Theorem ovolval2lem 40857

Description: The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^{^}. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypothesis

Ref	Expression
ovolval2lem.1	|- ( ph -> F : NN --> ( <_ i^i ( RR X. RR ) ) )

Assertion

Ref	Expression
ovolval2lem	|- ( ph -> ran seq 1 ( + , ( ( abs o. - ) o. F ) ) = ran ( n e. NN |-> sum_ k e. ( 1 ... n ) ( vol ` ( ( [,) o. F ) ` k ) ) ) )

Distinct variable groups:   k, F, n   ph, k

Allowed substitution hint:   ph( n)

Proof of Theorem ovolval2lem

Step	Hyp	Ref	Expression
1		reex 10027	. . . . . . 7 |- RR e. _V
2	1, 1	xpex 6962	. . . . . 6 |- ( RR X. RR ) e. _V
3		inss2 3834	. . . . . 6 |- ( <_ i^i ( RR X. RR ) ) C_ ( RR X. RR )
4		mapss 7900	. . . . . 6 |- ( ( ( RR X. RR ) e. _V /\ ( <_ i^i ( RR X. RR ) ) C_ ( RR X. RR ) ) -> ( ( <_ i^i ( RR X. RR ) ) ^m NN ) C_ ( ( RR X. RR ) ^m NN ) )
5	2, 3, 4	mp2an 708	. . . . 5 |- ( ( <_ i^i ( RR X. RR ) ) ^m NN ) C_ ( ( RR X. RR ) ^m NN )
6		ovolval2lem.1	. . . . . 6 |- ( ph -> F : NN --> ( <_ i^i ( RR X. RR ) ) )
7	2	inex2 4800	. . . . . . . 8 |- ( <_ i^i ( RR X. RR ) ) e. _V
8	7	a1i 11	. . . . . . 7 |- ( ph -> ( <_ i^i ( RR X. RR ) ) e. _V )
9		nnex 11026	. . . . . . . 8 |- NN e. _V
10	9	a1i 11	. . . . . . 7 |- ( ph -> NN e. _V )
11	8, 10	elmapd 7871	. . . . . 6 |- ( ph -> ( F e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) <-> F : NN --> ( <_ i^i ( RR X. RR ) ) ) )
12	6, 11	mpbird 247	. . . . 5 |- ( ph -> F e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) )
13	5, 12	sseldi 3601	. . . 4 |- ( ph -> F e. ( ( RR X. RR ) ^m NN ) )
14		1zzd 11408	. . . . 5 |- ( F e. ( ( RR X. RR ) ^m NN ) -> 1 e. ZZ )
15		nnuz 11723	. . . . 5 |- NN = ( ZZ>= ` 1 )
16		elmapi 7879	. . . . . . . . . 10 |- ( F e. ( ( RR X. RR ) ^m NN ) -> F : NN --> ( RR X. RR ) )
17	16	adantr 481	. . . . . . . . 9 |- ( ( F e. ( ( RR X. RR ) ^m NN ) /\ k e. NN ) -> F : NN --> ( RR X. RR ) )
18		simpr 477	. . . . . . . . 9 |- ( ( F e. ( ( RR X. RR ) ^m NN ) /\ k e. NN ) -> k e. NN )
19	17, 18	fvovco 39381	. . . . . . . 8 |- ( ( F e. ( ( RR X. RR ) ^m NN ) /\ k e. NN ) -> ( ( [,) o. F ) ` k ) = ( ( 1st ` ( F ` k ) ) [,) ( 2nd ` ( F ` k ) ) ) )
20	19	fveq2d 6195	. . . . . . 7 |- ( ( F e. ( ( RR X. RR ) ^m NN ) /\ k e. NN ) -> ( vol ` ( ( [,) o. F ) ` k ) ) = ( vol ` ( ( 1st ` ( F ` k ) ) [,) ( 2nd ` ( F ` k ) ) ) ) )
21	16	ffvelrnda 6359	. . . . . . . . 9 |- ( ( F e. ( ( RR X. RR ) ^m NN ) /\ k e. NN ) -> ( F ` k ) e. ( RR X. RR ) )
22		xp1st 7198	. . . . . . . . 9 |- ( ( F ` k ) e. ( RR X. RR ) -> ( 1st ` ( F ` k ) ) e. RR )
23	21, 22	syl 17	. . . . . . . 8 |- ( ( F e. ( ( RR X. RR ) ^m NN ) /\ k e. NN ) -> ( 1st ` ( F ` k ) ) e. RR )
24		xp2nd 7199	. . . . . . . . 9 |- ( ( F ` k ) e. ( RR X. RR ) -> ( 2nd ` ( F ` k ) ) e. RR )
25	21, 24	syl 17	. . . . . . . 8 |- ( ( F e. ( ( RR X. RR ) ^m NN ) /\ k e. NN ) -> ( 2nd ` ( F ` k ) ) e. RR )
26		volicore 40795	. . . . . . . 8 |- ( ( ( 1st ` ( F ` k ) ) e. RR /\ ( 2nd ` ( F ` k ) ) e. RR ) -> ( vol ` ( ( 1st ` ( F ` k ) ) [,) ( 2nd ` ( F ` k ) ) ) ) e. RR )
27	23, 25, 26	syl2anc 693	. . . . . . 7 |- ( ( F e. ( ( RR X. RR ) ^m NN ) /\ k e. NN ) -> ( vol ` ( ( 1st ` ( F ` k ) ) [,) ( 2nd ` ( F ` k ) ) ) ) e. RR )
28	20, 27	eqeltrd 2701	. . . . . 6 |- ( ( F e. ( ( RR X. RR ) ^m NN ) /\ k e. NN ) -> ( vol ` ( ( [,) o. F ) ` k ) ) e. RR )
29	28	recnd 10068	. . . . 5 |- ( ( F e. ( ( RR X. RR ) ^m NN ) /\ k e. NN ) -> ( vol ` ( ( [,) o. F ) ` k ) ) e. CC )
30		eqid 2622	. . . . 5 |- ( n e. NN |-> sum_ k e. ( 1 ... n ) ( vol ` ( ( [,) o. F ) ` k ) ) ) = ( n e. NN |-> sum_ k e. ( 1 ... n ) ( vol ` ( ( [,) o. F ) ` k ) ) )
31		eqid 2622	. . . . 5 |- seq 1 ( + , ( k e. NN |-> ( vol ` ( ( [,) o. F ) ` k ) ) ) ) = seq 1 ( + , ( k e. NN |-> ( vol ` ( ( [,) o. F ) ` k ) ) ) )
32	14, 15, 29, 30, 31	fsumsermpt 39811	. . . 4 |- ( F e. ( ( RR X. RR ) ^m NN ) -> ( n e. NN |-> sum_ k e. ( 1 ... n ) ( vol ` ( ( [,) o. F ) ` k ) ) ) = seq 1 ( + , ( k e. NN |-> ( vol ` ( ( [,) o. F ) ` k ) ) ) ) )
33	13, 32	syl 17	. . 3 |- ( ph -> ( n e. NN |-> sum_ k e. ( 1 ... n ) ( vol ` ( ( [,) o. F ) ` k ) ) ) = seq 1 ( + , ( k e. NN |-> ( vol ` ( ( [,) o. F ) ` k ) ) ) ) )
34		simpr 477	. . . . . . . . . 10 |- ( ( ( ph /\ k e. NN ) /\ ( 1st ` ( F ` k ) ) < ( 2nd ` ( F ` k ) ) ) -> ( 1st ` ( F ` k ) ) < ( 2nd ` ( F ` k ) ) )
35	34	iftrued 4094	. . . . . . . . 9 |- ( ( ( ph /\ k e. NN ) /\ ( 1st ` ( F ` k ) ) < ( 2nd ` ( F ` k ) ) ) -> if ( ( 1st ` ( F ` k ) ) < ( 2nd ` ( F ` k ) ) , ( ( 2nd ` ( F ` k ) ) - ( 1st ` ( F ` k ) ) ) , 0 ) = ( ( 2nd ` ( F ` k ) ) - ( 1st ` ( F ` k ) ) ) )
36	13, 23	sylan 488	. . . . . . . . . . 11 |- ( ( ph /\ k e. NN ) -> ( 1st ` ( F ` k ) ) e. RR )
37	36	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ k e. NN ) /\ -. ( 1st ` ( F ` k ) ) < ( 2nd ` ( F ` k ) ) ) -> ( 1st ` ( F ` k ) ) e. RR )
38	13, 25	sylan 488	. . . . . . . . . . 11 |- ( ( ph /\ k e. NN ) -> ( 2nd ` ( F ` k ) ) e. RR )
39	38	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ k e. NN ) /\ -. ( 1st ` ( F ` k ) ) < ( 2nd ` ( F ` k ) ) ) -> ( 2nd ` ( F ` k ) ) e. RR )
40		ressxr 10083	. . . . . . . . . . . 12 |- RR C_ RR*
41	40, 37	sseldi 3601	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. NN ) /\ -. ( 1st ` ( F ` k ) ) < ( 2nd ` ( F ` k ) ) ) -> ( 1st ` ( F ` k ) ) e. RR* )
42	40, 39	sseldi 3601	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. NN ) /\ -. ( 1st ` ( F ` k ) ) < ( 2nd ` ( F ` k ) ) ) -> ( 2nd ` ( F ` k ) ) e. RR* )
43		xpss 5226	. . . . . . . . . . . . . . . . . 18 |- ( RR X. RR ) C_ ( _V X. _V )
44	43, 21	sseldi 3601	. . . . . . . . . . . . . . . . 17 |- ( ( F e. ( ( RR X. RR ) ^m NN ) /\ k e. NN ) -> ( F ` k ) e. ( _V X. _V ) )
45		1st2ndb 7206	. . . . . . . . . . . . . . . . 17 |- ( ( F ` k ) e. ( _V X. _V ) <-> ( F ` k ) = <. ( 1st ` ( F ` k ) ) , ( 2nd ` ( F ` k ) ) >. )
46	44, 45	sylib 208	. . . . . . . . . . . . . . . 16 |- ( ( F e. ( ( RR X. RR ) ^m NN ) /\ k e. NN ) -> ( F ` k ) = <. ( 1st ` ( F ` k ) ) , ( 2nd ` ( F ` k ) ) >. )
47	13, 46	sylan 488	. . . . . . . . . . . . . . 15 |- ( ( ph /\ k e. NN ) -> ( F ` k ) = <. ( 1st ` ( F ` k ) ) , ( 2nd ` ( F ` k ) ) >. )
48	47	eqcomd 2628	. . . . . . . . . . . . . 14 |- ( ( ph /\ k e. NN ) -> <. ( 1st ` ( F ` k ) ) , ( 2nd ` ( F ` k ) ) >. = ( F ` k ) )
49		inss1 3833	. . . . . . . . . . . . . . . . 17 |- ( <_ i^i ( RR X. RR ) ) C_ <_
50	49	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ph -> ( <_ i^i ( RR X. RR ) ) C_ <_ )
51	6, 50	fssd 6057	. . . . . . . . . . . . . . 15 |- ( ph -> F : NN --> <_ )
52	51	ffvelrnda 6359	. . . . . . . . . . . . . 14 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. <_ )
53	48, 52	eqeltrd 2701	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. NN ) -> <. ( 1st ` ( F ` k ) ) , ( 2nd ` ( F ` k ) ) >. e. <_ )
54		df-br 4654	. . . . . . . . . . . . 13 |- ( ( 1st ` ( F ` k ) ) <_ ( 2nd ` ( F ` k ) ) <-> <. ( 1st ` ( F ` k ) ) , ( 2nd ` ( F ` k ) ) >. e. <_ )
55	53, 54	sylibr 224	. . . . . . . . . . . 12 |- ( ( ph /\ k e. NN ) -> ( 1st ` ( F ` k ) ) <_ ( 2nd ` ( F ` k ) ) )
56	55	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. NN ) /\ -. ( 1st ` ( F ` k ) ) < ( 2nd ` ( F ` k ) ) ) -> ( 1st ` ( F ` k ) ) <_ ( 2nd ` ( F ` k ) ) )
57		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. NN ) /\ -. ( 1st ` ( F ` k ) ) < ( 2nd ` ( F ` k ) ) ) -> -. ( 1st ` ( F ` k ) ) < ( 2nd ` ( F ` k ) ) )
58	39, 37	lenltd 10183	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. NN ) /\ -. ( 1st ` ( F ` k ) ) < ( 2nd ` ( F ` k ) ) ) -> ( ( 2nd ` ( F ` k ) ) <_ ( 1st ` ( F ` k ) ) <-> -. ( 1st ` ( F ` k ) ) < ( 2nd ` ( F ` k ) ) ) )
59	57, 58	mpbird 247	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. NN ) /\ -. ( 1st ` ( F ` k ) ) < ( 2nd ` ( F ` k ) ) ) -> ( 2nd ` ( F ` k ) ) <_ ( 1st ` ( F ` k ) ) )
60	41, 42, 56, 59	xrletrid 11986	. . . . . . . . . 10 |- ( ( ( ph /\ k e. NN ) /\ -. ( 1st ` ( F ` k ) ) < ( 2nd ` ( F ` k ) ) ) -> ( 1st ` ( F ` k ) ) = ( 2nd ` ( F ` k ) ) )
61		simp3 1063	. . . . . . . . . . . . . 14 |- ( ( ( 1st ` ( F ` k ) ) e. RR /\ ( 2nd ` ( F ` k ) ) e. RR /\ ( 1st ` ( F ` k ) ) = ( 2nd ` ( F ` k ) ) ) -> ( 1st ` ( F ` k ) ) = ( 2nd ` ( F ` k ) ) )
62		simp1 1061	. . . . . . . . . . . . . . 15 |- ( ( ( 1st ` ( F ` k ) ) e. RR /\ ( 2nd ` ( F ` k ) ) e. RR /\ ( 1st ` ( F ` k ) ) = ( 2nd ` ( F ` k ) ) ) -> ( 1st ` ( F ` k ) ) e. RR )
63		simp2 1062	. . . . . . . . . . . . . . 15 |- ( ( ( 1st ` ( F ` k ) ) e. RR /\ ( 2nd ` ( F ` k ) ) e. RR /\ ( 1st ` ( F ` k ) ) = ( 2nd ` ( F ` k ) ) ) -> ( 2nd ` ( F ` k ) ) e. RR )
64	62, 63	eqleltd 10181	. . . . . . . . . . . . . 14 |- ( ( ( 1st ` ( F ` k ) ) e. RR /\ ( 2nd ` ( F ` k ) ) e. RR /\ ( 1st ` ( F ` k ) ) = ( 2nd ` ( F ` k ) ) ) -> ( ( 1st ` ( F ` k ) ) = ( 2nd ` ( F ` k ) ) <-> ( ( 1st ` ( F ` k ) ) <_ ( 2nd ` ( F ` k ) ) /\ -. ( 1st ` ( F ` k ) ) < ( 2nd ` ( F ` k ) ) ) ) )
65	61, 64	mpbid 222	. . . . . . . . . . . . 13 |- ( ( ( 1st ` ( F ` k ) ) e. RR /\ ( 2nd ` ( F ` k ) ) e. RR /\ ( 1st ` ( F ` k ) ) = ( 2nd ` ( F ` k ) ) ) -> ( ( 1st ` ( F ` k ) ) <_ ( 2nd ` ( F ` k ) ) /\ -. ( 1st ` ( F ` k ) ) < ( 2nd ` ( F ` k ) ) ) )
66	65	simprd 479	. . . . . . . . . . . 12 |- ( ( ( 1st ` ( F ` k ) ) e. RR /\ ( 2nd ` ( F ` k ) ) e. RR /\ ( 1st ` ( F ` k ) ) = ( 2nd ` ( F ` k ) ) ) -> -. ( 1st ` ( F ` k ) ) < ( 2nd ` ( F ` k ) ) )
67	66	iffalsed 4097	. . . . . . . . . . 11 |- ( ( ( 1st ` ( F ` k ) ) e. RR /\ ( 2nd ` ( F ` k ) ) e. RR /\ ( 1st ` ( F ` k ) ) = ( 2nd ` ( F ` k ) ) ) -> if ( ( 1st ` ( F ` k ) ) < ( 2nd ` ( F ` k ) ) , ( ( 2nd ` ( F ` k ) ) - ( 1st ` ( F ` k ) ) ) , 0 ) = 0 )
68	63	recnd 10068	. . . . . . . . . . . 12 |- ( ( ( 1st ` ( F ` k ) ) e. RR /\ ( 2nd ` ( F ` k ) ) e. RR /\ ( 1st ` ( F ` k ) ) = ( 2nd ` ( F ` k ) ) ) -> ( 2nd ` ( F ` k ) ) e. CC )
69	61	eqcomd 2628	. . . . . . . . . . . 12 |- ( ( ( 1st ` ( F ` k ) ) e. RR /\ ( 2nd ` ( F ` k ) ) e. RR /\ ( 1st ` ( F ` k ) ) = ( 2nd ` ( F ` k ) ) ) -> ( 2nd ` ( F ` k ) ) = ( 1st ` ( F ` k ) ) )
70	68, 69	subeq0bd 10456	. . . . . . . . . . 11 |- ( ( ( 1st ` ( F ` k ) ) e. RR /\ ( 2nd ` ( F ` k ) ) e. RR /\ ( 1st ` ( F ` k ) ) = ( 2nd ` ( F ` k ) ) ) -> ( ( 2nd ` ( F ` k ) ) - ( 1st ` ( F ` k ) ) ) = 0 )
71	67, 70	eqtr4d 2659	. . . . . . . . . 10 |- ( ( ( 1st ` ( F ` k ) ) e. RR /\ ( 2nd ` ( F ` k ) ) e. RR /\ ( 1st ` ( F ` k ) ) = ( 2nd ` ( F ` k ) ) ) -> if ( ( 1st ` ( F ` k ) ) < ( 2nd ` ( F ` k ) ) , ( ( 2nd ` ( F ` k ) ) - ( 1st ` ( F ` k ) ) ) , 0 ) = ( ( 2nd ` ( F ` k ) ) - ( 1st ` ( F ` k ) ) ) )
72	37, 39, 60, 71	syl3anc 1326	. . . . . . . . 9 |- ( ( ( ph /\ k e. NN ) /\ -. ( 1st ` ( F ` k ) ) < ( 2nd ` ( F ` k ) ) ) -> if ( ( 1st ` ( F ` k ) ) < ( 2nd ` ( F ` k ) ) , ( ( 2nd ` ( F ` k ) ) - ( 1st ` ( F ` k ) ) ) , 0 ) = ( ( 2nd ` ( F ` k ) ) - ( 1st ` ( F ` k ) ) ) )
73	35, 72	pm2.61dan 832	. . . . . . . 8 |- ( ( ph /\ k e. NN ) -> if ( ( 1st ` ( F ` k ) ) < ( 2nd ` ( F ` k ) ) , ( ( 2nd ` ( F ` k ) ) - ( 1st ` ( F ` k ) ) ) , 0 ) = ( ( 2nd ` ( F ` k ) ) - ( 1st ` ( F ` k ) ) ) )
74		volico 40200	. . . . . . . . 9 |- ( ( ( 1st ` ( F ` k ) ) e. RR /\ ( 2nd ` ( F ` k ) ) e. RR ) -> ( vol ` ( ( 1st ` ( F ` k ) ) [,) ( 2nd ` ( F ` k ) ) ) ) = if ( ( 1st ` ( F ` k ) ) < ( 2nd ` ( F ` k ) ) , ( ( 2nd ` ( F ` k ) ) - ( 1st ` ( F ` k ) ) ) , 0 ) )
75	36, 38, 74	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ k e. NN ) -> ( vol ` ( ( 1st ` ( F ` k ) ) [,) ( 2nd ` ( F ` k ) ) ) ) = if ( ( 1st ` ( F ` k ) ) < ( 2nd ` ( F ` k ) ) , ( ( 2nd ` ( F ` k ) ) - ( 1st ` ( F ` k ) ) ) , 0 ) )
76	36, 38, 55	abssuble0d 14171	. . . . . . . 8 |- ( ( ph /\ k e. NN ) -> ( abs ` ( ( 1st ` ( F ` k ) ) - ( 2nd ` ( F ` k ) ) ) ) = ( ( 2nd ` ( F ` k ) ) - ( 1st ` ( F ` k ) ) ) )
77	73, 75, 76	3eqtr4d 2666	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( vol ` ( ( 1st ` ( F ` k ) ) [,) ( 2nd ` ( F ` k ) ) ) ) = ( abs ` ( ( 1st ` ( F ` k ) ) - ( 2nd ` ( F ` k ) ) ) ) )
78	13	adantr 481	. . . . . . . 8 |- ( ( ph /\ k e. NN ) -> F e. ( ( RR X. RR ) ^m NN ) )
79		simpr 477	. . . . . . . 8 |- ( ( ph /\ k e. NN ) -> k e. NN )
80	78, 79, 20	syl2anc 693	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( vol ` ( ( [,) o. F ) ` k ) ) = ( vol ` ( ( 1st ` ( F ` k ) ) [,) ( 2nd ` ( F ` k ) ) ) ) )
81	46	fveq2d 6195	. . . . . . . . 9 |- ( ( F e. ( ( RR X. RR ) ^m NN ) /\ k e. NN ) -> ( ( abs o. - ) ` ( F ` k ) ) = ( ( abs o. - ) ` <. ( 1st ` ( F ` k ) ) , ( 2nd ` ( F ` k ) ) >. ) )
82		df-ov 6653	. . . . . . . . . . 11 |- ( ( 1st ` ( F ` k ) ) ( abs o. - ) ( 2nd ` ( F ` k ) ) ) = ( ( abs o. - ) ` <. ( 1st ` ( F ` k ) ) , ( 2nd ` ( F ` k ) ) >. )
83	82	eqcomi 2631	. . . . . . . . . 10 |- ( ( abs o. - ) ` <. ( 1st ` ( F ` k ) ) , ( 2nd ` ( F ` k ) ) >. ) = ( ( 1st ` ( F ` k ) ) ( abs o. - ) ( 2nd ` ( F ` k ) ) )
84	83	a1i 11	. . . . . . . . 9 |- ( ( F e. ( ( RR X. RR ) ^m NN ) /\ k e. NN ) -> ( ( abs o. - ) ` <. ( 1st ` ( F ` k ) ) , ( 2nd ` ( F ` k ) ) >. ) = ( ( 1st ` ( F ` k ) ) ( abs o. - ) ( 2nd ` ( F ` k ) ) ) )
85	23	recnd 10068	. . . . . . . . . 10 |- ( ( F e. ( ( RR X. RR ) ^m NN ) /\ k e. NN ) -> ( 1st ` ( F ` k ) ) e. CC )
86	25	recnd 10068	. . . . . . . . . 10 |- ( ( F e. ( ( RR X. RR ) ^m NN ) /\ k e. NN ) -> ( 2nd ` ( F ` k ) ) e. CC )
87		eqid 2622	. . . . . . . . . . 11 |- ( abs o. - ) = ( abs o. - )
88	87	cnmetdval 22574	. . . . . . . . . 10 |- ( ( ( 1st ` ( F ` k ) ) e. CC /\ ( 2nd ` ( F ` k ) ) e. CC ) -> ( ( 1st ` ( F ` k ) ) ( abs o. - ) ( 2nd ` ( F ` k ) ) ) = ( abs ` ( ( 1st ` ( F ` k ) ) - ( 2nd ` ( F ` k ) ) ) ) )
89	85, 86, 88	syl2anc 693	. . . . . . . . 9 |- ( ( F e. ( ( RR X. RR ) ^m NN ) /\ k e. NN ) -> ( ( 1st ` ( F ` k ) ) ( abs o. - ) ( 2nd ` ( F ` k ) ) ) = ( abs ` ( ( 1st ` ( F ` k ) ) - ( 2nd ` ( F ` k ) ) ) ) )
90	81, 84, 89	3eqtrd 2660	. . . . . . . 8 |- ( ( F e. ( ( RR X. RR ) ^m NN ) /\ k e. NN ) -> ( ( abs o. - ) ` ( F ` k ) ) = ( abs ` ( ( 1st ` ( F ` k ) ) - ( 2nd ` ( F ` k ) ) ) ) )
91	78, 79, 90	syl2anc 693	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( ( abs o. - ) ` ( F ` k ) ) = ( abs ` ( ( 1st ` ( F ` k ) ) - ( 2nd ` ( F ` k ) ) ) ) )
92	77, 80, 91	3eqtr4d 2666	. . . . . 6 |- ( ( ph /\ k e. NN ) -> ( vol ` ( ( [,) o. F ) ` k ) ) = ( ( abs o. - ) ` ( F ` k ) ) )
93	92	mpteq2dva 4744	. . . . 5 |- ( ph -> ( k e. NN |-> ( vol ` ( ( [,) o. F ) ` k ) ) ) = ( k e. NN |-> ( ( abs o. - ) ` ( F ` k ) ) ) )
94	13, 16	syl 17	. . . . . 6 |- ( ph -> F : NN --> ( RR X. RR ) )
95		rr2sscn2 39582	. . . . . . 7 |- ( RR X. RR ) C_ ( CC X. CC )
96	95	a1i 11	. . . . . 6 |- ( ph -> ( RR X. RR ) C_ ( CC X. CC ) )
97		absf 14077	. . . . . . . 8 |- abs : CC --> RR
98		subf 10283	. . . . . . . 8 |- - : ( CC X. CC ) --> CC
99		fco 6058	. . . . . . . 8 |- ( ( abs : CC --> RR /\ - : ( CC X. CC ) --> CC ) -> ( abs o. - ) : ( CC X. CC ) --> RR )
100	97, 98, 99	mp2an 708	. . . . . . 7 |- ( abs o. - ) : ( CC X. CC ) --> RR
101	100	a1i 11	. . . . . 6 |- ( ph -> ( abs o. - ) : ( CC X. CC ) --> RR )
102	94, 96, 101	fcomptss 39395	. . . . 5 |- ( ph -> ( ( abs o. - ) o. F ) = ( k e. NN |-> ( ( abs o. - ) ` ( F ` k ) ) ) )
103	93, 102	eqtr4d 2659	. . . 4 |- ( ph -> ( k e. NN |-> ( vol ` ( ( [,) o. F ) ` k ) ) ) = ( ( abs o. - ) o. F ) )
104	103	seqeq3d 12809	. . 3 |- ( ph -> seq 1 ( + , ( k e. NN |-> ( vol ` ( ( [,) o. F ) ` k ) ) ) ) = seq 1 ( + , ( ( abs o. - ) o. F ) ) )
105	33, 104	eqtr2d 2657	. 2 |- ( ph -> seq 1 ( + , ( ( abs o. - ) o. F ) ) = ( n e. NN |-> sum_ k e. ( 1 ... n ) ( vol ` ( ( [,) o. F ) ` k ) ) ) )
106	105	rneqd 5353	1 |- ( ph -> ran seq 1 ( + , ( ( abs o. - ) o. F ) ) = ran ( n e. NN |-> sum_ k e. ( 1 ... n ) ( vol ` ( ( [,) o. F ) ` k ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ifcif 4086   <.cop 4183   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   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   [,)cico 12177   ...cfz 12326   seqcseq 12801   abscabs 13974   sum_csu 14416   volcvol 23232

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

This theorem is referenced by: (None)