |- ( n e. NN -> ( 2 ^ n ) e. RR )
21	20	adantl 482	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( 2 ^ n ) e. RR )
22	19	nnne0d 11065	. . . . . . . . . . . 12 |- ( n e. NN -> ( 2 ^ n ) =/= 0 )
23	22	adantl 482	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( 2 ^ n ) =/= 0 )
24	15, 21, 23	redivcld 10853	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( W / ( 2 ^ n ) ) e. RR )
25	12, 24	resubcld 10458	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) e. RR )
26		xp2nd 7199	. . . . . . . . . 10 |- ( ( F ` n ) e. ( RR X. RR ) -> ( 2nd ` ( F ` n ) ) e. RR )
27	10, 26	syl 17	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( 2nd ` ( F ` n ) ) e. RR )
28	25, 27	opelxpd 5149	. . . . . . . 8 |- ( ( ph /\ n e. NN ) -> <. ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) , ( 2nd ` ( F ` n ) ) >. e. ( RR X. RR ) )
29		ovolval5lem2.g	. . . . . . . 8 |- G = ( n e. NN |-> <. ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) , ( 2nd ` ( F ` n ) ) >. )
30	28, 29	fmptd 6385	. . . . . . 7 |- ( ph -> G : NN --> ( RR X. RR ) )
31	6, 8, 30	fcoss 39402	. . . . . 6 |- ( ph -> ( ( vol o. (,) ) o. G ) : NN --> ( 0 [,] +oo ) )
32	4, 31	sge0xrcl 40602	. . . . 5 |- ( ph -> (Σ^ ` ( ( vol o. (,) ) o. G ) ) e. RR* )
33	2, 32	eqeltrd 2701	. . . 4 |- ( ph -> Z e. RR* )
34		reex 10027	. . . . . . . . 9 |- RR e. _V
35	34, 34	xpex 6962	. . . . . . . 8 |- ( RR X. RR ) e. _V
36	35	a1i 11	. . . . . . 7 |- ( ph -> ( RR X. RR ) e. _V )
37	36, 4	elmapd 7871	. . . . . 6 |- ( ph -> ( G e. ( ( RR X. RR ) ^m NN ) <-> G : NN --> ( RR X. RR ) ) )
38	30, 37	mpbird 247	. . . . 5 |- ( ph -> G e. ( ( RR X. RR ) ^m NN ) )
39		ovolval5lem2.s	. . . . . . 7 |- ( ph -> A C_ U. ran ( [,) o. F ) )
40	30	ffvelrnda 6359	. . . . . . . . . . . . . 14 |- ( ( ph /\ n e. NN ) -> ( G ` n ) e. ( RR X. RR ) )
41		xp1st 7198	. . . . . . . . . . . . . 14 |- ( ( G ` n ) e. ( RR X. RR ) -> ( 1st ` ( G ` n ) ) e. RR )
42	40, 41	syl 17	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( G ` n ) ) e. RR )
43	42	rexrd 10089	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( G ` n ) ) e. RR* )
44		xp2nd 7199	. . . . . . . . . . . . . 14 |- ( ( G ` n ) e. ( RR X. RR ) -> ( 2nd ` ( G ` n ) ) e. RR )
45	40, 44	syl 17	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN ) -> ( 2nd ` ( G ` n ) ) e. RR )
46	45	rexrd 10089	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( 2nd ` ( G ` n ) ) e. RR* )
47	13	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ n e. NN ) -> W e. RR+ )
48	19	nnrpd 11870	. . . . . . . . . . . . . . . 16 |- ( n e. NN -> ( 2 ^ n ) e. RR+ )
49	48	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( ph /\ n e. NN ) -> ( 2 ^ n ) e. RR+ )
50	47, 49	rpdivcld 11889	. . . . . . . . . . . . . 14 |- ( ( ph /\ n e. NN ) -> ( W / ( 2 ^ n ) ) e. RR+ )
51	12, 50	ltsubrpd 11904	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) < ( 1st ` ( F ` n ) ) )
52		id 22	. . . . . . . . . . . . . . . . . 18 |- ( n e. NN -> n e. NN )
53		opex 4932	. . . . . . . . . . . . . . . . . . 19 |- <. ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) , ( 2nd ` ( F ` n ) ) >. e. _V
54	53	a1i 11	. . . . . . . . . . . . . . . . . 18 |- ( n e. NN -> <. ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) , ( 2nd ` ( F ` n ) ) >. e. _V )
55	29	fvmpt2 6291	. . . . . . . . . . . . . . . . . 18 |- ( ( n e. NN /\ <. ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) , ( 2nd ` ( F ` n ) ) >. e. _V ) -> ( G ` n ) = <. ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) , ( 2nd ` ( F ` n ) ) >. )
56	52, 54, 55	syl2anc 693	. . . . . . . . . . . . . . . . 17 |- ( n e. NN -> ( G ` n ) = <. ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) , ( 2nd ` ( F ` n ) ) >. )
57	56	fveq2d 6195	. . . . . . . . . . . . . . . 16 |- ( n e. NN -> ( 1st ` ( G ` n ) ) = ( 1st ` <. ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) , ( 2nd ` ( F ` n ) ) >. ) )
58		ovex 6678	. . . . . . . . . . . . . . . . . 18 |- ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) e. _V
59		fvex 6201	. . . . . . . . . . . . . . . . . 18 |- ( 2nd ` ( F ` n ) ) e. _V
60		op1stg 7180	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) e. _V /\ ( 2nd ` ( F ` n ) ) e. _V ) -> ( 1st ` <. ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) , ( 2nd ` ( F ` n ) ) >. ) = ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) )
61	58, 59, 60	mp2an 708	. . . . . . . . . . . . . . . . 17 |- ( 1st ` <. ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) , ( 2nd ` ( F ` n ) ) >. ) = ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) )
62	61	a1i 11	. . . . . . . . . . . . . . . 16 |- ( n e. NN -> ( 1st ` <. ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) , ( 2nd ` ( F ` n ) ) >. ) = ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) )
63	57, 62	eqtrd 2656	. . . . . . . . . . . . . . 15 |- ( n e. NN -> ( 1st ` ( G ` n ) ) = ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) )
64	63	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( G ` n ) ) = ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) )
65	64	breq1d 4663	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN ) -> ( ( 1st ` ( G ` n ) ) < ( 1st ` ( F ` n ) ) <-> ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) < ( 1st ` ( F ` n ) ) ) )
66	51, 65	mpbird 247	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( 1st ` ( G ` n ) ) < ( 1st ` ( F ` n ) ) )
67	56	fveq2d 6195	. . . . . . . . . . . . . . . 16 |- ( n e. NN -> ( 2nd ` ( G ` n ) ) = ( 2nd ` <. ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) , ( 2nd ` ( F ` n ) ) >. ) )
68	58, 59	op2nd 7177	. . . . . . . . . . . . . . . . 17 |- ( 2nd ` <. ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) , ( 2nd ` ( F ` n ) ) >. ) = ( 2nd ` ( F ` n ) )
69	68	a1i 11	. . . . . . . . . . . . . . . 16 |- ( n e. NN -> ( 2nd ` <. ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) , ( 2nd ` ( F ` n ) ) >. ) = ( 2nd ` ( F ` n ) ) )
70	67, 69	eqtrd 2656	. . . . . . . . . . . . . . 15 |- ( n e. NN -> ( 2nd ` ( G ` n ) ) = ( 2nd ` ( F ` n ) ) )
71	70	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ph /\ n e. NN ) -> ( 2nd ` ( G ` n ) ) = ( 2nd ` ( F ` n ) ) )
72	71	eqcomd 2628	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN ) -> ( 2nd ` ( F ` n ) ) = ( 2nd ` ( G ` n ) ) )
73	27, 72	eqled 10140	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( 2nd ` ( F ` n ) ) <_ ( 2nd ` ( G ` n ) ) )
74		icossioo 12264	. . . . . . . . . . . 12 |- ( ( ( ( 1st ` ( G ` n ) ) e. RR* /\ ( 2nd ` ( G ` n ) ) e. RR* ) /\ ( ( 1st ` ( G ` n ) ) < ( 1st ` ( F ` n ) ) /\ ( 2nd ` ( F ` n ) ) <_ ( 2nd ` ( G ` n ) ) ) ) -> ( ( 1st ` ( F ` n ) ) [,) ( 2nd ` ( F ` n ) ) ) C_ ( ( 1st ` ( G ` n ) ) (,) ( 2nd ` ( G ` n ) ) ) )
75	43, 46, 66, 73, 74	syl22anc 1327	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) [,) ( 2nd ` ( F ` n ) ) ) C_ ( ( 1st ` ( G ` n ) ) (,) ( 2nd ` ( G ` n ) ) ) )
76		1st2nd2 7205	. . . . . . . . . . . . . . 15 |- ( ( F ` n ) e. ( RR X. RR ) -> ( F ` n ) = <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. )
77	10, 76	syl 17	. . . . . . . . . . . . . 14 |- ( ( ph /\ n e. NN ) -> ( F ` n ) = <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. )
78	77	fveq2d 6195	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN ) -> ( [,) ` ( F ` n ) ) = ( [,) ` <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. ) )
79		df-ov 6653	. . . . . . . . . . . . . 14 |- ( ( 1st ` ( F ` n ) ) [,) ( 2nd ` ( F ` n ) ) ) = ( [,) ` <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. )
80	79	a1i 11	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) [,) ( 2nd ` ( F ` n ) ) ) = ( [,) ` <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. ) )
81	78, 80	eqtr4d 2659	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( [,) ` ( F ` n ) ) = ( ( 1st ` ( F ` n ) ) [,) ( 2nd ` ( F ` n ) ) ) )
82		1st2nd2 7205	. . . . . . . . . . . . . . 15 |- ( ( G ` n ) e. ( RR X. RR ) -> ( G ` n ) = <. ( 1st ` ( G ` n ) ) , ( 2nd ` ( G ` n ) ) >. )
83	40, 82	syl 17	. . . . . . . . . . . . . 14 |- ( ( ph /\ n e. NN ) -> ( G ` n ) = <. ( 1st ` ( G ` n ) ) , ( 2nd ` ( G ` n ) ) >. )
84	83	fveq2d 6195	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN ) -> ( (,) ` ( G ` n ) ) = ( (,) ` <. ( 1st ` ( G ` n ) ) , ( 2nd ` ( G ` n ) ) >. ) )
85		df-ov 6653	. . . . . . . . . . . . . 14 |- ( ( 1st ` ( G ` n ) ) (,) ( 2nd ` ( G ` n ) ) ) = ( (,) ` <. ( 1st ` ( G ` n ) ) , ( 2nd ` ( G ` n ) ) >. )
86	85	a1i 11	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN ) -> ( ( 1st ` ( G ` n ) ) (,) ( 2nd ` ( G ` n ) ) ) = ( (,) ` <. ( 1st ` ( G ` n ) ) , ( 2nd ` ( G ` n ) ) >. ) )
87	84, 86	eqtr4d 2659	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> ( (,) ` ( G ` n ) ) = ( ( 1st ` ( G ` n ) ) (,) ( 2nd ` ( G ` n ) ) ) )
88	81, 87	sseq12d 3634	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> ( ( [,) ` ( F ` n ) ) C_ ( (,) ` ( G ` n ) ) <-> ( ( 1st ` ( F ` n ) ) [,) ( 2nd ` ( F ` n ) ) ) C_ ( ( 1st ` ( G ` n ) ) (,) ( 2nd ` ( G ` n ) ) ) ) )
89	75, 88	mpbird 247	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( [,) ` ( F ` n ) ) C_ ( (,) ` ( G ` n ) ) )
90	89	ralrimiva 2966	. . . . . . . . 9 |- ( ph -> A. n e. NN ( [,) ` ( F ` n ) ) C_ ( (,) ` ( G ` n ) ) )
91		ss2iun 4536	. . . . . . . . 9 |- ( A. n e. NN ( [,) ` ( F ` n ) ) C_ ( (,) ` ( G ` n ) ) -> U_ n e. NN ( [,) ` ( F ` n ) ) C_ U_ n e. NN ( (,) ` ( G ` n ) ) )
92	90, 91	syl 17	. . . . . . . 8 |- ( ph -> U_ n e. NN ( [,) ` ( F ` n ) ) C_ U_ n e. NN ( (,) ` ( G ` n ) ) )
93		fvex 6201	. . . . . . . . . . . . 13 |- ( [,) ` ( F ` n ) ) e. _V
94	93	rgenw 2924	. . . . . . . . . . . 12 |- A. n e. NN ( [,) ` ( F ` n ) ) e. _V
95	94	a1i 11	. . . . . . . . . . 11 |- ( ph -> A. n e. NN ( [,) ` ( F ` n ) ) e. _V )
96		dfiun3g 5378	. . . . . . . . . . 11 |- ( A. n e. NN ( [,) ` ( F ` n ) ) e. _V -> U_ n e. NN ( [,) ` ( F ` n ) ) = U. ran ( n e. NN |-> ( [,) ` ( F ` n ) ) ) )
97	95, 96	syl 17	. . . . . . . . . 10 |- ( ph -> U_ n e. NN ( [,) ` ( F ` n ) ) = U. ran ( n e. NN |-> ( [,) ` ( F ` n ) ) ) )
98		icof 39411	. . . . . . . . . . . . . . 15 |- [,) : ( RR* X. RR* ) --> ~P RR*
99	98	a1i 11	. . . . . . . . . . . . . 14 |- ( ph -> [,) : ( RR* X. RR* ) --> ~P RR* )
100	9, 8, 99	fcomptss 39395	. . . . . . . . . . . . 13 |- ( ph -> ( [,) o. F ) = ( n e. NN |-> ( [,) ` ( F ` n ) ) ) )
101	100	eqcomd 2628	. . . . . . . . . . . 12 |- ( ph -> ( n e. NN |-> ( [,) ` ( F ` n ) ) ) = ( [,) o. F ) )
102	101	rneqd 5353	. . . . . . . . . . 11 |- ( ph -> ran ( n e. NN |-> ( [,) ` ( F ` n ) ) ) = ran ( [,) o. F ) )
103	102	unieqd 4446	. . . . . . . . . 10 |- ( ph -> U. ran ( n e. NN |-> ( [,) ` ( F ` n ) ) ) = U. ran ( [,) o. F ) )
104	97, 103	eqtr2d 2657	. . . . . . . . 9 |- ( ph -> U. ran ( [,) o. F ) = U_ n e. NN ( [,) ` ( F ` n ) ) )
105		fvex 6201	. . . . . . . . . . . . 13 |- ( (,) ` ( G ` n ) ) e. _V
106	105	rgenw 2924	. . . . . . . . . . . 12 |- A. n e. NN ( (,) ` ( G ` n ) ) e. _V
107	106	a1i 11	. . . . . . . . . . 11 |- ( ph -> A. n e. NN ( (,) ` ( G ` n ) ) e. _V )
108		dfiun3g 5378	. . . . . . . . . . 11 |- ( A. n e. NN ( (,) ` ( G ` n ) ) e. _V -> U_ n e. NN ( (,) ` ( G ` n ) ) = U. ran ( n e. NN |-> ( (,) ` ( G ` n ) ) ) )
109	107, 108	syl 17	. . . . . . . . . 10 |- ( ph -> U_ n e. NN ( (,) ` ( G ` n ) ) = U. ran ( n e. NN |-> ( (,) ` ( G ` n ) ) ) )
110		ioof 12271	. . . . . . . . . . . . . . 15 |- (,) : ( RR* X. RR* ) --> ~P RR
111	110	a1i 11	. . . . . . . . . . . . . 14 |- ( ph -> (,) : ( RR* X. RR* ) --> ~P RR )
112	30, 8, 111	fcomptss 39395	. . . . . . . . . . . . 13 |- ( ph -> ( (,) o. G ) = ( n e. NN |-> ( (,) ` ( G ` n ) ) ) )
113	112	eqcomd 2628	. . . . . . . . . . . 12 |- ( ph -> ( n e. NN |-> ( (,) ` ( G ` n ) ) ) = ( (,) o. G ) )
114	113	rneqd 5353	. . . . . . . . . . 11 |- ( ph -> ran ( n e. NN |-> ( (,) ` ( G ` n ) ) ) = ran ( (,) o. G ) )
115	114	unieqd 4446	. . . . . . . . . 10 |- ( ph -> U. ran ( n e. NN |-> ( (,) ` ( G ` n ) ) ) = U. ran ( (,) o. G ) )
116	109, 115	eqtr2d 2657	. . . . . . . . 9 |- ( ph -> U. ran ( (,) o. G ) = U_ n e. NN ( (,) ` ( G ` n ) ) )
117	104, 116	sseq12d 3634	. . . . . . . 8 |- ( ph -> ( U. ran ( [,) o. F ) C_ U. ran ( (,) o. G ) <-> U_ n e. NN ( [,) ` ( F ` n ) ) C_ U_ n e. NN ( (,) ` ( G ` n ) ) ) )
118	92, 117	mpbird 247	. . . . . . 7 |- ( ph -> U. ran ( [,) o. F ) C_ U. ran ( (,) o. G ) )
119	39, 118	sstrd 3613	. . . . . 6 |- ( ph -> A C_ U. ran ( (,) o. G ) )
120	119, 2	jca 554	. . . . 5 |- ( ph -> ( A C_ U. ran ( (,) o. G ) /\ Z = (Σ^ ` ( ( vol o. (,) ) o. G ) ) ) )
121		coeq2 5280	. . . . . . . . . 10 |- ( f = G -> ( (,) o. f ) = ( (,) o. G ) )
122	121	rneqd 5353	. . . . . . . . 9 |- ( f = G -> ran ( (,) o. f ) = ran ( (,) o. G ) )
123	122	unieqd 4446	. . . . . . . 8 |- ( f = G -> U. ran ( (,) o. f ) = U. ran ( (,) o. G ) )
124	123	sseq2d 3633	. . . . . . 7 |- ( f = G -> ( A C_ U. ran ( (,) o. f ) <-> A C_ U. ran ( (,) o. G ) ) )
125		coeq2 5280	. . . . . . . . 9 |- ( f = G -> ( ( vol o. (,) ) o. f ) = ( ( vol o. (,) ) o. G ) )
126	125	fveq2d 6195	. . . . . . . 8 |- ( f = G -> (Σ^ ` ( ( vol o. (,) ) o. f ) ) = (Σ^ ` ( ( vol o. (,) ) o. G ) ) )
127	126	eqeq2d 2632	. . . . . . 7 |- ( f = G -> ( Z = (Σ^ ` ( ( vol o. (,) ) o. f ) ) <-> Z = (Σ^ ` ( ( vol o. (,) ) o. G ) ) ) )
128	124, 127	anbi12d 747	. . . . . 6 |- ( f = G -> ( ( A C_ U. ran ( (,) o. f ) /\ Z = (Σ^ ` ( ( vol o. (,) ) o. f ) ) ) <-> ( A C_ U. ran ( (,) o. G ) /\ Z = (Σ^ ` ( ( vol o. (,) ) o. G ) ) ) ) )
129	128	rspcev 3309	. . . . 5 |- ( ( G e. ( ( RR X. RR ) ^m NN ) /\ ( A C_ U. ran ( (,) o. G ) /\ Z = (Σ^ ` ( ( vol o. (,) ) o. G ) ) ) ) -> E. f e. ( ( RR X. RR ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ Z = (Σ^ ` ( ( vol o. (,) ) o. f ) ) ) )
130	38, 120, 129	syl2anc 693	. . . 4 |- ( ph -> E. f e. ( ( RR X. RR ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ Z = (Σ^ ` ( ( vol o. (,) ) o. f ) ) ) )
131	33, 130	jca 554	. . 3 |- ( ph -> ( Z e. RR* /\ E. f e. ( ( RR X. RR ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ Z = (Σ^ ` ( ( vol o. (,) ) o. f ) ) ) ) )
132		eqeq1 2626	. . . . . 6 |- ( z = Z -> ( z = (Σ^ ` ( ( vol o. (,) ) o. f ) ) <-> Z = (Σ^ ` ( ( vol o. (,) ) o. f ) ) ) )
133	132	anbi2d 740	. . . . 5 |- ( z = Z -> ( ( A C_ U. ran ( (,) o. f ) /\ z = (Σ^ ` ( ( vol o. (,) ) o. f ) ) ) <-> ( A C_ U. ran ( (,) o. f ) /\ Z = (Σ^ ` ( ( vol o. (,) ) o. f ) ) ) ) )
134	133	rexbidv 3052	. . . 4 |- ( z = Z -> ( E. f e. ( ( RR X. RR ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ z = (Σ^ ` ( ( vol o. (,) ) o. f ) ) ) <-> E. f e. ( ( RR X. RR ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ Z = (Σ^ ` ( ( vol o. (,) ) o. f ) ) ) ) )
135		ovolval5lem2.q	. . . 4 |- Q = { z e. RR* | E. f e. ( ( RR X. RR ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ z = (Σ^ ` ( ( vol o. (,) ) o. f ) ) ) }
136	134, 135	elrab2 3366	. . 3 |- ( Z e. Q <-> ( Z e. RR* /\ E. f e. ( ( RR X. RR ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ Z = (Σ^ ` ( ( vol o. (,) ) o. f ) ) ) ) )
137	131, 136	sylibr 224	. 2 |- ( ph -> Z e. Q )
138		fveq2 6191	. . . . . . 7 |- ( m = n -> ( F ` m ) = ( F ` n ) )
139	138	fveq2d 6195	. . . . . 6 |- ( m = n -> ( 1st ` ( F ` m ) ) = ( 1st ` ( F ` n ) ) )
140	138	fveq2d 6195	. . . . . 6 |- ( m = n -> ( 2nd ` ( F ` m ) ) = ( 2nd ` ( F ` n ) ) )
141	139, 140	breq12d 4666	. . . . 5 |- ( m = n -> ( ( 1st ` ( F ` m ) ) < ( 2nd ` ( F ` m ) ) <-> ( 1st ` ( F ` n ) ) < ( 2nd ` ( F ` n ) ) ) )
142	141	cbvrabv 3199	. . . 4 |- { m e. NN | ( 1st ` ( F ` m ) ) < ( 2nd ` ( F ` m ) ) } = { n e. NN | ( 1st ` ( F ` n ) ) < ( 2nd ` ( F ` n ) ) }
143	12, 27, 13, 142	ovolval5lem1 40866	. . 3 |- ( ph -> (Σ^ ` ( n e. NN |-> ( vol ` ( ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) (,) ( 2nd ` ( F ` n ) ) ) ) ) ) <_ ( (Σ^ ` ( n e. NN |-> ( vol ` ( ( 1st ` ( F ` n ) ) [,) ( 2nd ` ( F ` n ) ) ) ) ) ) +e W ) )
144		nfcv 2764	. . . . . . . 8 |- F/_ n G
145	30, 8	fssd 6057	. . . . . . . 8 |- ( ph -> G : NN --> ( RR* X. RR* ) )
146	144, 145	volioofmpt 40211	. . . . . . 7 |- ( ph -> ( ( vol o. (,) ) o. G ) = ( n e. NN |-> ( vol ` ( ( 1st ` ( G ` n ) ) (,) ( 2nd ` ( G ` n ) ) ) ) ) )
147	64, 71	oveq12d 6668	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( ( 1st ` ( G ` n ) ) (,) ( 2nd ` ( G ` n ) ) ) = ( ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) (,) ( 2nd ` ( F ` n ) ) ) )
148	147	fveq2d 6195	. . . . . . . 8 |- ( ( ph /\ n e. NN ) -> ( vol ` ( ( 1st ` ( G ` n ) ) (,) ( 2nd ` ( G ` n ) ) ) ) = ( vol ` ( ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) (,) ( 2nd ` ( F ` n ) ) ) ) )
149	148	mpteq2dva 4744	. . . . . . 7 |- ( ph -> ( n e. NN |-> ( vol ` ( ( 1st ` ( G ` n ) ) (,) ( 2nd ` ( G ` n ) ) ) ) ) = ( n e. NN |-> ( vol ` ( ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) (,) ( 2nd ` ( F ` n ) ) ) ) ) )
150	146, 149	eqtrd 2656	. . . . . 6 |- ( ph -> ( ( vol o. (,) ) o. G ) = ( n e. NN |-> ( vol ` ( ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) (,) ( 2nd ` ( F ` n ) ) ) ) ) )
151	150	fveq2d 6195	. . . . 5 |- ( ph -> (Σ^ ` ( ( vol o. (,) ) o. G ) ) = (Σ^ ` ( n e. NN |-> ( vol ` ( ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) (,) ( 2nd ` ( F ` n ) ) ) ) ) ) )
152	2, 151	eqtrd 2656	. . . 4 |- ( ph -> Z = (Σ^ ` ( n e. NN |-> ( vol ` ( ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) (,) ( 2nd ` ( F ` n ) ) ) ) ) ) )
153		ovolval5lem2.y	. . . . . 6 |- ( ph -> Y = (Σ^ ` ( ( vol o. [,) ) o. F ) ) )
154		nfcv 2764	. . . . . . . 8 |- F/_ n F
155		ressxr 10083	. . . . . . . . . . 11 |- RR C_ RR*
156		xpss2 5229	. . . . . . . . . . 11 |- ( RR C_ RR* -> ( RR X. RR ) C_ ( RR X. RR* ) )
157	155, 156	ax-mp 5	. . . . . . . . . 10 |- ( RR X. RR ) C_ ( RR X. RR* )
158	157	a1i 11	. . . . . . . . 9 |- ( ph -> ( RR X. RR ) C_ ( RR X. RR* ) )
159	9, 158	fssd 6057	. . . . . . . 8 |- ( ph -> F : NN --> ( RR X. RR* ) )
160	154, 159	volicofmpt 40214	. . . . . . 7 |- ( ph -> ( ( vol o. [,) ) o. F ) = ( n e. NN |-> ( vol ` ( ( 1st ` ( F ` n ) ) [,) ( 2nd ` ( F ` n ) ) ) ) ) )
161	160	fveq2d 6195	. . . . . 6 |- ( ph -> (Σ^ ` ( ( vol o. [,) ) o. F ) ) = (Σ^ ` ( n e. NN |-> ( vol ` ( ( 1st ` ( F ` n ) ) [,) ( 2nd ` ( F ` n ) ) ) ) ) ) )
162	153, 161	eqtrd 2656	. . . . 5 |- ( ph -> Y = (Σ^ ` ( n e. NN |-> ( vol ` ( ( 1st ` ( F ` n ) ) [,) ( 2nd ` ( F ` n ) ) ) ) ) ) )
163	162	oveq1d 6665	. . . 4 |- ( ph -> ( Y +e W ) = ( (Σ^ ` ( n e. NN |-> ( vol ` ( ( 1st ` ( F ` n ) ) [,) ( 2nd ` ( F ` n ) ) ) ) ) ) +e W ) )
164	152, 163	breq12d 4666	. . 3 |- ( ph -> ( Z <_ ( Y +e W ) <-> (Σ^ ` ( n e. NN |-> ( vol ` ( ( ( 1st ` ( F ` n ) ) - ( W / ( 2 ^ n ) ) ) (,) ( 2nd ` ( F ` n ) ) ) ) ) ) <_ ( (Σ^ ` ( n e. NN |-> ( vol ` ( ( 1st ` ( F ` n ) ) [,) ( 2nd ` ( F ` n ) ) ) ) ) ) +e W ) ) )
165	143, 164	mpbird 247	. 2 |- ( ph -> Z <_ ( Y +e W ) )
166		breq1 4656	. . 3 |- ( z = Z -> ( z <_ ( Y +e W ) <-> Z <_ ( Y +e W ) ) )
167	166	rspcev 3309	. 2 |- ( ( Z e. Q /\ Z <_ ( Y +e W ) ) -> E. z e. Q z <_ ( Y +e W ) )
168	137, 165, 167	syl2anc 693	1 |- ( ph -> E. z e. Q z <_ ( Y +e W ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   <.cop 4183   U.cuni 4436   U_ciun 4520   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   ran crn 5115   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   ^m cmap 7857   RRcr 9935   0cc0 9936   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   RR+crp 11832   +ecxad 11944   (,)cioo 12175   [,)cico 12177   [,]cicc 12178   ^cexp 12860   volcvol 23232  Σ^{^}csumge0 40579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cmp 21190  df-ovol 23233  df-vol 23234  df-sumge0 40580

This theorem is referenced by:  ovolval5lem3  40868