Theorem uniioombllem5 23355

Description: Lemma for uniioombl 23357. (Contributed by Mario Carneiro, 25-Aug-2014.)

Hypotheses

Ref	Expression
uniioombl.1	|- ( ph -> F : NN --> ( <_ i^i ( RR X. RR ) ) )
uniioombl.2	|- ( ph -> Disj x e. NN ( (,) ` ( F ` x ) ) )
uniioombl.3	|- S = seq 1 ( + , ( ( abs o. - ) o. F ) )
uniioombl.a	|- A = U. ran ( (,) o. F )
uniioombl.e	|- ( ph -> ( vol* ` E ) e. RR )
uniioombl.c	|- ( ph -> C e. RR+ )
uniioombl.g	|- ( ph -> G : NN --> ( <_ i^i ( RR X. RR ) ) )
uniioombl.s	|- ( ph -> E C_ U. ran ( (,) o. G ) )
uniioombl.t	|- T = seq 1 ( + , ( ( abs o. - ) o. G ) )
uniioombl.v	|- ( ph -> sup ( ran T , RR* , < ) <_ ( ( vol* ` E ) + C ) )
uniioombl.m	|- ( ph -> M e. NN )
uniioombl.m2	|- ( ph -> ( abs ` ( ( T ` M ) - sup ( ran T , RR* , < ) ) ) < C )
uniioombl.k	|- K = U. ( ( (,) o. G ) " ( 1 ... M ) )
uniioombl.n	|- ( ph -> N e. NN )
uniioombl.n2	|- ( ph -> A. j e. ( 1 ... M ) ( abs ` ( sum_ i e. ( 1 ... N ) ( vol* ` ( ( (,) ` ( F ` i ) ) i^i ( (,) ` ( G ` j ) ) ) ) - ( vol* ` ( ( (,) ` ( G ` j ) ) i^i A ) ) ) ) < ( C / M ) )
uniioombl.l	|- L = U. ( ( (,) o. F ) " ( 1 ... N ) )

Assertion

Ref	Expression
uniioombllem5	|- ( ph -> ( ( vol* ` ( E i^i A ) ) + ( vol* ` ( E \ A ) ) ) <_ ( ( vol* ` E ) + ( 4 x. C ) ) )

Distinct variable groups:   i, j, x, F   i, G, j, x   j, K, x   A, j, x   C, i, j, x   i, M, j, x   i, N, j   ph, i, j, x   T, i, j, x

Allowed substitution hints:   A( i)   S( x, i, j)   E( x, i, j)   K( i)   L( x, i, j)   N( x)

Proof of Theorem uniioombllem5

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss1 3833	. . . . 5 |- ( E i^i A ) C_ E
2	1	a1i 11	. . . 4 |- ( ph -> ( E i^i A ) C_ E )
3		uniioombl.s	. . . . 5 |- ( ph -> E C_ U. ran ( (,) o. G ) )
4		uniioombl.g	. . . . . . . 8 |- ( ph -> G : NN --> ( <_ i^i ( RR X. RR ) ) )
5	4	uniiccdif 23346	. . . . . . 7 |- ( ph -> ( U. ran ( (,) o. G ) C_ U. ran ( [,] o. G ) /\ ( vol* ` ( U. ran ( [,] o. G ) \ U. ran ( (,) o. G ) ) ) = 0 ) )
6	5	simpld 475	. . . . . 6 |- ( ph -> U. ran ( (,) o. G ) C_ U. ran ( [,] o. G ) )
7		ovolficcss 23238	. . . . . . 7 |- ( G : NN --> ( <_ i^i ( RR X. RR ) ) -> U. ran ( [,] o. G ) C_ RR )
8	4, 7	syl 17	. . . . . 6 |- ( ph -> U. ran ( [,] o. G ) C_ RR )
9	6, 8	sstrd 3613	. . . . 5 |- ( ph -> U. ran ( (,) o. G ) C_ RR )
10	3, 9	sstrd 3613	. . . 4 |- ( ph -> E C_ RR )
11		uniioombl.e	. . . 4 |- ( ph -> ( vol* ` E ) e. RR )
12		ovolsscl 23254	. . . 4 |- ( ( ( E i^i A ) C_ E /\ E C_ RR /\ ( vol* ` E ) e. RR ) -> ( vol* ` ( E i^i A ) ) e. RR )
13	2, 10, 11, 12	syl3anc 1326	. . 3 |- ( ph -> ( vol* ` ( E i^i A ) ) e. RR )
14		difssd 3738	. . . 4 |- ( ph -> ( E \ A ) C_ E )
15		ovolsscl 23254	. . . 4 |- ( ( ( E \ A ) C_ E /\ E C_ RR /\ ( vol* ` E ) e. RR ) -> ( vol* ` ( E \ A ) ) e. RR )
16	14, 10, 11, 15	syl3anc 1326	. . 3 |- ( ph -> ( vol* ` ( E \ A ) ) e. RR )
17	13, 16	readdcld 10069	. 2 |- ( ph -> ( ( vol* ` ( E i^i A ) ) + ( vol* ` ( E \ A ) ) ) e. RR )
18		inss1 3833	. . . . . 6 |- ( K i^i A ) C_ K
19	18	a1i 11	. . . . 5 |- ( ph -> ( K i^i A ) C_ K )
20		uniioombl.k	. . . . . . . 8 |- K = U. ( ( (,) o. G ) " ( 1 ... M ) )
21		imassrn 5477	. . . . . . . . 9 |- ( ( (,) o. G ) " ( 1 ... M ) ) C_ ran ( (,) o. G )
22	21	unissi 4461	. . . . . . . 8 |- U. ( ( (,) o. G ) " ( 1 ... M ) ) C_ U. ran ( (,) o. G )
23	20, 22	eqsstri 3635	. . . . . . 7 |- K C_ U. ran ( (,) o. G )
24	23	a1i 11	. . . . . 6 |- ( ph -> K C_ U. ran ( (,) o. G ) )
25	24, 9	sstrd 3613	. . . . 5 |- ( ph -> K C_ RR )
26		uniioombl.1	. . . . . . . 8 |- ( ph -> F : NN --> ( <_ i^i ( RR X. RR ) ) )
27		uniioombl.2	. . . . . . . 8 |- ( ph -> Disj x e. NN ( (,) ` ( F ` x ) ) )
28		uniioombl.3	. . . . . . . 8 |- S = seq 1 ( + , ( ( abs o. - ) o. F ) )
29		uniioombl.a	. . . . . . . 8 |- A = U. ran ( (,) o. F )
30		uniioombl.c	. . . . . . . 8 |- ( ph -> C e. RR+ )
31		uniioombl.t	. . . . . . . 8 |- T = seq 1 ( + , ( ( abs o. - ) o. G ) )
32		uniioombl.v	. . . . . . . 8 |- ( ph -> sup ( ran T , RR* , < ) <_ ( ( vol* ` E ) + C ) )
33	26, 27, 28, 29, 11, 30, 4, 3, 31, 32	uniioombllem1 23349	. . . . . . 7 |- ( ph -> sup ( ran T , RR* , < ) e. RR )
34		ssid 3624	. . . . . . . 8 |- U. ran ( (,) o. G ) C_ U. ran ( (,) o. G )
35	31	ovollb 23247	. . . . . . . 8 |- ( ( G : NN --> ( <_ i^i ( RR X. RR ) ) /\ U. ran ( (,) o. G ) C_ U. ran ( (,) o. G ) ) -> ( vol* ` U. ran ( (,) o. G ) ) <_ sup ( ran T , RR* , < ) )
36	4, 34, 35	sylancl 694	. . . . . . 7 |- ( ph -> ( vol* ` U. ran ( (,) o. G ) ) <_ sup ( ran T , RR* , < ) )
37		ovollecl 23251	. . . . . . 7 |- ( ( U. ran ( (,) o. G ) C_ RR /\ sup ( ran T , RR* , < ) e. RR /\ ( vol* ` U. ran ( (,) o. G ) ) <_ sup ( ran T , RR* , < ) ) -> ( vol* ` U. ran ( (,) o. G ) ) e. RR )
38	9, 33, 36, 37	syl3anc 1326	. . . . . 6 |- ( ph -> ( vol* ` U. ran ( (,) o. G ) ) e. RR )
39		ovolsscl 23254	. . . . . 6 |- ( ( K C_ U. ran ( (,) o. G ) /\ U. ran ( (,) o. G ) C_ RR /\ ( vol* ` U. ran ( (,) o. G ) ) e. RR ) -> ( vol* ` K ) e. RR )
40	24, 9, 38, 39	syl3anc 1326	. . . . 5 |- ( ph -> ( vol* ` K ) e. RR )
41		ovolsscl 23254	. . . . 5 |- ( ( ( K i^i A ) C_ K /\ K C_ RR /\ ( vol* ` K ) e. RR ) -> ( vol* ` ( K i^i A ) ) e. RR )
42	19, 25, 40, 41	syl3anc 1326	. . . 4 |- ( ph -> ( vol* ` ( K i^i A ) ) e. RR )
43		difssd 3738	. . . . 5 |- ( ph -> ( K \ A ) C_ K )
44		ovolsscl 23254	. . . . 5 |- ( ( ( K \ A ) C_ K /\ K C_ RR /\ ( vol* ` K ) e. RR ) -> ( vol* ` ( K \ A ) ) e. RR )
45	43, 25, 40, 44	syl3anc 1326	. . . 4 |- ( ph -> ( vol* ` ( K \ A ) ) e. RR )
46	42, 45	readdcld 10069	. . 3 |- ( ph -> ( ( vol* ` ( K i^i A ) ) + ( vol* ` ( K \ A ) ) ) e. RR )
47	30	rpred 11872	. . . 4 |- ( ph -> C e. RR )
48	47, 47	readdcld 10069	. . 3 |- ( ph -> ( C + C ) e. RR )
49	46, 48	readdcld 10069	. 2 |- ( ph -> ( ( ( vol* ` ( K i^i A ) ) + ( vol* ` ( K \ A ) ) ) + ( C + C ) ) e. RR )
50		4re 11097	. . . 4 |- 4 e. RR
51		remulcl 10021	. . . 4 |- ( ( 4 e. RR /\ C e. RR ) -> ( 4 x. C ) e. RR )
52	50, 47, 51	sylancr 695	. . 3 |- ( ph -> ( 4 x. C ) e. RR )
53	11, 52	readdcld 10069	. 2 |- ( ph -> ( ( vol* ` E ) + ( 4 x. C ) ) e. RR )
54		uniioombl.m	. . . 4 |- ( ph -> M e. NN )
55		uniioombl.m2	. . . 4 |- ( ph -> ( abs ` ( ( T ` M ) - sup ( ran T , RR* , < ) ) ) < C )
56	26, 27, 28, 29, 11, 30, 4, 3, 31, 32, 54, 55, 20	uniioombllem3 23353	. . 3 |- ( ph -> ( ( vol* ` ( E i^i A ) ) + ( vol* ` ( E \ A ) ) ) < ( ( ( vol* ` ( K i^i A ) ) + ( vol* ` ( K \ A ) ) ) + ( C + C ) ) )
57	17, 49, 56	ltled 10185	. 2 |- ( ph -> ( ( vol* ` ( E i^i A ) ) + ( vol* ` ( E \ A ) ) ) <_ ( ( ( vol* ` ( K i^i A ) ) + ( vol* ` ( K \ A ) ) ) + ( C + C ) ) )
58	11, 48	readdcld 10069	. . . 4 |- ( ph -> ( ( vol* ` E ) + ( C + C ) ) e. RR )
59	40, 47	readdcld 10069	. . . . 5 |- ( ph -> ( ( vol* ` K ) + C ) e. RR )
60		inss1 3833	. . . . . . . . . 10 |- ( K i^i L ) C_ K
61	60	a1i 11	. . . . . . . . 9 |- ( ph -> ( K i^i L ) C_ K )
62		ovolsscl 23254	. . . . . . . . 9 |- ( ( ( K i^i L ) C_ K /\ K C_ RR /\ ( vol* ` K ) e. RR ) -> ( vol* ` ( K i^i L ) ) e. RR )
63	61, 25, 40, 62	syl3anc 1326	. . . . . . . 8 |- ( ph -> ( vol* ` ( K i^i L ) ) e. RR )
64	63, 47	readdcld 10069	. . . . . . 7 |- ( ph -> ( ( vol* ` ( K i^i L ) ) + C ) e. RR )
65		difssd 3738	. . . . . . . 8 |- ( ph -> ( K \ L ) C_ K )
66		ovolsscl 23254	. . . . . . . 8 |- ( ( ( K \ L ) C_ K /\ K C_ RR /\ ( vol* ` K ) e. RR ) -> ( vol* ` ( K \ L ) ) e. RR )
67	65, 25, 40, 66	syl3anc 1326	. . . . . . 7 |- ( ph -> ( vol* ` ( K \ L ) ) e. RR )
68		uniioombl.n	. . . . . . . 8 |- ( ph -> N e. NN )
69		uniioombl.n2	. . . . . . . 8 |- ( ph -> A. j e. ( 1 ... M ) ( abs ` ( sum_ i e. ( 1 ... N ) ( vol* ` ( ( (,) ` ( F ` i ) ) i^i ( (,) ` ( G ` j ) ) ) ) - ( vol* ` ( ( (,) ` ( G ` j ) ) i^i A ) ) ) ) < ( C / M ) )
70		uniioombl.l	. . . . . . . 8 |- L = U. ( ( (,) o. F ) " ( 1 ... N ) )
71	26, 27, 28, 29, 11, 30, 4, 3, 31, 32, 54, 55, 20, 68, 69, 70	uniioombllem4 23354	. . . . . . 7 |- ( ph -> ( vol* ` ( K i^i A ) ) <_ ( ( vol* ` ( K i^i L ) ) + C ) )
72		imassrn 5477	. . . . . . . . . . 11 |- ( ( (,) o. F ) " ( 1 ... N ) ) C_ ran ( (,) o. F )
73	72	unissi 4461	. . . . . . . . . 10 |- U. ( ( (,) o. F ) " ( 1 ... N ) ) C_ U. ran ( (,) o. F )
74	73, 70, 29	3sstr4i 3644	. . . . . . . . 9 |- L C_ A
75		sscon 3744	. . . . . . . . 9 |- ( L C_ A -> ( K \ A ) C_ ( K \ L ) )
76	74, 75	mp1i 13	. . . . . . . 8 |- ( ph -> ( K \ A ) C_ ( K \ L ) )
77	65, 25	sstrd 3613	. . . . . . . 8 |- ( ph -> ( K \ L ) C_ RR )
78		ovolss 23253	. . . . . . . 8 |- ( ( ( K \ A ) C_ ( K \ L ) /\ ( K \ L ) C_ RR ) -> ( vol* ` ( K \ A ) ) <_ ( vol* ` ( K \ L ) ) )
79	76, 77, 78	syl2anc 693	. . . . . . 7 |- ( ph -> ( vol* ` ( K \ A ) ) <_ ( vol* ` ( K \ L ) ) )
80	42, 45, 64, 67, 71, 79	le2addd 10646	. . . . . 6 |- ( ph -> ( ( vol* ` ( K i^i A ) ) + ( vol* ` ( K \ A ) ) ) <_ ( ( ( vol* ` ( K i^i L ) ) + C ) + ( vol* ` ( K \ L ) ) ) )
81	63	recnd 10068	. . . . . . . 8 |- ( ph -> ( vol* ` ( K i^i L ) ) e. CC )
82	47	recnd 10068	. . . . . . . 8 |- ( ph -> C e. CC )
83	67	recnd 10068	. . . . . . . 8 |- ( ph -> ( vol* ` ( K \ L ) ) e. CC )
84	81, 82, 83	add32d 10263	. . . . . . 7 |- ( ph -> ( ( ( vol* ` ( K i^i L ) ) + C ) + ( vol* ` ( K \ L ) ) ) = ( ( ( vol* ` ( K i^i L ) ) + ( vol* ` ( K \ L ) ) ) + C ) )
85		ioof 12271	. . . . . . . . . . . . 13 |- (,) : ( RR* X. RR* ) --> ~P RR
86		inss2 3834	. . . . . . . . . . . . . . 15 |- ( <_ i^i ( RR X. RR ) ) C_ ( RR X. RR )
87		rexpssxrxp 10084	. . . . . . . . . . . . . . 15 |- ( RR X. RR ) C_ ( RR* X. RR* )
88	86, 87	sstri 3612	. . . . . . . . . . . . . 14 |- ( <_ i^i ( RR X. RR ) ) C_ ( RR* X. RR* )
89		fss 6056	. . . . . . . . . . . . . 14 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ ( <_ i^i ( RR X. RR ) ) C_ ( RR* X. RR* ) ) -> F : NN --> ( RR* X. RR* ) )
90	26, 88, 89	sylancl 694	. . . . . . . . . . . . 13 |- ( ph -> F : NN --> ( RR* X. RR* ) )
91		fco 6058	. . . . . . . . . . . . 13 |- ( ( (,) : ( RR* X. RR* ) --> ~P RR /\ F : NN --> ( RR* X. RR* ) ) -> ( (,) o. F ) : NN --> ~P RR )
92	85, 90, 91	sylancr 695	. . . . . . . . . . . 12 |- ( ph -> ( (,) o. F ) : NN --> ~P RR )
93		ffun 6048	. . . . . . . . . . . 12 |- ( ( (,) o. F ) : NN --> ~P RR -> Fun ( (,) o. F ) )
94		funiunfv 6506	. . . . . . . . . . . 12 |- ( Fun ( (,) o. F ) -> U_ n e. ( 1 ... N ) ( ( (,) o. F ) ` n ) = U. ( ( (,) o. F ) " ( 1 ... N ) ) )
95	92, 93, 94	3syl 18	. . . . . . . . . . 11 |- ( ph -> U_ n e. ( 1 ... N ) ( ( (,) o. F ) ` n ) = U. ( ( (,) o. F ) " ( 1 ... N ) ) )
96	95, 70	syl6eqr 2674	. . . . . . . . . 10 |- ( ph -> U_ n e. ( 1 ... N ) ( ( (,) o. F ) ` n ) = L )
97		fzfid 12772	. . . . . . . . . . 11 |- ( ph -> ( 1 ... N ) e. Fin )
98		elfznn 12370	. . . . . . . . . . . . . . 15 |- ( n e. ( 1 ... N ) -> n e. NN )
99		fvco3 6275	. . . . . . . . . . . . . . 15 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ n e. NN ) -> ( ( (,) o. F ) ` n ) = ( (,) ` ( F ` n ) ) )
100	26, 98, 99	syl2an 494	. . . . . . . . . . . . . 14 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( (,) o. F ) ` n ) = ( (,) ` ( F ` n ) ) )
101		ffvelrn 6357	. . . . . . . . . . . . . . . . . . 19 |- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ n e. NN ) -> ( F ` n ) e. ( <_ i^i ( RR X. RR ) ) )
102	26, 98, 101	syl2an 494	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( F ` n ) e. ( <_ i^i ( RR X. RR ) ) )
103	86, 102	sseldi 3601	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( F ` n ) e. ( RR X. RR ) )
104		1st2nd2 7205	. . . . . . . . . . . . . . . . 17 |- ( ( F ` n ) e. ( RR X. RR ) -> ( F ` n ) = <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. )
105	103, 104	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( F ` n ) = <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. )
106	105	fveq2d 6195	. . . . . . . . . . . . . . 15 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( (,) ` ( F ` n ) ) = ( (,) ` <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. ) )
107		df-ov 6653	. . . . . . . . . . . . . . 15 |- ( ( 1st ` ( F ` n ) ) (,) ( 2nd ` ( F ` n ) ) ) = ( (,) ` <. ( 1st ` ( F ` n ) ) , ( 2nd ` ( F ` n ) ) >. )
108	106, 107	syl6eqr 2674	. . . . . . . . . . . . . 14 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( (,) ` ( F ` n ) ) = ( ( 1st ` ( F ` n ) ) (,) ( 2nd ` ( F ` n ) ) ) )
109	100, 108	eqtrd 2656	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( (,) o. F ) ` n ) = ( ( 1st ` ( F ` n ) ) (,) ( 2nd ` ( F ` n ) ) ) )
110		ioombl 23333	. . . . . . . . . . . . 13 |- ( ( 1st ` ( F ` n ) ) (,) ( 2nd ` ( F ` n ) ) ) e. dom vol
111	109, 110	syl6eqel 2709	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( (,) o. F ) ` n ) e. dom vol )
112	111	ralrimiva 2966	. . . . . . . . . . 11 |- ( ph -> A. n e. ( 1 ... N ) ( ( (,) o. F ) ` n ) e. dom vol )
113		finiunmbl 23312	. . . . . . . . . . 11 |- ( ( ( 1 ... N ) e. Fin /\ A. n e. ( 1 ... N ) ( ( (,) o. F ) ` n ) e. dom vol ) -> U_ n e. ( 1 ... N ) ( ( (,) o. F ) ` n ) e. dom vol )
114	97, 112, 113	syl2anc 693	. . . . . . . . . 10 |- ( ph -> U_ n e. ( 1 ... N ) ( ( (,) o. F ) ` n ) e. dom vol )
115	96, 114	eqeltrrd 2702	. . . . . . . . 9 |- ( ph -> L e. dom vol )
116		mblsplit 23300	. . . . . . . . 9 |- ( ( L e. dom vol /\ K C_ RR /\ ( vol* ` K ) e. RR ) -> ( vol* ` K ) = ( ( vol* ` ( K i^i L ) ) + ( vol* ` ( K \ L ) ) ) )
117	115, 25, 40, 116	syl3anc 1326	. . . . . . . 8 |- ( ph -> ( vol* ` K ) = ( ( vol* ` ( K i^i L ) ) + ( vol* ` ( K \ L ) ) ) )
118	117	oveq1d 6665	. . . . . . 7 |- ( ph -> ( ( vol* ` K ) + C ) = ( ( ( vol* ` ( K i^i L ) ) + ( vol* ` ( K \ L ) ) ) + C ) )
119	84, 118	eqtr4d 2659	. . . . . 6 |- ( ph -> ( ( ( vol* ` ( K i^i L ) ) + C ) + ( vol* ` ( K \ L ) ) ) = ( ( vol* ` K ) + C ) )
120	80, 119	breqtrd 4679	. . . . 5 |- ( ph -> ( ( vol* ` ( K i^i A ) ) + ( vol* ` ( K \ A ) ) ) <_ ( ( vol* ` K ) + C ) )
121	11, 47	readdcld 10069	. . . . . . 7 |- ( ph -> ( ( vol* ` E ) + C ) e. RR )
122	31	ovollb 23247	. . . . . . . . 9 |- ( ( G : NN --> ( <_ i^i ( RR X. RR ) ) /\ K C_ U. ran ( (,) o. G ) ) -> ( vol* ` K ) <_ sup ( ran T , RR* , < ) )
123	4, 23, 122	sylancl 694	. . . . . . . 8 |- ( ph -> ( vol* ` K ) <_ sup ( ran T , RR* , < ) )
124	40, 33, 121, 123, 32	letrd 10194	. . . . . . 7 |- ( ph -> ( vol* ` K ) <_ ( ( vol* ` E ) + C ) )
125	40, 121, 47, 124	leadd1dd 10641	. . . . . 6 |- ( ph -> ( ( vol* ` K ) + C ) <_ ( ( ( vol* ` E ) + C ) + C ) )
126	11	recnd 10068	. . . . . . 7 |- ( ph -> ( vol* ` E ) e. CC )
127	126, 82, 82	addassd 10062	. . . . . 6 |- ( ph -> ( ( ( vol* ` E ) + C ) + C ) = ( ( vol* ` E ) + ( C + C ) ) )
128	125, 127	breqtrd 4679	. . . . 5 |- ( ph -> ( ( vol* ` K ) + C ) <_ ( ( vol* ` E ) + ( C + C ) ) )
129	46, 59, 58, 120, 128	letrd 10194	. . . 4 |- ( ph -> ( ( vol* ` ( K i^i A ) ) + ( vol* ` ( K \ A ) ) ) <_ ( ( vol* ` E ) + ( C + C ) ) )
130	46, 58, 48, 129	leadd1dd 10641	. . 3 |- ( ph -> ( ( ( vol* ` ( K i^i A ) ) + ( vol* ` ( K \ A ) ) ) + ( C + C ) ) <_ ( ( ( vol* ` E ) + ( C + C ) ) + ( C + C ) ) )
131	48	recnd 10068	. . . . 5 |- ( ph -> ( C + C ) e. CC )
132	126, 131, 131	addassd 10062	. . . 4 |- ( ph -> ( ( ( vol* ` E ) + ( C + C ) ) + ( C + C ) ) = ( ( vol* ` E ) + ( ( C + C ) + ( C + C ) ) ) )
133		2t2e4 11177	. . . . . . 7 |- ( 2 x. 2 ) = 4
134	133	oveq1i 6660	. . . . . 6 |- ( ( 2 x. 2 ) x. C ) = ( 4 x. C )
135		2cnd 11093	. . . . . . . 8 |- ( ph -> 2 e. CC )
136	135, 135, 82	mulassd 10063	. . . . . . 7 |- ( ph -> ( ( 2 x. 2 ) x. C ) = ( 2 x. ( 2 x. C ) ) )
137	82	2timesd 11275	. . . . . . . 8 |- ( ph -> ( 2 x. C ) = ( C + C ) )
138	137	oveq2d 6666	. . . . . . 7 |- ( ph -> ( 2 x. ( 2 x. C ) ) = ( 2 x. ( C + C ) ) )
139	131	2timesd 11275	. . . . . . 7 |- ( ph -> ( 2 x. ( C + C ) ) = ( ( C + C ) + ( C + C ) ) )
140	136, 138, 139	3eqtrd 2660	. . . . . 6 |- ( ph -> ( ( 2 x. 2 ) x. C ) = ( ( C + C ) + ( C + C ) ) )
141	134, 140	syl5eqr 2670	. . . . 5 |- ( ph -> ( 4 x. C ) = ( ( C + C ) + ( C + C ) ) )
142	141	oveq2d 6666	. . . 4 |- ( ph -> ( ( vol* ` E ) + ( 4 x. C ) ) = ( ( vol* ` E ) + ( ( C + C ) + ( C + C ) ) ) )
143	132, 142	eqtr4d 2659	. . 3 |- ( ph -> ( ( ( vol* ` E ) + ( C + C ) ) + ( C + C ) ) = ( ( vol* ` E ) + ( 4 x. C ) ) )
144	130, 143	breqtrd 4679	. 2 |- ( ph -> ( ( ( vol* ` ( K i^i A ) ) + ( vol* ` ( K \ A ) ) ) + ( C + C ) ) <_ ( ( vol* ` E ) + ( 4 x. C ) ) )
145	17, 49, 53, 57, 144	letrd 10194	1 |- ( ph -> ( ( vol* ` ( E i^i A ) ) + ( vol* ` ( E \ A ) ) ) <_ ( ( vol* ` E ) + ( 4 x. C ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   \ cdif 3571   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   <.cop 4183   U.cuni 4436   U_ciun 4520  Disj wdisj 4620   class class class wbr 4653   X. cxp 5112   dom cdm 5114   ran crn 5115   "cima 5117   o. ccom 5118   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   Fincfn 7955   supcsup 8346   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   4c4 11072   RR+crp 11832   (,)cioo 12175   [,]cicc 12178   ...cfz 12326   seqcseq 12801   abscabs 13974   sum_csu 14416   vol*covol 23231   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-disj 4621  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-acn 8768  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-4 11081  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:  uniioombllem6  23356