Theorem itg1addlem4 23466

Description: Lemma for itg1add . (Contributed by Mario Carneiro, 28-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
itg1addlem4	|- ( ph -> ( S.1 ` ( F oF + G ) ) = sum_ y e. ran F sum_ z e. ran G ( ( y + z ) x. ( y I z ) ) )

Distinct variable groups:   i, j, y, z   y, I   y, P, z   i, F, j, y, z   i, G, j, y, z   ph, i, j, y, z

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

Proof of Theorem itg1addlem4

Dummy variables w v x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		i1fadd.1	. . . . 5 |- ( ph -> F e. dom S.1 )
2		i1fadd.2	. . . . 5 |- ( ph -> G e. dom S.1 )
3	1, 2	i1fadd 23462	. . . 4 |- ( ph -> ( F oF + G ) e. dom S.1 )
4		i1frn 23444	. . . . . . . 8 |- ( F e. dom S.1 -> ran F e. Fin )
5	1, 4	syl 17	. . . . . . 7 |- ( ph -> ran F e. Fin )
6		i1frn 23444	. . . . . . . 8 |- ( G e. dom S.1 -> ran G e. Fin )
7	2, 6	syl 17	. . . . . . 7 |- ( ph -> ran G e. Fin )
8		xpfi 8231	. . . . . . 7 |- ( ( ran F e. Fin /\ ran G e. Fin ) -> ( ran F X. ran G ) e. Fin )
9	5, 7, 8	syl2anc 693	. . . . . 6 |- ( ph -> ( ran F X. ran G ) e. Fin )
10		ax-addf 10015	. . . . . . . . . 10 |- + : ( CC X. CC ) --> CC
11		ffn 6045	. . . . . . . . . 10 |- ( + : ( CC X. CC ) --> CC -> + Fn ( CC X. CC ) )
12	10, 11	ax-mp 5	. . . . . . . . 9 |- + Fn ( CC X. CC )
13		i1ff 23443	. . . . . . . . . . . . 13 |- ( F e. dom S.1 -> F : RR --> RR )
14	1, 13	syl 17	. . . . . . . . . . . 12 |- ( ph -> F : RR --> RR )
15		frn 6053	. . . . . . . . . . . 12 |- ( F : RR --> RR -> ran F C_ RR )
16	14, 15	syl 17	. . . . . . . . . . 11 |- ( ph -> ran F C_ RR )
17		ax-resscn 9993	. . . . . . . . . . 11 |- RR C_ CC
18	16, 17	syl6ss 3615	. . . . . . . . . 10 |- ( ph -> ran F C_ CC )
19		i1ff 23443	. . . . . . . . . . . . 13 |- ( G e. dom S.1 -> G : RR --> RR )
20	2, 19	syl 17	. . . . . . . . . . . 12 |- ( ph -> G : RR --> RR )
21		frn 6053	. . . . . . . . . . . 12 |- ( G : RR --> RR -> ran G C_ RR )
22	20, 21	syl 17	. . . . . . . . . . 11 |- ( ph -> ran G C_ RR )
23	22, 17	syl6ss 3615	. . . . . . . . . 10 |- ( ph -> ran G C_ CC )
24		xpss12 5225	. . . . . . . . . 10 |- ( ( ran F C_ CC /\ ran G C_ CC ) -> ( ran F X. ran G ) C_ ( CC X. CC ) )
25	18, 23, 24	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( ran F X. ran G ) C_ ( CC X. CC ) )
26		fnssres 6004	. . . . . . . . 9 |- ( ( + Fn ( CC X. CC ) /\ ( ran F X. ran G ) C_ ( CC X. CC ) ) -> ( + |` ( ran F X. ran G ) ) Fn ( ran F X. ran G ) )
27	12, 25, 26	sylancr 695	. . . . . . . 8 |- ( ph -> ( + |` ( ran F X. ran G ) ) Fn ( ran F X. ran G ) )
28		itg1add.4	. . . . . . . . 9 |- P = ( + |` ( ran F X. ran G ) )
29	28	fneq1i 5985	. . . . . . . 8 |- ( P Fn ( ran F X. ran G ) <-> ( + |` ( ran F X. ran G ) ) Fn ( ran F X. ran G ) )
30	27, 29	sylibr 224	. . . . . . 7 |- ( ph -> P Fn ( ran F X. ran G ) )
31		dffn4 6121	. . . . . . 7 |- ( P Fn ( ran F X. ran G ) <-> P : ( ran F X. ran G ) -onto-> ran P )
32	30, 31	sylib 208	. . . . . 6 |- ( ph -> P : ( ran F X. ran G ) -onto-> ran P )
33		fofi 8252	. . . . . 6 |- ( ( ( ran F X. ran G ) e. Fin /\ P : ( ran F X. ran G ) -onto-> ran P ) -> ran P e. Fin )
34	9, 32, 33	syl2anc 693	. . . . 5 |- ( ph -> ran P e. Fin )
35		difss 3737	. . . . 5 |- ( ran P \ { 0 } ) C_ ran P
36		ssfi 8180	. . . . 5 |- ( ( ran P e. Fin /\ ( ran P \ { 0 } ) C_ ran P ) -> ( ran P \ { 0 } ) e. Fin )
37	34, 35, 36	sylancl 694	. . . 4 |- ( ph -> ( ran P \ { 0 } ) e. Fin )
38		opelxpi 5148	. . . . . . . . . 10 |- ( ( x e. ran F /\ y e. ran G ) -> <. x , y >. e. ( ran F X. ran G ) )
39		ffun 6048	. . . . . . . . . . . 12 |- ( + : ( CC X. CC ) --> CC -> Fun + )
40	10, 39	ax-mp 5	. . . . . . . . . . 11 |- Fun +
41	10	fdmi 6052	. . . . . . . . . . . 12 |- dom + = ( CC X. CC )
42	25, 41	syl6sseqr 3652	. . . . . . . . . . 11 |- ( ph -> ( ran F X. ran G ) C_ dom + )
43		funfvima2 6493	. . . . . . . . . . 11 |- ( ( Fun + /\ ( ran F X. ran G ) C_ dom + ) -> ( <. x , y >. e. ( ran F X. ran G ) -> ( + ` <. x , y >. ) e. ( + " ( ran F X. ran G ) ) ) )
44	40, 42, 43	sylancr 695	. . . . . . . . . 10 |- ( ph -> ( <. x , y >. e. ( ran F X. ran G ) -> ( + ` <. x , y >. ) e. ( + " ( ran F X. ran G ) ) ) )
45	38, 44	syl5 34	. . . . . . . . 9 |- ( ph -> ( ( x e. ran F /\ y e. ran G ) -> ( + ` <. x , y >. ) e. ( + " ( ran F X. ran G ) ) ) )
46	45	imp 445	. . . . . . . 8 |- ( ( ph /\ ( x e. ran F /\ y e. ran G ) ) -> ( + ` <. x , y >. ) e. ( + " ( ran F X. ran G ) ) )
47		df-ov 6653	. . . . . . . 8 |- ( x + y ) = ( + ` <. x , y >. )
48	28	rneqi 5352	. . . . . . . . 9 |- ran P = ran ( + |` ( ran F X. ran G ) )
49		df-ima 5127	. . . . . . . . 9 |- ( + " ( ran F X. ran G ) ) = ran ( + |` ( ran F X. ran G ) )
50	48, 49	eqtr4i 2647	. . . . . . . 8 |- ran P = ( + " ( ran F X. ran G ) )
51	46, 47, 50	3eltr4g 2718	. . . . . . 7 |- ( ( ph /\ ( x e. ran F /\ y e. ran G ) ) -> ( x + y ) e. ran P )
52		ffn 6045	. . . . . . . . 9 |- ( F : RR --> RR -> F Fn RR )
53	14, 52	syl 17	. . . . . . . 8 |- ( ph -> F Fn RR )
54		dffn3 6054	. . . . . . . 8 |- ( F Fn RR <-> F : RR --> ran F )
55	53, 54	sylib 208	. . . . . . 7 |- ( ph -> F : RR --> ran F )
56		ffn 6045	. . . . . . . . 9 |- ( G : RR --> RR -> G Fn RR )
57	20, 56	syl 17	. . . . . . . 8 |- ( ph -> G Fn RR )
58		dffn3 6054	. . . . . . . 8 |- ( G Fn RR <-> G : RR --> ran G )
59	57, 58	sylib 208	. . . . . . 7 |- ( ph -> G : RR --> ran G )
60		reex 10027	. . . . . . . 8 |- RR e. _V
61	60	a1i 11	. . . . . . 7 |- ( ph -> RR e. _V )
62		inidm 3822	. . . . . . 7 |- ( RR i^i RR ) = RR
63	51, 55, 59, 61, 61, 62	off 6912	. . . . . 6 |- ( ph -> ( F oF + G ) : RR --> ran P )
64		frn 6053	. . . . . 6 |- ( ( F oF + G ) : RR --> ran P -> ran ( F oF + G ) C_ ran P )
65	63, 64	syl 17	. . . . 5 |- ( ph -> ran ( F oF + G ) C_ ran P )
66	65	ssdifd 3746	. . . 4 |- ( ph -> ( ran ( F oF + G ) \ { 0 } ) C_ ( ran P \ { 0 } ) )
67	16	sselda 3603	. . . . . . . . . 10 |- ( ( ph /\ y e. ran F ) -> y e. RR )
68	22	sselda 3603	. . . . . . . . . 10 |- ( ( ph /\ z e. ran G ) -> z e. RR )
69	67, 68	anim12dan 882	. . . . . . . . 9 |- ( ( ph /\ ( y e. ran F /\ z e. ran G ) ) -> ( y e. RR /\ z e. RR ) )
70		readdcl 10019	. . . . . . . . 9 |- ( ( y e. RR /\ z e. RR ) -> ( y + z ) e. RR )
71	69, 70	syl 17	. . . . . . . 8 |- ( ( ph /\ ( y e. ran F /\ z e. ran G ) ) -> ( y + z ) e. RR )
72	71	ralrimivva 2971	. . . . . . 7 |- ( ph -> A. y e. ran F A. z e. ran G ( y + z ) e. RR )
73		funimassov 6811	. . . . . . . 8 |- ( ( Fun + /\ ( ran F X. ran G ) C_ dom + ) -> ( ( + " ( ran F X. ran G ) ) C_ RR <-> A. y e. ran F A. z e. ran G ( y + z ) e. RR ) )
74	40, 42, 73	sylancr 695	. . . . . . 7 |- ( ph -> ( ( + " ( ran F X. ran G ) ) C_ RR <-> A. y e. ran F A. z e. ran G ( y + z ) e. RR ) )
75	72, 74	mpbird 247	. . . . . 6 |- ( ph -> ( + " ( ran F X. ran G ) ) C_ RR )
76	50, 75	syl5eqss 3649	. . . . 5 |- ( ph -> ran P C_ RR )
77	76	ssdifd 3746	. . . 4 |- ( ph -> ( ran P \ { 0 } ) C_ ( RR \ { 0 } ) )
78		itg1val2 23451	. . . 4 |- ( ( ( F oF + G ) e. dom S.1 /\ ( ( ran P \ { 0 } ) e. Fin /\ ( ran ( F oF + G ) \ { 0 } ) C_ ( ran P \ { 0 } ) /\ ( ran P \ { 0 } ) C_ ( RR \ { 0 } ) ) ) -> ( S.1 ` ( F oF + G ) ) = sum_ w e. ( ran P \ { 0 } ) ( w x. ( vol ` ( `' ( F oF + G ) " { w } ) ) ) )
79	3, 37, 66, 77, 78	syl13anc 1328	. . 3 |- ( ph -> ( S.1 ` ( F oF + G ) ) = sum_ w e. ( ran P \ { 0 } ) ( w x. ( vol ` ( `' ( F oF + G ) " { w } ) ) ) )
80	20	adantr 481	. . . . . . . 8 |- ( ( ph /\ w e. ( ran P \ { 0 } ) ) -> G : RR --> RR )
81	7	adantr 481	. . . . . . . 8 |- ( ( ph /\ w e. ( ran P \ { 0 } ) ) -> ran G e. Fin )
82		inss2 3834	. . . . . . . . 9 |- ( ( `' F " { ( w - z ) } ) i^i ( `' G " { z } ) ) C_ ( `' G " { z } )
83	82	a1i 11	. . . . . . . 8 |- ( ( ( ph /\ w e. ( ran P \ { 0 } ) ) /\ z e. ran G ) -> ( ( `' F " { ( w - z ) } ) i^i ( `' G " { z } ) ) C_ ( `' G " { z } ) )
84		i1fima 23445	. . . . . . . . . . 11 |- ( F e. dom S.1 -> ( `' F " { ( w - z ) } ) e. dom vol )
85	1, 84	syl 17	. . . . . . . . . 10 |- ( ph -> ( `' F " { ( w - z ) } ) e. dom vol )
86		i1fima 23445	. . . . . . . . . . 11 |- ( G e. dom S.1 -> ( `' G " { z } ) e. dom vol )
87	2, 86	syl 17	. . . . . . . . . 10 |- ( ph -> ( `' G " { z } ) e. dom vol )
88		inmbl 23310	. . . . . . . . . 10 |- ( ( ( `' F " { ( w - z ) } ) e. dom vol /\ ( `' G " { z } ) e. dom vol ) -> ( ( `' F " { ( w - z ) } ) i^i ( `' G " { z } ) ) e. dom vol )
89	85, 87, 88	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( ( `' F " { ( w - z ) } ) i^i ( `' G " { z } ) ) e. dom vol )
90	89	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ w e. ( ran P \ { 0 } ) ) /\ z e. ran G ) -> ( ( `' F " { ( w - z ) } ) i^i ( `' G " { z } ) ) e. dom vol )
91	35, 76	syl5ss 3614	. . . . . . . . . . . . 13 |- ( ph -> ( ran P \ { 0 } ) C_ RR )
92	91	sselda 3603	. . . . . . . . . . . 12 |- ( ( ph /\ w e. ( ran P \ { 0 } ) ) -> w e. RR )
93	92	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ w e. ( ran P \ { 0 } ) ) /\ z e. ran G ) -> w e. RR )
94	68	adantlr 751	. . . . . . . . . . 11 |- ( ( ( ph /\ w e. ( ran P \ { 0 } ) ) /\ z e. ran G ) -> z e. RR )
95	93, 94	resubcld 10458	. . . . . . . . . 10 |- ( ( ( ph /\ w e. ( ran P \ { 0 } ) ) /\ z e. ran G ) -> ( w - z ) e. RR )
96	93	recnd 10068	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w e. ( ran P \ { 0 } ) ) /\ z e. ran G ) -> w e. CC )
97	94	recnd 10068	. . . . . . . . . . . . 13 |- ( ( ( ph /\ w e. ( ran P \ { 0 } ) ) /\ z e. ran G ) -> z e. CC )
98	96, 97	npcand 10396	. . . . . . . . . . . 12 |- ( ( ( ph /\ w e. ( ran P \ { 0 } ) ) /\ z e. ran G ) -> ( ( w - z ) + z ) = w )
99		eldifsni 4320	. . . . . . . . . . . . 13 |- ( w e. ( ran P \ { 0 } ) -> w =/= 0 )
100	99	ad2antlr 763	. . . . . . . . . . . 12 |- ( ( ( ph /\ w e. ( ran P \ { 0 } ) ) /\ z e. ran G ) -> w =/= 0 )
101	98, 100	eqnetrd 2861	. . . . . . . . . . 11 |- ( ( ( ph /\ w e. ( ran P \ { 0 } ) ) /\ z e. ran G ) -> ( ( w - z ) + z ) =/= 0 )
102		oveq12 6659	. . . . . . . . . . . . 13 |- ( ( ( w - z ) = 0 /\ z = 0 ) -> ( ( w - z ) + z ) = ( 0 + 0 ) )
103		00id 10211	. . . . . . . . . . . . 13 |- ( 0 + 0 ) = 0
104	102, 103	syl6eq 2672	. . . . . . . . . . . 12 |- ( ( ( w - z ) = 0 /\ z = 0 ) -> ( ( w - z ) + z ) = 0 )
105	104	necon3ai 2819	. . . . . . . . . . 11 |- ( ( ( w - z ) + z ) =/= 0 -> -. ( ( w - z ) = 0 /\ z = 0 ) )
106	101, 105	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ w e. ( ran P \ { 0 } ) ) /\ z e. ran G ) -> -. ( ( w - z ) = 0 /\ z = 0 ) )
107		itg1add.3	. . . . . . . . . . 11 |- I = ( i e. RR , j e. RR |-> if ( ( i = 0 /\ j = 0 ) , 0 , ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) ) )
108	1, 2, 107	itg1addlem3 23465	. . . . . . . . . 10 |- ( ( ( ( w - z ) e. RR /\ z e. RR ) /\ -. ( ( w - z ) = 0 /\ z = 0 ) ) -> ( ( w - z ) I z ) = ( vol ` ( ( `' F " { ( w - z ) } ) i^i ( `' G " { z } ) ) ) )
109	95, 94, 106, 108	syl21anc 1325	. . . . . . . . 9 |- ( ( ( ph /\ w e. ( ran P \ { 0 } ) ) /\ z e. ran G ) -> ( ( w - z ) I z ) = ( vol ` ( ( `' F " { ( w - z ) } ) i^i ( `' G " { z } ) ) ) )
110	1, 2, 107	itg1addlem2 23464	. . . . . . . . . . 11 |- ( ph -> I : ( RR X. RR ) --> RR )
111	110	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ w e. ( ran P \ { 0 } ) ) /\ z e. ran G ) -> I : ( RR X. RR ) --> RR )
112	111, 95, 94	fovrnd 6806	. . . . . . . . 9 |- ( ( ( ph /\ w e. ( ran P \ { 0 } ) ) /\ z e. ran G ) -> ( ( w - z ) I z ) e. RR )
113	109, 112	eqeltrrd 2702	. . . . . . . 8 |- ( ( ( ph /\ w e. ( ran P \ { 0 } ) ) /\ z e. ran G ) -> ( vol ` ( ( `' F " { ( w - z ) } ) i^i ( `' G " { z } ) ) ) e. RR )
114	80, 81, 83, 90, 113	itg1addlem1 23459	. . . . . . 7 |- ( ( ph /\ w e. ( ran P \ { 0 } ) ) -> ( vol ` U_ z e. ran G ( ( `' F " { ( w - z ) } ) i^i ( `' G " { z } ) ) ) = sum_ z e. ran G ( vol ` ( ( `' F " { ( w - z ) } ) i^i ( `' G " { z } ) ) ) )
115	92	recnd 10068	. . . . . . . . 9 |- ( ( ph /\ w e. ( ran P \ { 0 } ) ) -> w e. CC )
116	1, 2	i1faddlem 23460	. . . . . . . . 9 |- ( ( ph /\ w e. CC ) -> ( `' ( F oF + G ) " { w } ) = U_ z e. ran G ( ( `' F " { ( w - z ) } ) i^i ( `' G " { z } ) ) )
117	115, 116	syldan 487	. . . . . . . 8 |- ( ( ph /\ w e. ( ran P \ { 0 } ) ) -> ( `' ( F oF + G ) " { w } ) = U_ z e. ran G ( ( `' F " { ( w - z ) } ) i^i ( `' G " { z } ) ) )
118	117	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ w e. ( ran P \ { 0 } ) ) -> ( vol ` ( `' ( F oF + G ) " { w } ) ) = ( vol ` U_ z e. ran G ( ( `' F " { ( w - z ) } ) i^i ( `' G " { z } ) ) ) )
119	109	sumeq2dv 14433	. . . . . . 7 |- ( ( ph /\ w e. ( ran P \ { 0 } ) ) -> sum_ z e. ran G ( ( w - z ) I z ) = sum_ z e. ran G ( vol ` ( ( `' F " { ( w - z ) } ) i^i ( `' G " { z } ) ) ) )
120	114, 118, 119	3eqtr4d 2666	. . . . . 6 |- ( ( ph /\ w e. ( ran P \ { 0 } ) ) -> ( vol ` ( `' ( F oF + G ) " { w } ) ) = sum_ z e. ran G ( ( w - z ) I z ) )
121	120	oveq2d 6666	. . . . 5 |- ( ( ph /\ w e. ( ran P \ { 0 } ) ) -> ( w x. ( vol ` ( `' ( F oF + G ) " { w } ) ) ) = ( w x. sum_ z e. ran G ( ( w - z ) I z ) ) )
122	112	recnd 10068	. . . . . 6 |- ( ( ( ph /\ w e. ( ran P \ { 0 } ) ) /\ z e. ran G ) -> ( ( w - z ) I z ) e. CC )
123	81, 115, 122	fsummulc2 14516	. . . . 5 |- ( ( ph /\ w e. ( ran P \ { 0 } ) ) -> ( w x. sum_ z e. ran G ( ( w - z ) I z ) ) = sum_ z e. ran G ( w x. ( ( w - z ) I z ) ) )
124	121, 123	eqtrd 2656	. . . 4 |- ( ( ph /\ w e. ( ran P \ { 0 } ) ) -> ( w x. ( vol ` ( `' ( F oF + G ) " { w } ) ) ) = sum_ z e. ran G ( w x. ( ( w - z ) I z ) ) )
125	124	sumeq2dv 14433	. . 3 |- ( ph -> sum_ w e. ( ran P \ { 0 } ) ( w x. ( vol ` ( `' ( F oF + G ) " { w } ) ) ) = sum_ w e. ( ran P \ { 0 } ) sum_ z e. ran G ( w x. ( ( w - z ) I z ) ) )
126	96, 122	mulcld 10060	. . . . 5 |- ( ( ( ph /\ w e. ( ran P \ { 0 } ) ) /\ z e. ran G ) -> ( w x. ( ( w - z ) I z ) ) e. CC )
127	126	anasss 679	. . . 4 |- ( ( ph /\ ( w e. ( ran P \ { 0 } ) /\ z e. ran G ) ) -> ( w x. ( ( w - z ) I z ) ) e. CC )
128	37, 7, 127	fsumcom 14507	. . 3 |- ( ph -> sum_ w e. ( ran P \ { 0 } ) sum_ z e. ran G ( w x. ( ( w - z ) I z ) ) = sum_ z e. ran G sum_ w e. ( ran P \ { 0 } ) ( w x. ( ( w - z ) I z ) ) )
129	79, 125, 128	3eqtrd 2660	. 2 |- ( ph -> ( S.1 ` ( F oF + G ) ) = sum_ z e. ran G sum_ w e. ( ran P \ { 0 } ) ( w x. ( ( w - z ) I z ) ) )
130		oveq1 6657	. . . . . . 7 |- ( y = ( w - z ) -> ( y + z ) = ( ( w - z ) + z ) )
131		oveq1 6657	. . . . . . 7 |- ( y = ( w - z ) -> ( y I z ) = ( ( w - z ) I z ) )
132	130, 131	oveq12d 6668	. . . . . 6 |- ( y = ( w - z ) -> ( ( y + z ) x. ( y I z ) ) = ( ( ( w - z ) + z ) x. ( ( w - z ) I z ) ) )
133	34	adantr 481	. . . . . 6 |- ( ( ph /\ z e. ran G ) -> ran P e. Fin )
134	76	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ z e. ran G ) -> ran P C_ RR )
135	134	sselda 3603	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ran G ) /\ v e. ran P ) -> v e. RR )
136	68	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ran G ) /\ v e. ran P ) -> z e. RR )
137	135, 136	resubcld 10458	. . . . . . . . 9 |- ( ( ( ph /\ z e. ran G ) /\ v e. ran P ) -> ( v - z ) e. RR )
138	137	ex 450	. . . . . . . 8 |- ( ( ph /\ z e. ran G ) -> ( v e. ran P -> ( v - z ) e. RR ) )
139	135	recnd 10068	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ran G ) /\ v e. ran P ) -> v e. CC )
140	139	adantrr 753	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ran G ) /\ ( v e. ran P /\ y e. ran P ) ) -> v e. CC )
141	76	sselda 3603	. . . . . . . . . . . 12 |- ( ( ph /\ y e. ran P ) -> y e. RR )
142	141	ad2ant2rl 785	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ran G ) /\ ( v e. ran P /\ y e. ran P ) ) -> y e. RR )
143	142	recnd 10068	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ran G ) /\ ( v e. ran P /\ y e. ran P ) ) -> y e. CC )
144	68	recnd 10068	. . . . . . . . . . 11 |- ( ( ph /\ z e. ran G ) -> z e. CC )
145	144	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ran G ) /\ ( v e. ran P /\ y e. ran P ) ) -> z e. CC )
146	140, 143, 145	subcan2ad 10437	. . . . . . . . 9 |- ( ( ( ph /\ z e. ran G ) /\ ( v e. ran P /\ y e. ran P ) ) -> ( ( v - z ) = ( y - z ) <-> v = y ) )
147	146	ex 450	. . . . . . . 8 |- ( ( ph /\ z e. ran G ) -> ( ( v e. ran P /\ y e. ran P ) -> ( ( v - z ) = ( y - z ) <-> v = y ) ) )
148	138, 147	dom2lem 7995	. . . . . . 7 |- ( ( ph /\ z e. ran G ) -> ( v e. ran P |-> ( v - z ) ) : ran P -1-1-> RR )
149		f1f1orn 6148	. . . . . . 7 |- ( ( v e. ran P |-> ( v - z ) ) : ran P -1-1-> RR -> ( v e. ran P |-> ( v - z ) ) : ran P -1-1-onto-> ran ( v e. ran P |-> ( v - z ) ) )
150	148, 149	syl 17	. . . . . 6 |- ( ( ph /\ z e. ran G ) -> ( v e. ran P |-> ( v - z ) ) : ran P -1-1-onto-> ran ( v e. ran P |-> ( v - z ) ) )
151		oveq1 6657	. . . . . . . 8 |- ( v = w -> ( v - z ) = ( w - z ) )
152		eqid 2622	. . . . . . . 8 |- ( v e. ran P |-> ( v - z ) ) = ( v e. ran P |-> ( v - z ) )
153		ovex 6678	. . . . . . . 8 |- ( w - z ) e. _V
154	151, 152, 153	fvmpt 6282	. . . . . . 7 |- ( w e. ran P -> ( ( v e. ran P |-> ( v - z ) ) ` w ) = ( w - z ) )
155	154	adantl 482	. . . . . 6 |- ( ( ( ph /\ z e. ran G ) /\ w e. ran P ) -> ( ( v e. ran P |-> ( v - z ) ) ` w ) = ( w - z ) )
156		f1f 6101	. . . . . . . . . . 11 |- ( ( v e. ran P |-> ( v - z ) ) : ran P -1-1-> RR -> ( v e. ran P |-> ( v - z ) ) : ran P --> RR )
157		frn 6053	. . . . . . . . . . 11 |- ( ( v e. ran P |-> ( v - z ) ) : ran P --> RR -> ran ( v e. ran P |-> ( v - z ) ) C_ RR )
158	148, 156, 157	3syl 18	. . . . . . . . . 10 |- ( ( ph /\ z e. ran G ) -> ran ( v e. ran P |-> ( v - z ) ) C_ RR )
159	158	sselda 3603	. . . . . . . . 9 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran ( v e. ran P |-> ( v - z ) ) ) -> y e. RR )
160	68	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran ( v e. ran P |-> ( v - z ) ) ) -> z e. RR )
161	159, 160	readdcld 10069	. . . . . . . 8 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran ( v e. ran P |-> ( v - z ) ) ) -> ( y + z ) e. RR )
162	110	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran ( v e. ran P |-> ( v - z ) ) ) -> I : ( RR X. RR ) --> RR )
163	162, 159, 160	fovrnd 6806	. . . . . . . 8 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran ( v e. ran P |-> ( v - z ) ) ) -> ( y I z ) e. RR )
164	161, 163	remulcld 10070	. . . . . . 7 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran ( v e. ran P |-> ( v - z ) ) ) -> ( ( y + z ) x. ( y I z ) ) e. RR )
165	164	recnd 10068	. . . . . 6 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran ( v e. ran P |-> ( v - z ) ) ) -> ( ( y + z ) x. ( y I z ) ) e. CC )
166	132, 133, 150, 155, 165	fsumf1o 14454	. . . . 5 |- ( ( ph /\ z e. ran G ) -> sum_ y e. ran ( v e. ran P |-> ( v - z ) ) ( ( y + z ) x. ( y I z ) ) = sum_ w e. ran P ( ( ( w - z ) + z ) x. ( ( w - z ) I z ) ) )
167	134	sselda 3603	. . . . . . . . 9 |- ( ( ( ph /\ z e. ran G ) /\ w e. ran P ) -> w e. RR )
168	167	recnd 10068	. . . . . . . 8 |- ( ( ( ph /\ z e. ran G ) /\ w e. ran P ) -> w e. CC )
169	144	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ z e. ran G ) /\ w e. ran P ) -> z e. CC )
170	168, 169	npcand 10396	. . . . . . 7 |- ( ( ( ph /\ z e. ran G ) /\ w e. ran P ) -> ( ( w - z ) + z ) = w )
171	170	oveq1d 6665	. . . . . 6 |- ( ( ( ph /\ z e. ran G ) /\ w e. ran P ) -> ( ( ( w - z ) + z ) x. ( ( w - z ) I z ) ) = ( w x. ( ( w - z ) I z ) ) )
172	171	sumeq2dv 14433	. . . . 5 |- ( ( ph /\ z e. ran G ) -> sum_ w e. ran P ( ( ( w - z ) + z ) x. ( ( w - z ) I z ) ) = sum_ w e. ran P ( w x. ( ( w - z ) I z ) ) )
173	166, 172	eqtrd 2656	. . . 4 |- ( ( ph /\ z e. ran G ) -> sum_ y e. ran ( v e. ran P |-> ( v - z ) ) ( ( y + z ) x. ( y I z ) ) = sum_ w e. ran P ( w x. ( ( w - z ) I z ) ) )
174	42	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran F ) -> ( ran F X. ran G ) C_ dom + )
175		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran F ) -> y e. ran F )
176		simplr 792	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran F ) -> z e. ran G )
177		opelxpi 5148	. . . . . . . . . . . 12 |- ( ( y e. ran F /\ z e. ran G ) -> <. y , z >. e. ( ran F X. ran G ) )
178	175, 176, 177	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran F ) -> <. y , z >. e. ( ran F X. ran G ) )
179		funfvima2 6493	. . . . . . . . . . . 12 |- ( ( Fun + /\ ( ran F X. ran G ) C_ dom + ) -> ( <. y , z >. e. ( ran F X. ran G ) -> ( + ` <. y , z >. ) e. ( + " ( ran F X. ran G ) ) ) )
180	40, 179	mpan 706	. . . . . . . . . . 11 |- ( ( ran F X. ran G ) C_ dom + -> ( <. y , z >. e. ( ran F X. ran G ) -> ( + ` <. y , z >. ) e. ( + " ( ran F X. ran G ) ) ) )
181	174, 178, 180	sylc 65	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran F ) -> ( + ` <. y , z >. ) e. ( + " ( ran F X. ran G ) ) )
182		df-ov 6653	. . . . . . . . . 10 |- ( y + z ) = ( + ` <. y , z >. )
183	181, 182, 50	3eltr4g 2718	. . . . . . . . 9 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran F ) -> ( y + z ) e. ran P )
184	67	adantlr 751	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran F ) -> y e. RR )
185	184	recnd 10068	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran F ) -> y e. CC )
186	144	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran F ) -> z e. CC )
187	185, 186	pncand 10393	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran F ) -> ( ( y + z ) - z ) = y )
188	187	eqcomd 2628	. . . . . . . . 9 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran F ) -> y = ( ( y + z ) - z ) )
189		oveq1 6657	. . . . . . . . . . 11 |- ( v = ( y + z ) -> ( v - z ) = ( ( y + z ) - z ) )
190	189	eqeq2d 2632	. . . . . . . . . 10 |- ( v = ( y + z ) -> ( y = ( v - z ) <-> y = ( ( y + z ) - z ) ) )
191	190	rspcev 3309	. . . . . . . . 9 |- ( ( ( y + z ) e. ran P /\ y = ( ( y + z ) - z ) ) -> E. v e. ran P y = ( v - z ) )
192	183, 188, 191	syl2anc 693	. . . . . . . 8 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran F ) -> E. v e. ran P y = ( v - z ) )
193	192	ralrimiva 2966	. . . . . . 7 |- ( ( ph /\ z e. ran G ) -> A. y e. ran F E. v e. ran P y = ( v - z ) )
194		ssabral 3673	. . . . . . 7 |- ( ran F C_ { y | E. v e. ran P y = ( v - z ) } <-> A. y e. ran F E. v e. ran P y = ( v - z ) )
195	193, 194	sylibr 224	. . . . . 6 |- ( ( ph /\ z e. ran G ) -> ran F C_ { y | E. v e. ran P y = ( v - z ) } )
196	152	rnmpt 5371	. . . . . 6 |- ran ( v e. ran P |-> ( v - z ) ) = { y | E. v e. ran P y = ( v - z ) }
197	195, 196	syl6sseqr 3652	. . . . 5 |- ( ( ph /\ z e. ran G ) -> ran F C_ ran ( v e. ran P |-> ( v - z ) ) )
198	68	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran F ) -> z e. RR )
199	184, 198	readdcld 10069	. . . . . . 7 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran F ) -> ( y + z ) e. RR )
200	110	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran F ) -> I : ( RR X. RR ) --> RR )
201	200, 184, 198	fovrnd 6806	. . . . . . 7 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran F ) -> ( y I z ) e. RR )
202	199, 201	remulcld 10070	. . . . . 6 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran F ) -> ( ( y + z ) x. ( y I z ) ) e. RR )
203	202	recnd 10068	. . . . 5 |- ( ( ( ph /\ z e. ran G ) /\ y e. ran F ) -> ( ( y + z ) x. ( y I z ) ) e. CC )
204	158	ssdifd 3746	. . . . . . 7 |- ( ( ph /\ z e. ran G ) -> ( ran ( v e. ran P |-> ( v - z ) ) \ ran F ) C_ ( RR \ ran F ) )
205	204	sselda 3603	. . . . . 6 |- ( ( ( ph /\ z e. ran G ) /\ y e. ( ran ( v e. ran P |-> ( v - z ) ) \ ran F ) ) -> y e. ( RR \ ran F ) )
206		eldifi 3732	. . . . . . . . . . . . 13 |- ( y e. ( RR \ ran F ) -> y e. RR )
207	206	ad2antrl 764	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. ran G ) /\ ( y e. ( RR \ ran F ) /\ -. ( y = 0 /\ z = 0 ) ) ) -> y e. RR )
208	68	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. ran G ) /\ ( y e. ( RR \ ran F ) /\ -. ( y = 0 /\ z = 0 ) ) ) -> z e. RR )
209		simprr 796	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. ran G ) /\ ( y e. ( RR \ ran F ) /\ -. ( y = 0 /\ z = 0 ) ) ) -> -. ( y = 0 /\ z = 0 ) )
210	1, 2, 107	itg1addlem3 23465	. . . . . . . . . . . 12 |- ( ( ( y e. RR /\ z e. RR ) /\ -. ( y = 0 /\ z = 0 ) ) -> ( y I z ) = ( vol ` ( ( `' F " { y } ) i^i ( `' G " { z } ) ) ) )
211	207, 208, 209, 210	syl21anc 1325	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ran G ) /\ ( y e. ( RR \ ran F ) /\ -. ( y = 0 /\ z = 0 ) ) ) -> ( y I z ) = ( vol ` ( ( `' F " { y } ) i^i ( `' G " { z } ) ) ) )
212		inss1 3833	. . . . . . . . . . . . . . 15 |- ( ( `' F " { y } ) i^i ( `' G " { z } ) ) C_ ( `' F " { y } )
213		eldifn 3733	. . . . . . . . . . . . . . . . . . 19 |- ( y e. ( RR \ ran F ) -> -. y e. ran F )
214	213	ad2antrl 764	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ z e. ran G ) /\ ( y e. ( RR \ ran F ) /\ -. ( y = 0 /\ z = 0 ) ) ) -> -. y e. ran F )
215		vex 3203	. . . . . . . . . . . . . . . . . . . 20 |- y e. _V
216		vex 3203	. . . . . . . . . . . . . . . . . . . . 21 |- v e. _V
217	216	eliniseg 5494	. . . . . . . . . . . . . . . . . . . 20 |- ( y e. _V -> ( v e. ( `' F " { y } ) <-> v F y ) )
218	215, 217	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 |- ( v e. ( `' F " { y } ) <-> v F y )
219	216, 215	brelrn 5356	. . . . . . . . . . . . . . . . . . 19 |- ( v F y -> y e. ran F )
220	218, 219	sylbi 207	. . . . . . . . . . . . . . . . . 18 |- ( v e. ( `' F " { y } ) -> y e. ran F )
221	214, 220	nsyl 135	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ z e. ran G ) /\ ( y e. ( RR \ ran F ) /\ -. ( y = 0 /\ z = 0 ) ) ) -> -. v e. ( `' F " { y } ) )
222	221	pm2.21d 118	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ z e. ran G ) /\ ( y e. ( RR \ ran F ) /\ -. ( y = 0 /\ z = 0 ) ) ) -> ( v e. ( `' F " { y } ) -> v e. (/) ) )
223	222	ssrdv 3609	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ z e. ran G ) /\ ( y e. ( RR \ ran F ) /\ -. ( y = 0 /\ z = 0 ) ) ) -> ( `' F " { y } ) C_ (/) )
224	212, 223	syl5ss 3614	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ z e. ran G ) /\ ( y e. ( RR \ ran F ) /\ -. ( y = 0 /\ z = 0 ) ) ) -> ( ( `' F " { y } ) i^i ( `' G " { z } ) ) C_ (/) )
225		ss0 3974	. . . . . . . . . . . . . 14 |- ( ( ( `' F " { y } ) i^i ( `' G " { z } ) ) C_ (/) -> ( ( `' F " { y } ) i^i ( `' G " { z } ) ) = (/) )
226	224, 225	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ph /\ z e. ran G ) /\ ( y e. ( RR \ ran F ) /\ -. ( y = 0 /\ z = 0 ) ) ) -> ( ( `' F " { y } ) i^i ( `' G " { z } ) ) = (/) )
227	226	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. ran G ) /\ ( y e. ( RR \ ran F ) /\ -. ( y = 0 /\ z = 0 ) ) ) -> ( vol ` ( ( `' F " { y } ) i^i ( `' G " { z } ) ) ) = ( vol ` (/) ) )
228		0mbl 23307	. . . . . . . . . . . . . 14 |- (/) e. dom vol
229		mblvol 23298	. . . . . . . . . . . . . 14 |- ( (/) e. dom vol -> ( vol ` (/) ) = ( vol* ` (/) ) )
230	228, 229	ax-mp 5	. . . . . . . . . . . . 13 |- ( vol ` (/) ) = ( vol* ` (/) )
231		ovol0 23261	. . . . . . . . . . . . 13 |- ( vol* ` (/) ) = 0
232	230, 231	eqtri 2644	. . . . . . . . . . . 12 |- ( vol ` (/) ) = 0
233	227, 232	syl6eq 2672	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ran G ) /\ ( y e. ( RR \ ran F ) /\ -. ( y = 0 /\ z = 0 ) ) ) -> ( vol ` ( ( `' F " { y } ) i^i ( `' G " { z } ) ) ) = 0 )
234	211, 233	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ran G ) /\ ( y e. ( RR \ ran F ) /\ -. ( y = 0 /\ z = 0 ) ) ) -> ( y I z ) = 0 )
235	234	oveq2d 6666	. . . . . . . . 9 |- ( ( ( ph /\ z e. ran G ) /\ ( y e. ( RR \ ran F ) /\ -. ( y = 0 /\ z = 0 ) ) ) -> ( ( y + z ) x. ( y I z ) ) = ( ( y + z ) x. 0 ) )
236	207, 208	readdcld 10069	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ran G ) /\ ( y e. ( RR \ ran F ) /\ -. ( y = 0 /\ z = 0 ) ) ) -> ( y + z ) e. RR )
237	236	recnd 10068	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ran G ) /\ ( y e. ( RR \ ran F ) /\ -. ( y = 0 /\ z = 0 ) ) ) -> ( y + z ) e. CC )
238	237	mul01d 10235	. . . . . . . . 9 |- ( ( ( ph /\ z e. ran G ) /\ ( y e. ( RR \ ran F ) /\ -. ( y = 0 /\ z = 0 ) ) ) -> ( ( y + z ) x. 0 ) = 0 )
239	235, 238	eqtrd 2656	. . . . . . . 8 |- ( ( ( ph /\ z e. ran G ) /\ ( y e. ( RR \ ran F ) /\ -. ( y = 0 /\ z = 0 ) ) ) -> ( ( y + z ) x. ( y I z ) ) = 0 )
240	239	expr 643	. . . . . . 7 |- ( ( ( ph /\ z e. ran G ) /\ y e. ( RR \ ran F ) ) -> ( -. ( y = 0 /\ z = 0 ) -> ( ( y + z ) x. ( y I z ) ) = 0 ) )
241		oveq12 6659	. . . . . . . . . 10 |- ( ( y = 0 /\ z = 0 ) -> ( y + z ) = ( 0 + 0 ) )
242	241, 103	syl6eq 2672	. . . . . . . . 9 |- ( ( y = 0 /\ z = 0 ) -> ( y + z ) = 0 )
243		oveq12 6659	. . . . . . . . . 10 |- ( ( y = 0 /\ z = 0 ) -> ( y I z ) = ( 0 I 0 ) )
244		0re 10040	. . . . . . . . . . 11 |- 0 e. RR
245		iftrue 4092	. . . . . . . . . . . 12 |- ( ( i = 0 /\ j = 0 ) -> if ( ( i = 0 /\ j = 0 ) , 0 , ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) ) = 0 )
246		c0ex 10034	. . . . . . . . . . . 12 |- 0 e. _V
247	245, 107, 246	ovmpt2a 6791	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ 0 e. RR ) -> ( 0 I 0 ) = 0 )
248	244, 244, 247	mp2an 708	. . . . . . . . . 10 |- ( 0 I 0 ) = 0
249	243, 248	syl6eq 2672	. . . . . . . . 9 |- ( ( y = 0 /\ z = 0 ) -> ( y I z ) = 0 )
250	242, 249	oveq12d 6668	. . . . . . . 8 |- ( ( y = 0 /\ z = 0 ) -> ( ( y + z ) x. ( y I z ) ) = ( 0 x. 0 ) )
251		0cn 10032	. . . . . . . . 9 |- 0 e. CC
252	251	mul01i 10226	. . . . . . . 8 |- ( 0 x. 0 ) = 0
253	250, 252	syl6eq 2672	. . . . . . 7 |- ( ( y = 0 /\ z = 0 ) -> ( ( y + z ) x. ( y I z ) ) = 0 )
254	240, 253	pm2.61d2 172	. . . . . 6 |- ( ( ( ph /\ z e. ran G ) /\ y e. ( RR \ ran F ) ) -> ( ( y + z ) x. ( y I z ) ) = 0 )
255	205, 254	syldan 487	. . . . 5 |- ( ( ( ph /\ z e. ran G ) /\ y e. ( ran ( v e. ran P |-> ( v - z ) ) \ ran F ) ) -> ( ( y + z ) x. ( y I z ) ) = 0 )
256		f1ofo 6144	. . . . . . 7 |- ( ( v e. ran P |-> ( v - z ) ) : ran P -1-1-onto-> ran ( v e. ran P |-> ( v - z ) ) -> ( v e. ran P |-> ( v - z ) ) : ran P -onto-> ran ( v e. ran P |-> ( v - z ) ) )
257	150, 256	syl 17	. . . . . 6 |- ( ( ph /\ z e. ran G ) -> ( v e. ran P |-> ( v - z ) ) : ran P -onto-> ran ( v e. ran P |-> ( v - z ) ) )
258		fofi 8252	. . . . . 6 |- ( ( ran P e. Fin /\ ( v e. ran P |-> ( v - z ) ) : ran P -onto-> ran ( v e. ran P |-> ( v - z ) ) ) -> ran ( v e. ran P |-> ( v - z ) ) e. Fin )
259	133, 257, 258	syl2anc 693	. . . . 5 |- ( ( ph /\ z e. ran G ) -> ran ( v e. ran P |-> ( v - z ) ) e. Fin )
260	197, 203, 255, 259	fsumss 14456	. . . 4 |- ( ( ph /\ z e. ran G ) -> sum_ y e. ran F ( ( y + z ) x. ( y I z ) ) = sum_ y e. ran ( v e. ran P |-> ( v - z ) ) ( ( y + z ) x. ( y I z ) ) )
261	35	a1i 11	. . . . 5 |- ( ( ph /\ z e. ran G ) -> ( ran P \ { 0 } ) C_ ran P )
262	126	an32s 846	. . . . 5 |- ( ( ( ph /\ z e. ran G ) /\ w e. ( ran P \ { 0 } ) ) -> ( w x. ( ( w - z ) I z ) ) e. CC )
263		dfin4 3867	. . . . . . . 8 |- ( ran P i^i { 0 } ) = ( ran P \ ( ran P \ { 0 } ) )
264		inss2 3834	. . . . . . . 8 |- ( ran P i^i { 0 } ) C_ { 0 }
265	263, 264	eqsstr3i 3636	. . . . . . 7 |- ( ran P \ ( ran P \ { 0 } ) ) C_ { 0 }
266	265	sseli 3599	. . . . . 6 |- ( w e. ( ran P \ ( ran P \ { 0 } ) ) -> w e. { 0 } )
267		elsni 4194	. . . . . . . . 9 |- ( w e. { 0 } -> w = 0 )
268	267	adantl 482	. . . . . . . 8 |- ( ( ( ph /\ z e. ran G ) /\ w e. { 0 } ) -> w = 0 )
269	268	oveq1d 6665	. . . . . . 7 |- ( ( ( ph /\ z e. ran G ) /\ w e. { 0 } ) -> ( w x. ( ( w - z ) I z ) ) = ( 0 x. ( ( w - z ) I z ) ) )
270	110	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ran G ) /\ w e. { 0 } ) -> I : ( RR X. RR ) --> RR )
271	268, 244	syl6eqel 2709	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ran G ) /\ w e. { 0 } ) -> w e. RR )
272	68	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ran G ) /\ w e. { 0 } ) -> z e. RR )
273	271, 272	resubcld 10458	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ran G ) /\ w e. { 0 } ) -> ( w - z ) e. RR )
274	270, 273, 272	fovrnd 6806	. . . . . . . . 9 |- ( ( ( ph /\ z e. ran G ) /\ w e. { 0 } ) -> ( ( w - z ) I z ) e. RR )
275	274	recnd 10068	. . . . . . . 8 |- ( ( ( ph /\ z e. ran G ) /\ w e. { 0 } ) -> ( ( w - z ) I z ) e. CC )
276	275	mul02d 10234	. . . . . . 7 |- ( ( ( ph /\ z e. ran G ) /\ w e. { 0 } ) -> ( 0 x. ( ( w - z ) I z ) ) = 0 )
277	269, 276	eqtrd 2656	. . . . . 6 |- ( ( ( ph /\ z e. ran G ) /\ w e. { 0 } ) -> ( w x. ( ( w - z ) I z ) ) = 0 )
278	266, 277	sylan2 491	. . . . 5 |- ( ( ( ph /\ z e. ran G ) /\ w e. ( ran P \ ( ran P \ { 0 } ) ) ) -> ( w x. ( ( w - z ) I z ) ) = 0 )
279	261, 262, 278, 133	fsumss 14456	. . . 4 |- ( ( ph /\ z e. ran G ) -> sum_ w e. ( ran P \ { 0 } ) ( w x. ( ( w - z ) I z ) ) = sum_ w e. ran P ( w x. ( ( w - z ) I z ) ) )
280	173, 260, 279	3eqtr4d 2666	. . 3 |- ( ( ph /\ z e. ran G ) -> sum_ y e. ran F ( ( y + z ) x. ( y I z ) ) = sum_ w e. ( ran P \ { 0 } ) ( w x. ( ( w - z ) I z ) ) )
281	280	sumeq2dv 14433	. 2 |- ( ph -> sum_ z e. ran G sum_ y e. ran F ( ( y + z ) x. ( y I z ) ) = sum_ z e. ran G sum_ w e. ( ran P \ { 0 } ) ( w x. ( ( w - z ) I z ) ) )
282	203	anasss 679	. . 3 |- ( ( ph /\ ( z e. ran G /\ y e. ran F ) ) -> ( ( y + z ) x. ( y I z ) ) e. CC )
283	7, 5, 282	fsumcom 14507	. 2 |- ( ph -> sum_ z e. ran G sum_ y e. ran F ( ( y + z ) x. ( y I z ) ) = sum_ y e. ran F sum_ z e. ran G ( ( y + z ) x. ( y I z ) ) )
284	129, 281, 283	3eqtr2d 2662	1 |- ( ph -> ( S.1 ` ( F oF + G ) ) = sum_ y e. ran F sum_ z e. ran G ( ( y + z ) x. ( y I z ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ifcif 4086   {csn 4177   <.cop 4183   U_ciun 4520   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   oFcof 6895   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   - cmin 10266   sum_csu 14416   vol*covol 23231   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:  itg1addlem5  23467