Theorem itg1addlem5 23467

Description: Lemma for itg1add . (Contributed by Mario Carneiro, 27-Jun-2014.)

Hypotheses

Ref	Expression
i1fadd.1	|- ( ph -> F e. dom S.1 )
i1fadd.2	|- ( ph -> G e. dom S.1 )
itg1add.3	|- I = ( i e. RR , j e. RR |-> if ( ( i = 0 /\ j = 0 ) , 0 , ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) ) )
itg1add.4	|- P = ( + |` ( ran F X. ran G ) )

Assertion

Ref	Expression
itg1addlem5	|- ( ph -> ( S.1 ` ( F oF + G ) ) = ( ( S.1 ` F ) + ( S.1 ` G ) ) )

Distinct variable groups:   i, j, F   i, G, j   ph, i, j

Allowed substitution hints:   P( i, j)   I( i, j)

Proof of Theorem itg1addlem5

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		i1fadd.1	. . . 4 |- ( ph -> F e. dom S.1 )
2		i1frn 23444	. . . 4 |- ( F e. dom S.1 -> ran F e. Fin )
3	1, 2	syl 17	. . 3 |- ( ph -> ran F e. Fin )
4		i1fadd.2	. . . . . 6 |- ( ph -> G e. dom S.1 )
5		i1frn 23444	. . . . . 6 |- ( G e. dom S.1 -> ran G e. Fin )
6	4, 5	syl 17	. . . . 5 |- ( ph -> ran G e. Fin )
7	6	adantr 481	. . . 4 |- ( ( ph /\ y e. ran F ) -> ran G e. Fin )
8		i1ff 23443	. . . . . . . . . 10 |- ( F e. dom S.1 -> F : RR --> RR )
9	1, 8	syl 17	. . . . . . . . 9 |- ( ph -> F : RR --> RR )
10		frn 6053	. . . . . . . . 9 |- ( F : RR --> RR -> ran F C_ RR )
11	9, 10	syl 17	. . . . . . . 8 |- ( ph -> ran F C_ RR )
12	11	sselda 3603	. . . . . . 7 |- ( ( ph /\ y e. ran F ) -> y e. RR )
13	12	adantr 481	. . . . . 6 |- ( ( ( ph /\ y e. ran F ) /\ z e. ran G ) -> y e. RR )
14	13	recnd 10068	. . . . 5 |- ( ( ( ph /\ y e. ran F ) /\ z e. ran G ) -> y e. CC )
15		itg1add.3	. . . . . . . . 9 |- I = ( i e. RR , j e. RR |-> if ( ( i = 0 /\ j = 0 ) , 0 , ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) ) )
16	1, 4, 15	itg1addlem2 23464	. . . . . . . 8 |- ( ph -> I : ( RR X. RR ) --> RR )
17	16	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ y e. ran F ) /\ z e. ran G ) -> I : ( RR X. RR ) --> RR )
18		i1ff 23443	. . . . . . . . . . 11 |- ( G e. dom S.1 -> G : RR --> RR )
19	4, 18	syl 17	. . . . . . . . . 10 |- ( ph -> G : RR --> RR )
20		frn 6053	. . . . . . . . . 10 |- ( G : RR --> RR -> ran G C_ RR )
21	19, 20	syl 17	. . . . . . . . 9 |- ( ph -> ran G C_ RR )
22	21	sselda 3603	. . . . . . . 8 |- ( ( ph /\ z e. ran G ) -> z e. RR )
23	22	adantlr 751	. . . . . . 7 |- ( ( ( ph /\ y e. ran F ) /\ z e. ran G ) -> z e. RR )
24	17, 13, 23	fovrnd 6806	. . . . . 6 |- ( ( ( ph /\ y e. ran F ) /\ z e. ran G ) -> ( y I z ) e. RR )
25	24	recnd 10068	. . . . 5 |- ( ( ( ph /\ y e. ran F ) /\ z e. ran G ) -> ( y I z ) e. CC )
26	14, 25	mulcld 10060	. . . 4 |- ( ( ( ph /\ y e. ran F ) /\ z e. ran G ) -> ( y x. ( y I z ) ) e. CC )
27	7, 26	fsumcl 14464	. . 3 |- ( ( ph /\ y e. ran F ) -> sum_ z e. ran G ( y x. ( y I z ) ) e. CC )
28	23	recnd 10068	. . . . 5 |- ( ( ( ph /\ y e. ran F ) /\ z e. ran G ) -> z e. CC )
29	28, 25	mulcld 10060	. . . 4 |- ( ( ( ph /\ y e. ran F ) /\ z e. ran G ) -> ( z x. ( y I z ) ) e. CC )
30	7, 29	fsumcl 14464	. . 3 |- ( ( ph /\ y e. ran F ) -> sum_ z e. ran G ( z x. ( y I z ) ) e. CC )
31	3, 27, 30	fsumadd 14470	. 2 |- ( ph -> sum_ y e. ran F ( sum_ z e. ran G ( y x. ( y I z ) ) + sum_ z e. ran G ( z x. ( y I z ) ) ) = ( sum_ y e. ran F sum_ z e. ran G ( y x. ( y I z ) ) + sum_ y e. ran F sum_ z e. ran G ( z x. ( y I z ) ) ) )
32		itg1add.4	. . . 4 |- P = ( + |` ( ran F X. ran G ) )
33	1, 4, 15, 32	itg1addlem4 23466	. . 3 |- ( ph -> ( S.1 ` ( F oF + G ) ) = sum_ y e. ran F sum_ z e. ran G ( ( y + z ) x. ( y I z ) ) )
34	14, 28, 25	adddird 10065	. . . . . 6 |- ( ( ( ph /\ y e. ran F ) /\ z e. ran G ) -> ( ( y + z ) x. ( y I z ) ) = ( ( y x. ( y I z ) ) + ( z x. ( y I z ) ) ) )
35	34	sumeq2dv 14433	. . . . 5 |- ( ( ph /\ y e. ran F ) -> sum_ z e. ran G ( ( y + z ) x. ( y I z ) ) = sum_ z e. ran G ( ( y x. ( y I z ) ) + ( z x. ( y I z ) ) ) )
36	7, 26, 29	fsumadd 14470	. . . . 5 |- ( ( ph /\ y e. ran F ) -> sum_ z e. ran G ( ( y x. ( y I z ) ) + ( z x. ( y I z ) ) ) = ( sum_ z e. ran G ( y x. ( y I z ) ) + sum_ z e. ran G ( z x. ( y I z ) ) ) )
37	35, 36	eqtrd 2656	. . . 4 |- ( ( ph /\ y e. ran F ) -> sum_ z e. ran G ( ( y + z ) x. ( y I z ) ) = ( sum_ z e. ran G ( y x. ( y I z ) ) + sum_ z e. ran G ( z x. ( y I z ) ) ) )
38	37	sumeq2dv 14433	. . 3 |- ( ph -> sum_ y e. ran F sum_ z e. ran G ( ( y + z ) x. ( y I z ) ) = sum_ y e. ran F ( sum_ z e. ran G ( y x. ( y I z ) ) + sum_ z e. ran G ( z x. ( y I z ) ) ) )
39	33, 38	eqtrd 2656	. 2 |- ( ph -> ( S.1 ` ( F oF + G ) ) = sum_ y e. ran F ( sum_ z e. ran G ( y x. ( y I z ) ) + sum_ z e. ran G ( z x. ( y I z ) ) ) )
40		itg1val 23450	. . . . 5 |- ( F e. dom S.1 -> ( S.1 ` F ) = sum_ y e. ( ran F \ { 0 } ) ( y x. ( vol ` ( `' F " { y } ) ) ) )
41	1, 40	syl 17	. . . 4 |- ( ph -> ( S.1 ` F ) = sum_ y e. ( ran F \ { 0 } ) ( y x. ( vol ` ( `' F " { y } ) ) ) )
42	19	adantr 481	. . . . . . . . 9 |- ( ( ph /\ y e. ( ran F \ { 0 } ) ) -> G : RR --> RR )
43	6	adantr 481	. . . . . . . . 9 |- ( ( ph /\ y e. ( ran F \ { 0 } ) ) -> ran G e. Fin )
44		inss2 3834	. . . . . . . . . 10 |- ( ( `' F " { y } ) i^i ( `' G " { z } ) ) C_ ( `' G " { z } )
45	44	a1i 11	. . . . . . . . 9 |- ( ( ( ph /\ y e. ( ran F \ { 0 } ) ) /\ z e. ran G ) -> ( ( `' F " { y } ) i^i ( `' G " { z } ) ) C_ ( `' G " { z } ) )
46		i1fima 23445	. . . . . . . . . . . 12 |- ( F e. dom S.1 -> ( `' F " { y } ) e. dom vol )
47	1, 46	syl 17	. . . . . . . . . . 11 |- ( ph -> ( `' F " { y } ) e. dom vol )
48	47	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ y e. ( ran F \ { 0 } ) ) /\ z e. ran G ) -> ( `' F " { y } ) e. dom vol )
49		i1fima 23445	. . . . . . . . . . . 12 |- ( G e. dom S.1 -> ( `' G " { z } ) e. dom vol )
50	4, 49	syl 17	. . . . . . . . . . 11 |- ( ph -> ( `' G " { z } ) e. dom vol )
51	50	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ y e. ( ran F \ { 0 } ) ) /\ z e. ran G ) -> ( `' G " { z } ) e. dom vol )
52		inmbl 23310	. . . . . . . . . 10 |- ( ( ( `' F " { y } ) e. dom vol /\ ( `' G " { z } ) e. dom vol ) -> ( ( `' F " { y } ) i^i ( `' G " { z } ) ) e. dom vol )
53	48, 51, 52	syl2anc 693	. . . . . . . . 9 |- ( ( ( ph /\ y e. ( ran F \ { 0 } ) ) /\ z e. ran G ) -> ( ( `' F " { y } ) i^i ( `' G " { z } ) ) e. dom vol )
54	11	ssdifssd 3748	. . . . . . . . . . . . 13 |- ( ph -> ( ran F \ { 0 } ) C_ RR )
55	54	sselda 3603	. . . . . . . . . . . 12 |- ( ( ph /\ y e. ( ran F \ { 0 } ) ) -> y e. RR )
56	55	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. ( ran F \ { 0 } ) ) /\ z e. ran G ) -> y e. RR )
57	21	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ y e. ( ran F \ { 0 } ) ) -> ran G C_ RR )
58	57	sselda 3603	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. ( ran F \ { 0 } ) ) /\ z e. ran G ) -> z e. RR )
59		eldifsni 4320	. . . . . . . . . . . . 13 |- ( y e. ( ran F \ { 0 } ) -> y =/= 0 )
60	59	ad2antlr 763	. . . . . . . . . . . 12 |- ( ( ( ph /\ y e. ( ran F \ { 0 } ) ) /\ z e. ran G ) -> y =/= 0 )
61		simpl 473	. . . . . . . . . . . . 13 |- ( ( y = 0 /\ z = 0 ) -> y = 0 )
62	61	necon3ai 2819	. . . . . . . . . . . 12 |- ( y =/= 0 -> -. ( y = 0 /\ z = 0 ) )
63	60, 62	syl 17	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. ( ran F \ { 0 } ) ) /\ z e. ran G ) -> -. ( y = 0 /\ z = 0 ) )
64	1, 4, 15	itg1addlem3 23465	. . . . . . . . . . 11 |- ( ( ( y e. RR /\ z e. RR ) /\ -. ( y = 0 /\ z = 0 ) ) -> ( y I z ) = ( vol ` ( ( `' F " { y } ) i^i ( `' G " { z } ) ) ) )
65	56, 58, 63, 64	syl21anc 1325	. . . . . . . . . 10 |- ( ( ( ph /\ y e. ( ran F \ { 0 } ) ) /\ z e. ran G ) -> ( y I z ) = ( vol ` ( ( `' F " { y } ) i^i ( `' G " { z } ) ) ) )
66	16	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. ( ran F \ { 0 } ) ) /\ z e. ran G ) -> I : ( RR X. RR ) --> RR )
67	66, 56, 58	fovrnd 6806	. . . . . . . . . 10 |- ( ( ( ph /\ y e. ( ran F \ { 0 } ) ) /\ z e. ran G ) -> ( y I z ) e. RR )
68	65, 67	eqeltrrd 2702	. . . . . . . . 9 |- ( ( ( ph /\ y e. ( ran F \ { 0 } ) ) /\ z e. ran G ) -> ( vol ` ( ( `' F " { y } ) i^i ( `' G " { z } ) ) ) e. RR )
69	42, 43, 45, 53, 68	itg1addlem1 23459	. . . . . . . 8 |- ( ( ph /\ y e. ( ran F \ { 0 } ) ) -> ( vol ` U_ z e. ran G ( ( `' F " { y } ) i^i ( `' G " { z } ) ) ) = sum_ z e. ran G ( vol ` ( ( `' F " { y } ) i^i ( `' G " { z } ) ) ) )
70		iunin2 4584	. . . . . . . . . 10 |- U_ z e. ran G ( ( `' F " { y } ) i^i ( `' G " { z } ) ) = ( ( `' F " { y } ) i^i U_ z e. ran G ( `' G " { z } ) )
71	1	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ y e. ( ran F \ { 0 } ) ) -> F e. dom S.1 )
72	71, 46	syl 17	. . . . . . . . . . . . 13 |- ( ( ph /\ y e. ( ran F \ { 0 } ) ) -> ( `' F " { y } ) e. dom vol )
73		mblss 23299	. . . . . . . . . . . . 13 |- ( ( `' F " { y } ) e. dom vol -> ( `' F " { y } ) C_ RR )
74	72, 73	syl 17	. . . . . . . . . . . 12 |- ( ( ph /\ y e. ( ran F \ { 0 } ) ) -> ( `' F " { y } ) C_ RR )
75		iunid 4575	. . . . . . . . . . . . . . 15 |- U_ z e. ran G { z } = ran G
76	75	imaeq2i 5464	. . . . . . . . . . . . . 14 |- ( `' G " U_ z e. ran G { z } ) = ( `' G " ran G )
77		imaiun 6503	. . . . . . . . . . . . . 14 |- ( `' G " U_ z e. ran G { z } ) = U_ z e. ran G ( `' G " { z } )
78		cnvimarndm 5486	. . . . . . . . . . . . . 14 |- ( `' G " ran G ) = dom G
79	76, 77, 78	3eqtr3i 2652	. . . . . . . . . . . . 13 |- U_ z e. ran G ( `' G " { z } ) = dom G
80		fdm 6051	. . . . . . . . . . . . . 14 |- ( G : RR --> RR -> dom G = RR )
81	42, 80	syl 17	. . . . . . . . . . . . 13 |- ( ( ph /\ y e. ( ran F \ { 0 } ) ) -> dom G = RR )
82	79, 81	syl5eq 2668	. . . . . . . . . . . 12 |- ( ( ph /\ y e. ( ran F \ { 0 } ) ) -> U_ z e. ran G ( `' G " { z } ) = RR )
83	74, 82	sseqtr4d 3642	. . . . . . . . . . 11 |- ( ( ph /\ y e. ( ran F \ { 0 } ) ) -> ( `' F " { y } ) C_ U_ z e. ran G ( `' G " { z } ) )
84		df-ss 3588	. . . . . . . . . . 11 |- ( ( `' F " { y } ) C_ U_ z e. ran G ( `' G " { z } ) <-> ( ( `' F " { y } ) i^i U_ z e. ran G ( `' G " { z } ) ) = ( `' F " { y } ) )
85	83, 84	sylib 208	. . . . . . . . . 10 |- ( ( ph /\ y e. ( ran F \ { 0 } ) ) -> ( ( `' F " { y } ) i^i U_ z e. ran G ( `' G " { z } ) ) = ( `' F " { y } ) )
86	70, 85	syl5req 2669	. . . . . . . . 9 |- ( ( ph /\ y e. ( ran F \ { 0 } ) ) -> ( `' F " { y } ) = U_ z e. ran G ( ( `' F " { y } ) i^i ( `' G " { z } ) ) )
87	86	fveq2d 6195	. . . . . . . 8 |- ( ( ph /\ y e. ( ran F \ { 0 } ) ) -> ( vol ` ( `' F " { y } ) ) = ( vol ` U_ z e. ran G ( ( `' F " { y } ) i^i ( `' G " { z } ) ) ) )
88	65	sumeq2dv 14433	. . . . . . . 8 |- ( ( ph /\ y e. ( ran F \ { 0 } ) ) -> sum_ z e. ran G ( y I z ) = sum_ z e. ran G ( vol ` ( ( `' F " { y } ) i^i ( `' G " { z } ) ) ) )
89	69, 87, 88	3eqtr4d 2666	. . . . . . 7 |- ( ( ph /\ y e. ( ran F \ { 0 } ) ) -> ( vol ` ( `' F " { y } ) ) = sum_ z e. ran G ( y I z ) )
90	89	oveq2d 6666	. . . . . 6 |- ( ( ph /\ y e. ( ran F \ { 0 } ) ) -> ( y x. ( vol ` ( `' F " { y } ) ) ) = ( y x. sum_ z e. ran G ( y I z ) ) )
91	55	recnd 10068	. . . . . . 7 |- ( ( ph /\ y e. ( ran F \ { 0 } ) ) -> y e. CC )
92	67	recnd 10068	. . . . . . 7 |- ( ( ( ph /\ y e. ( ran F \ { 0 } ) ) /\ z e. ran G ) -> ( y I z ) e. CC )
93	43, 91, 92	fsummulc2 14516	. . . . . 6 |- ( ( ph /\ y e. ( ran F \ { 0 } ) ) -> ( y x. sum_ z e. ran G ( y I z ) ) = sum_ z e. ran G ( y x. ( y I z ) ) )
94	90, 93	eqtrd 2656	. . . . 5 |- ( ( ph /\ y e. ( ran F \ { 0 } ) ) -> ( y x. ( vol ` ( `' F " { y } ) ) ) = sum_ z e. ran G ( y x. ( y I z ) ) )
95	94	sumeq2dv 14433	. . . 4 |- ( ph -> sum_ y e. ( ran F \ { 0 } ) ( y x. ( vol ` ( `' F " { y } ) ) ) = sum_ y e. ( ran F \ { 0 } ) sum_ z e. ran G ( y x. ( y I z ) ) )
96		difssd 3738	. . . . 5 |- ( ph -> ( ran F \ { 0 } ) C_ ran F )
97	56	recnd 10068	. . . . . . 7 |- ( ( ( ph /\ y e. ( ran F \ { 0 } ) ) /\ z e. ran G ) -> y e. CC )
98	97, 92	mulcld 10060	. . . . . 6 |- ( ( ( ph /\ y e. ( ran F \ { 0 } ) ) /\ z e. ran G ) -> ( y x. ( y I z ) ) e. CC )
99	43, 98	fsumcl 14464	. . . . 5 |- ( ( ph /\ y e. ( ran F \ { 0 } ) ) -> sum_ z e. ran G ( y x. ( y I z ) ) e. CC )
100		dfin4 3867	. . . . . . . 8 |- ( ran F i^i { 0 } ) = ( ran F \ ( ran F \ { 0 } ) )
101		inss2 3834	. . . . . . . 8 |- ( ran F i^i { 0 } ) C_ { 0 }
102	100, 101	eqsstr3i 3636	. . . . . . 7 |- ( ran F \ ( ran F \ { 0 } ) ) C_ { 0 }
103	102	sseli 3599	. . . . . 6 |- ( y e. ( ran F \ ( ran F \ { 0 } ) ) -> y e. { 0 } )
104		elsni 4194	. . . . . . . . . . 11 |- ( y e. { 0 } -> y = 0 )
105	104	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( ph /\ y e. { 0 } ) /\ z e. ran G ) -> y = 0 )
106	105	oveq1d 6665	. . . . . . . . 9 |- ( ( ( ph /\ y e. { 0 } ) /\ z e. ran G ) -> ( y x. ( y I z ) ) = ( 0 x. ( y I z ) ) )
107	16	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ y e. { 0 } ) /\ z e. ran G ) -> I : ( RR X. RR ) --> RR )
108		0re 10040	. . . . . . . . . . . . 13 |- 0 e. RR
109	105, 108	syl6eqel 2709	. . . . . . . . . . . 12 |- ( ( ( ph /\ y e. { 0 } ) /\ z e. ran G ) -> y e. RR )
110	22	adantlr 751	. . . . . . . . . . . 12 |- ( ( ( ph /\ y e. { 0 } ) /\ z e. ran G ) -> z e. RR )
111	107, 109, 110	fovrnd 6806	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. { 0 } ) /\ z e. ran G ) -> ( y I z ) e. RR )
112	111	recnd 10068	. . . . . . . . . 10 |- ( ( ( ph /\ y e. { 0 } ) /\ z e. ran G ) -> ( y I z ) e. CC )
113	112	mul02d 10234	. . . . . . . . 9 |- ( ( ( ph /\ y e. { 0 } ) /\ z e. ran G ) -> ( 0 x. ( y I z ) ) = 0 )
114	106, 113	eqtrd 2656	. . . . . . . 8 |- ( ( ( ph /\ y e. { 0 } ) /\ z e. ran G ) -> ( y x. ( y I z ) ) = 0 )
115	114	sumeq2dv 14433	. . . . . . 7 |- ( ( ph /\ y e. { 0 } ) -> sum_ z e. ran G ( y x. ( y I z ) ) = sum_ z e. ran G 0 )
116	6	adantr 481	. . . . . . . . 9 |- ( ( ph /\ y e. { 0 } ) -> ran G e. Fin )
117	116	olcd 408	. . . . . . . 8 |- ( ( ph /\ y e. { 0 } ) -> ( ran G C_ ( ZZ>= ` 0 ) \/ ran G e. Fin ) )
118		sumz 14453	. . . . . . . 8 |- ( ( ran G C_ ( ZZ>= ` 0 ) \/ ran G e. Fin ) -> sum_ z e. ran G 0 = 0 )
119	117, 118	syl 17	. . . . . . 7 |- ( ( ph /\ y e. { 0 } ) -> sum_ z e. ran G 0 = 0 )
120	115, 119	eqtrd 2656	. . . . . 6 |- ( ( ph /\ y e. { 0 } ) -> sum_ z e. ran G ( y x. ( y I z ) ) = 0 )
121	103, 120	sylan2 491	. . . . 5 |- ( ( ph /\ y e. ( ran F \ ( ran F \ { 0 } ) ) ) -> sum_ z e. ran G ( y x. ( y I z ) ) = 0 )
122	96, 99, 121, 3	fsumss 14456	. . . 4 |- ( ph -> sum_ y e. ( ran F \ { 0 } ) sum_ z e. ran G ( y x. ( y I z ) ) = sum_ y e. ran F sum_ z e. ran G ( y x. ( y I z ) ) )
123	41, 95, 122	3eqtrd 2660	. . 3 |- ( ph -> ( S.1 ` F ) = sum_ y e. ran F sum_ z e. ran G ( y x. ( y I z ) ) )
124		itg1val 23450	. . . . 5 |- ( G e. dom S.1 -> ( S.1 ` G ) = sum_ z e. ( ran G \ { 0 } ) ( z x. ( vol ` ( `' G " { z } ) ) ) )
125	4, 124	syl 17	. . . 4 |- ( ph -> ( S.1 ` G ) = sum_ z e. ( ran G \ { 0 } ) ( z x. ( vol ` ( `' G " { z } ) ) ) )
126	9	adantr 481	. . . . . . . . 9 |- ( ( ph /\ z e. ( ran G \ { 0 } ) ) -> F : RR --> RR )
127	3	adantr 481	. . . . . . . . 9 |- ( ( ph /\ z e. ( ran G \ { 0 } ) ) -> ran F e. Fin )
128		inss1 3833	. . . . . . . . . 10 |- ( ( `' F " { y } ) i^i ( `' G " { z } ) ) C_ ( `' F " { y } )
129	128	a1i 11	. . . . . . . . 9 |- ( ( ( ph /\ z e. ( ran G \ { 0 } ) ) /\ y e. ran F ) -> ( ( `' F " { y } ) i^i ( `' G " { z } ) ) C_ ( `' F " { y } ) )
130	47	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ( ran G \ { 0 } ) ) /\ y e. ran F ) -> ( `' F " { y } ) e. dom vol )
131	50	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ( ran G \ { 0 } ) ) /\ y e. ran F ) -> ( `' G " { z } ) e. dom vol )
132	130, 131, 52	syl2anc 693	. . . . . . . . 9 |- ( ( ( ph /\ z e. ( ran G \ { 0 } ) ) /\ y e. ran F ) -> ( ( `' F " { y } ) i^i ( `' G " { z } ) ) e. dom vol )
133	11	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ z e. ( ran G \ { 0 } ) ) -> ran F C_ RR )
134	133	sselda 3603	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ( ran G \ { 0 } ) ) /\ y e. ran F ) -> y e. RR )
135	21	ssdifssd 3748	. . . . . . . . . . . . 13 |- ( ph -> ( ran G \ { 0 } ) C_ RR )
136	135	sselda 3603	. . . . . . . . . . . 12 |- ( ( ph /\ z e. ( ran G \ { 0 } ) ) -> z e. RR )
137	136	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ( ran G \ { 0 } ) ) /\ y e. ran F ) -> z e. RR )
138		eldifsni 4320	. . . . . . . . . . . . 13 |- ( z e. ( ran G \ { 0 } ) -> z =/= 0 )
139	138	ad2antlr 763	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. ( ran G \ { 0 } ) ) /\ y e. ran F ) -> z =/= 0 )
140		simpr 477	. . . . . . . . . . . . 13 |- ( ( y = 0 /\ z = 0 ) -> z = 0 )
141	140	necon3ai 2819	. . . . . . . . . . . 12 |- ( z =/= 0 -> -. ( y = 0 /\ z = 0 ) )
142	139, 141	syl 17	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ( ran G \ { 0 } ) ) /\ y e. ran F ) -> -. ( y = 0 /\ z = 0 ) )
143	134, 137, 142, 64	syl21anc 1325	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ( ran G \ { 0 } ) ) /\ y e. ran F ) -> ( y I z ) = ( vol ` ( ( `' F " { y } ) i^i ( `' G " { z } ) ) ) )
144	16	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ( ran G \ { 0 } ) ) /\ y e. ran F ) -> I : ( RR X. RR ) --> RR )
145	144, 134, 137	fovrnd 6806	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ( ran G \ { 0 } ) ) /\ y e. ran F ) -> ( y I z ) e. RR )
146	143, 145	eqeltrrd 2702	. . . . . . . . 9 |- ( ( ( ph /\ z e. ( ran G \ { 0 } ) ) /\ y e. ran F ) -> ( vol ` ( ( `' F " { y } ) i^i ( `' G " { z } ) ) ) e. RR )
147	126, 127, 129, 132, 146	itg1addlem1 23459	. . . . . . . 8 |- ( ( ph /\ z e. ( ran G \ { 0 } ) ) -> ( vol ` U_ y e. ran F ( ( `' F " { y } ) i^i ( `' G " { z } ) ) ) = sum_ y e. ran F ( vol ` ( ( `' F " { y } ) i^i ( `' G " { z } ) ) ) )
148		incom 3805	. . . . . . . . . . . . 13 |- ( ( `' F " { y } ) i^i ( `' G " { z } ) ) = ( ( `' G " { z } ) i^i ( `' F " { y } ) )
149	148	a1i 11	. . . . . . . . . . . 12 |- ( y e. ran F -> ( ( `' F " { y } ) i^i ( `' G " { z } ) ) = ( ( `' G " { z } ) i^i ( `' F " { y } ) ) )
150	149	iuneq2i 4539	. . . . . . . . . . 11 |- U_ y e. ran F ( ( `' F " { y } ) i^i ( `' G " { z } ) ) = U_ y e. ran F ( ( `' G " { z } ) i^i ( `' F " { y } ) )
151		iunin2 4584	. . . . . . . . . . 11 |- U_ y e. ran F ( ( `' G " { z } ) i^i ( `' F " { y } ) ) = ( ( `' G " { z } ) i^i U_ y e. ran F ( `' F " { y } ) )
152	150, 151	eqtri 2644	. . . . . . . . . 10 |- U_ y e. ran F ( ( `' F " { y } ) i^i ( `' G " { z } ) ) = ( ( `' G " { z } ) i^i U_ y e. ran F ( `' F " { y } ) )
153		cnvimass 5485	. . . . . . . . . . . . 13 |- ( `' G " { z } ) C_ dom G
154	19, 80	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> dom G = RR )
155	154	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. ( ran G \ { 0 } ) ) -> dom G = RR )
156	153, 155	syl5sseq 3653	. . . . . . . . . . . 12 |- ( ( ph /\ z e. ( ran G \ { 0 } ) ) -> ( `' G " { z } ) C_ RR )
157		iunid 4575	. . . . . . . . . . . . . . 15 |- U_ y e. ran F { y } = ran F
158	157	imaeq2i 5464	. . . . . . . . . . . . . 14 |- ( `' F " U_ y e. ran F { y } ) = ( `' F " ran F )
159		imaiun 6503	. . . . . . . . . . . . . 14 |- ( `' F " U_ y e. ran F { y } ) = U_ y e. ran F ( `' F " { y } )
160		cnvimarndm 5486	. . . . . . . . . . . . . 14 |- ( `' F " ran F ) = dom F
161	158, 159, 160	3eqtr3i 2652	. . . . . . . . . . . . 13 |- U_ y e. ran F ( `' F " { y } ) = dom F
162		fdm 6051	. . . . . . . . . . . . . . 15 |- ( F : RR --> RR -> dom F = RR )
163	9, 162	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> dom F = RR )
164	163	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. ( ran G \ { 0 } ) ) -> dom F = RR )
165	161, 164	syl5eq 2668	. . . . . . . . . . . 12 |- ( ( ph /\ z e. ( ran G \ { 0 } ) ) -> U_ y e. ran F ( `' F " { y } ) = RR )
166	156, 165	sseqtr4d 3642	. . . . . . . . . . 11 |- ( ( ph /\ z e. ( ran G \ { 0 } ) ) -> ( `' G " { z } ) C_ U_ y e. ran F ( `' F " { y } ) )
167		df-ss 3588	. . . . . . . . . . 11 |- ( ( `' G " { z } ) C_ U_ y e. ran F ( `' F " { y } ) <-> ( ( `' G " { z } ) i^i U_ y e. ran F ( `' F " { y } ) ) = ( `' G " { z } ) )
168	166, 167	sylib 208	. . . . . . . . . 10 |- ( ( ph /\ z e. ( ran G \ { 0 } ) ) -> ( ( `' G " { z } ) i^i U_ y e. ran F ( `' F " { y } ) ) = ( `' G " { z } ) )
169	152, 168	syl5req 2669	. . . . . . . . 9 |- ( ( ph /\ z e. ( ran G \ { 0 } ) ) -> ( `' G " { z } ) = U_ y e. ran F ( ( `' F " { y } ) i^i ( `' G " { z } ) ) )
170	169	fveq2d 6195	. . . . . . . 8 |- ( ( ph /\ z e. ( ran G \ { 0 } ) ) -> ( vol ` ( `' G " { z } ) ) = ( vol ` U_ y e. ran F ( ( `' F " { y } ) i^i ( `' G " { z } ) ) ) )
171	143	sumeq2dv 14433	. . . . . . . 8 |- ( ( ph /\ z e. ( ran G \ { 0 } ) ) -> sum_ y e. ran F ( y I z ) = sum_ y e. ran F ( vol ` ( ( `' F " { y } ) i^i ( `' G " { z } ) ) ) )
172	147, 170, 171	3eqtr4d 2666	. . . . . . 7 |- ( ( ph /\ z e. ( ran G \ { 0 } ) ) -> ( vol ` ( `' G " { z } ) ) = sum_ y e. ran F ( y I z ) )
173	172	oveq2d 6666	. . . . . 6 |- ( ( ph /\ z e. ( ran G \ { 0 } ) ) -> ( z x. ( vol ` ( `' G " { z } ) ) ) = ( z x. sum_ y e. ran F ( y I z ) ) )
174	136	recnd 10068	. . . . . . 7 |- ( ( ph /\ z e. ( ran G \ { 0 } ) ) -> z e. CC )
175	145	recnd 10068	. . . . . . 7 |- ( ( ( ph /\ z e. ( ran G \ { 0 } ) ) /\ y e. ran F ) -> ( y I z ) e. CC )
176	127, 174, 175	fsummulc2 14516	. . . . . 6 |- ( ( ph /\ z e. ( ran G \ { 0 } ) ) -> ( z x. sum_ y e. ran F ( y I z ) ) = sum_ y e. ran F ( z x. ( y I z ) ) )
177	173, 176	eqtrd 2656	. . . . 5 |- ( ( ph /\ z e. ( ran G \ { 0 } ) ) -> ( z x. ( vol ` ( `' G " { z } ) ) ) = sum_ y e. ran F ( z x. ( y I z ) ) )
178	177	sumeq2dv 14433	. . . 4 |- ( ph -> sum_ z e. ( ran G \ { 0 } ) ( z x. ( vol ` ( `' G " { z } ) ) ) = sum_ z e. ( ran G \ { 0 } ) sum_ y e. ran F ( z x. ( y I z ) ) )
179		difssd 3738	. . . . . 6 |- ( ph -> ( ran G \ { 0 } ) C_ ran G )
180	174	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ z e. ( ran G \ { 0 } ) ) /\ y e. ran F ) -> z e. CC )
181	180, 175	mulcld 10060	. . . . . . 7 |- ( ( ( ph /\ z e. ( ran G \ { 0 } ) ) /\ y e. ran F ) -> ( z x. ( y I z ) ) e. CC )
182	127, 181	fsumcl 14464	. . . . . 6 |- ( ( ph /\ z e. ( ran G \ { 0 } ) ) -> sum_ y e. ran F ( z x. ( y I z ) ) e. CC )
183		dfin4 3867	. . . . . . . . 9 |- ( ran G i^i { 0 } ) = ( ran G \ ( ran G \ { 0 } ) )
184		inss2 3834	. . . . . . . . 9 |- ( ran G i^i { 0 } ) C_ { 0 }
185	183, 184	eqsstr3i 3636	. . . . . . . 8 |- ( ran G \ ( ran G \ { 0 } ) ) C_ { 0 }
186	185	sseli 3599	. . . . . . 7 |- ( z e. ( ran G \ ( ran G \ { 0 } ) ) -> z e. { 0 } )
187		elsni 4194	. . . . . . . . . . . 12 |- ( z e. { 0 } -> z = 0 )
188	187	ad2antlr 763	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. { 0 } ) /\ y e. ran F ) -> z = 0 )
189	188	oveq1d 6665	. . . . . . . . . 10 |- ( ( ( ph /\ z e. { 0 } ) /\ y e. ran F ) -> ( z x. ( y I z ) ) = ( 0 x. ( y I z ) ) )
190	16	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ z e. { 0 } ) /\ y e. ran F ) -> I : ( RR X. RR ) --> RR )
191	12	adantlr 751	. . . . . . . . . . . . 13 |- ( ( ( ph /\ z e. { 0 } ) /\ y e. ran F ) -> y e. RR )
192	188, 108	syl6eqel 2709	. . . . . . . . . . . . 13 |- ( ( ( ph /\ z e. { 0 } ) /\ y e. ran F ) -> z e. RR )
193	190, 191, 192	fovrnd 6806	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. { 0 } ) /\ y e. ran F ) -> ( y I z ) e. RR )
194	193	recnd 10068	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. { 0 } ) /\ y e. ran F ) -> ( y I z ) e. CC )
195	194	mul02d 10234	. . . . . . . . . 10 |- ( ( ( ph /\ z e. { 0 } ) /\ y e. ran F ) -> ( 0 x. ( y I z ) ) = 0 )
196	189, 195	eqtrd 2656	. . . . . . . . 9 |- ( ( ( ph /\ z e. { 0 } ) /\ y e. ran F ) -> ( z x. ( y I z ) ) = 0 )
197	196	sumeq2dv 14433	. . . . . . . 8 |- ( ( ph /\ z e. { 0 } ) -> sum_ y e. ran F ( z x. ( y I z ) ) = sum_ y e. ran F 0 )
198	3	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ z e. { 0 } ) -> ran F e. Fin )
199	198	olcd 408	. . . . . . . . 9 |- ( ( ph /\ z e. { 0 } ) -> ( ran F C_ ( ZZ>= ` 0 ) \/ ran F e. Fin ) )
200		sumz 14453	. . . . . . . . 9 |- ( ( ran F C_ ( ZZ>= ` 0 ) \/ ran F e. Fin ) -> sum_ y e. ran F 0 = 0 )
201	199, 200	syl 17	. . . . . . . 8 |- ( ( ph /\ z e. { 0 } ) -> sum_ y e. ran F 0 = 0 )
202	197, 201	eqtrd 2656	. . . . . . 7 |- ( ( ph /\ z e. { 0 } ) -> sum_ y e. ran F ( z x. ( y I z ) ) = 0 )
203	186, 202	sylan2 491	. . . . . 6 |- ( ( ph /\ z e. ( ran G \ ( ran G \ { 0 } ) ) ) -> sum_ y e. ran F ( z x. ( y I z ) ) = 0 )
204	179, 182, 203, 6	fsumss 14456	. . . . 5 |- ( ph -> sum_ z e. ( ran G \ { 0 } ) sum_ y e. ran F ( z x. ( y I z ) ) = sum_ z e. ran G sum_ y e. ran F ( z x. ( y I z ) ) )
205	22	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran F ) -> z e. RR )
206	205	recnd 10068	. . . . . . . 8 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran F ) -> z e. CC )
207	16	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran F ) -> I : ( RR X. RR ) --> RR )
208	11	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ z e. ran G ) -> ran F C_ RR )
209	208	sselda 3603	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran F ) -> y e. RR )
210	207, 209, 205	fovrnd 6806	. . . . . . . . 9 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran F ) -> ( y I z ) e. RR )
211	210	recnd 10068	. . . . . . . 8 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran F ) -> ( y I z ) e. CC )
212	206, 211	mulcld 10060	. . . . . . 7 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran F ) -> ( z x. ( y I z ) ) e. CC )
213	212	anasss 679	. . . . . 6 |- ( ( ph /\ ( z e. ran G /\ y e. ran F ) ) -> ( z x. ( y I z ) ) e. CC )
214	6, 3, 213	fsumcom 14507	. . . . 5 |- ( ph -> sum_ z e. ran G sum_ y e. ran F ( z x. ( y I z ) ) = sum_ y e. ran F sum_ z e. ran G ( z x. ( y I z ) ) )
215	204, 214	eqtrd 2656	. . . 4 |- ( ph -> sum_ z e. ( ran G \ { 0 } ) sum_ y e. ran F ( z x. ( y I z ) ) = sum_ y e. ran F sum_ z e. ran G ( z x. ( y I z ) ) )
216	125, 178, 215	3eqtrd 2660	. . 3 |- ( ph -> ( S.1 ` G ) = sum_ y e. ran F sum_ z e. ran G ( z x. ( y I z ) ) )
217	123, 216	oveq12d 6668	. 2 |- ( ph -> ( ( S.1 ` F ) + ( S.1 ` G ) ) = ( sum_ y e. ran F sum_ z e. ran G ( y x. ( y I z ) ) + sum_ y e. ran F sum_ z e. ran G ( z x. ( y I z ) ) ) )
218	31, 39, 217	3eqtr4d 2666	1 |- ( ph -> ( S.1 ` ( F oF + G ) ) = ( ( S.1 ` F ) + ( S.1 ` G ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   i^i cin 3573   C_ wss 3574   ifcif 4086   {csn 4177   U_ciun 4520   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   oFcof 6895   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   ZZ>=cuz 11687   sum_csu 14416   volcvol 23232   S.1citg1 23384

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  ax-addf 10015

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-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-xadd 11947  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-sum 14417  df-xmet 19739  df-met 19740  df-ovol 23233  df-vol 23234  df-mbf 23388  df-itg1 23389

This theorem is referenced by:  itg1add  23468