Theorem i1fadd 21195

Description: The sum of two simple functions is a simple function. (Contributed by Mario Carneiro, 18-Jun-2014.)

Hypotheses

Ref	Expression
i1fadd.1	|- ( ph -> F e. dom S.1 )
i1fadd.2	|- ( ph -> G e. dom S.1 )

Assertion

Ref	Expression
i1fadd	|- ( ph -> ( F oF + G ) e. dom S.1 )

Proof of Theorem i1fadd

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

Step	Hyp	Ref	Expression
1		readdcl 9386	. . . 4 |- ( ( x e. RR /\ y e. RR ) -> ( x + y ) e. RR )
2	1	adantl 466	. . 3 |- ( ( ph /\ ( x e. RR /\ y e. RR ) ) -> ( x + y ) e. RR )
3		i1fadd.1	. . . 4 |- ( ph -> F e. dom S.1 )
4		i1ff 21176	. . . 4 |- ( F e. dom S.1 -> F : RR --> RR )
5	3, 4	syl 16	. . 3 |- ( ph -> F : RR --> RR )
6		i1fadd.2	. . . 4 |- ( ph -> G e. dom S.1 )
7		i1ff 21176	. . . 4 |- ( G e. dom S.1 -> G : RR --> RR )
8	6, 7	syl 16	. . 3 |- ( ph -> G : RR --> RR )
9		reex 9394	. . . 4 |- RR e. _V
10	9	a1i 11	. . 3 |- ( ph -> RR e. _V )
11		inidm 3580	. . 3 |- ( RR i^i RR ) = RR
12	2, 5, 8, 10, 10, 11	off 6355	. 2 |- ( ph -> ( F oF + G ) : RR --> RR )
13		i1frn 21177	. . . . . 6 |- ( F e. dom S.1 -> ran F e. Fin )
14	3, 13	syl 16	. . . . 5 |- ( ph -> ran F e. Fin )
15		i1frn 21177	. . . . . 6 |- ( G e. dom S.1 -> ran G e. Fin )
16	6, 15	syl 16	. . . . 5 |- ( ph -> ran G e. Fin )
17		xpfi 7604	. . . . 5 |- ( ( ran F e. Fin /\ ran G e. Fin ) -> ( ran F X. ran G ) e. Fin )
18	14, 16, 17	syl2anc 661	. . . 4 |- ( ph -> ( ran F X. ran G ) e. Fin )
19		eqid 2443	. . . . . 6 |- ( u e. ran F , v e. ran G |-> ( u + v ) ) = ( u e. ran F , v e. ran G |-> ( u + v ) )
20		ovex 6137	. . . . . 6 |- ( u + v ) e. _V
21	19, 20	fnmpt2i 6664	. . . . 5 |- ( u e. ran F , v e. ran G |-> ( u + v ) ) Fn ( ran F X. ran G )
22		dffn4 5647	. . . . 5 |- ( ( u e. ran F , v e. ran G |-> ( u + v ) ) Fn ( ran F X. ran G ) <-> ( u e. ran F , v e. ran G |-> ( u + v ) ) : ( ran F X. ran G ) -onto-> ran ( u e. ran F , v e. ran G |-> ( u + v ) ) )
23	21, 22	mpbi 208	. . . 4 |- ( u e. ran F , v e. ran G |-> ( u + v ) ) : ( ran F X. ran G ) -onto-> ran ( u e. ran F , v e. ran G |-> ( u + v ) )
24		fofi 7618	. . . 4 |- ( ( ( ran F X. ran G ) e. Fin /\ ( u e. ran F , v e. ran G |-> ( u + v ) ) : ( ran F X. ran G ) -onto-> ran ( u e. ran F , v e. ran G |-> ( u + v ) ) ) -> ran ( u e. ran F , v e. ran G |-> ( u + v ) ) e. Fin )
25	18, 23, 24	sylancl 662	. . 3 |- ( ph -> ran ( u e. ran F , v e. ran G |-> ( u + v ) ) e. Fin )
26		eqid 2443	. . . . . . . . 9 |- ( x + y ) = ( x + y )
27		rspceov 6149	. . . . . . . . 9 |- ( ( x e. ran F /\ y e. ran G /\ ( x + y ) = ( x + y ) ) -> E. u e. ran F E. v e. ran G ( x + y ) = ( u + v ) )
28	26, 27	mp3an3 1303	. . . . . . . 8 |- ( ( x e. ran F /\ y e. ran G ) -> E. u e. ran F E. v e. ran G ( x + y ) = ( u + v ) )
29		ovex 6137	. . . . . . . . 9 |- ( x + y ) e. _V
30		eqeq1 2449	. . . . . . . . . 10 |- ( w = ( x + y ) -> ( w = ( u + v ) <-> ( x + y ) = ( u + v ) ) )
31	30	2rexbidv 2779	. . . . . . . . 9 |- ( w = ( x + y ) -> ( E. u e. ran F E. v e. ran G w = ( u + v ) <-> E. u e. ran F E. v e. ran G ( x + y ) = ( u + v ) ) )
32	29, 31	elab 3127	. . . . . . . 8 |- ( ( x + y ) e. { w | E. u e. ran F E. v e. ran G w = ( u + v ) } <-> E. u e. ran F E. v e. ran G ( x + y ) = ( u + v ) )
33	28, 32	sylibr 212	. . . . . . 7 |- ( ( x e. ran F /\ y e. ran G ) -> ( x + y ) e. { w | E. u e. ran F E. v e. ran G w = ( u + v ) } )
34	33	adantl 466	. . . . . 6 |- ( ( ph /\ ( x e. ran F /\ y e. ran G ) ) -> ( x + y ) e. { w | E. u e. ran F E. v e. ran G w = ( u + v ) } )
35		ffn 5580	. . . . . . . 8 |- ( F : RR --> RR -> F Fn RR )
36	5, 35	syl 16	. . . . . . 7 |- ( ph -> F Fn RR )
37		dffn3 5587	. . . . . . 7 |- ( F Fn RR <-> F : RR --> ran F )
38	36, 37	sylib 196	. . . . . 6 |- ( ph -> F : RR --> ran F )
39		ffn 5580	. . . . . . . 8 |- ( G : RR --> RR -> G Fn RR )
40	8, 39	syl 16	. . . . . . 7 |- ( ph -> G Fn RR )
41		dffn3 5587	. . . . . . 7 |- ( G Fn RR <-> G : RR --> ran G )
42	40, 41	sylib 196	. . . . . 6 |- ( ph -> G : RR --> ran G )
43	34, 38, 42, 10, 10, 11	off 6355	. . . . 5 |- ( ph -> ( F oF + G ) : RR --> { w | E. u e. ran F E. v e. ran G w = ( u + v ) } )
44		frn 5586	. . . . 5 |- ( ( F oF + G ) : RR --> { w | E. u e. ran F E. v e. ran G w = ( u + v ) } -> ran ( F oF + G ) C_ { w | E. u e. ran F E. v e. ran G w = ( u + v ) } )
45	43, 44	syl 16	. . . 4 |- ( ph -> ran ( F oF + G ) C_ { w | E. u e. ran F E. v e. ran G w = ( u + v ) } )
46	19	rnmpt2 6221	. . . 4 |- ran ( u e. ran F , v e. ran G |-> ( u + v ) ) = { w | E. u e. ran F E. v e. ran G w = ( u + v ) }
47	45, 46	syl6sseqr 3424	. . 3 |- ( ph -> ran ( F oF + G ) C_ ran ( u e. ran F , v e. ran G |-> ( u + v ) ) )
48		ssfi 7554	. . 3 |- ( ( ran ( u e. ran F , v e. ran G |-> ( u + v ) ) e. Fin /\ ran ( F oF + G ) C_ ran ( u e. ran F , v e. ran G |-> ( u + v ) ) ) -> ran ( F oF + G ) e. Fin )
49	25, 47, 48	syl2anc 661	. 2 |- ( ph -> ran ( F oF + G ) e. Fin )
50		frn 5586	. . . . . . . 8 |- ( ( F oF + G ) : RR --> RR -> ran ( F oF + G ) C_ RR )
51	12, 50	syl 16	. . . . . . 7 |- ( ph -> ran ( F oF + G ) C_ RR )
52	51	ssdifssd 3515	. . . . . 6 |- ( ph -> ( ran ( F oF + G ) \ { 0 } ) C_ RR )
53	52	sselda 3377	. . . . 5 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> y e. RR )
54	53	recnd 9433	. . . 4 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> y e. CC )
55	3, 6	i1faddlem 21193	. . . 4 |- ( ( ph /\ y e. CC ) -> ( `' ( F oF + G ) " { y } ) = U_ z e. ran G ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) )
56	54, 55	syldan 470	. . 3 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ( `' ( F oF + G ) " { y } ) = U_ z e. ran G ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) )
57	16	adantr 465	. . . 4 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ran G e. Fin )
58	3	ad2antrr 725	. . . . . . . 8 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> F e. dom S.1 )
59		i1fmbf 21175	. . . . . . . 8 |- ( F e. dom S.1 -> F e. MblFn )
60	58, 59	syl 16	. . . . . . 7 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> F e. MblFn )
61	5	ad2antrr 725	. . . . . . 7 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> F : RR --> RR )
62	12	ad2antrr 725	. . . . . . . . . 10 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> ( F oF + G ) : RR --> RR )
63	62, 50	syl 16	. . . . . . . . 9 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> ran ( F oF + G ) C_ RR )
64		eldifi 3499	. . . . . . . . . 10 |- ( y e. ( ran ( F oF + G ) \ { 0 } ) -> y e. ran ( F oF + G ) )
65	64	ad2antlr 726	. . . . . . . . 9 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> y e. ran ( F oF + G ) )
66	63, 65	sseldd 3378	. . . . . . . 8 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> y e. RR )
67	8	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> G : RR --> RR )
68		frn 5586	. . . . . . . . . 10 |- ( G : RR --> RR -> ran G C_ RR )
69	67, 68	syl 16	. . . . . . . . 9 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ran G C_ RR )
70	69	sselda 3377	. . . . . . . 8 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> z e. RR )
71	66, 70	resubcld 9797	. . . . . . 7 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> ( y - z ) e. RR )
72		mbfimasn 21134	. . . . . . 7 |- ( ( F e. MblFn /\ F : RR --> RR /\ ( y - z ) e. RR ) -> ( `' F " { ( y - z ) } ) e. dom vol )
73	60, 61, 71, 72	syl3anc 1218	. . . . . 6 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> ( `' F " { ( y - z ) } ) e. dom vol )
74	6	ad2antrr 725	. . . . . . . 8 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> G e. dom S.1 )
75		i1fmbf 21175	. . . . . . . 8 |- ( G e. dom S.1 -> G e. MblFn )
76	74, 75	syl 16	. . . . . . 7 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> G e. MblFn )
77	8	ad2antrr 725	. . . . . . 7 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> G : RR --> RR )
78		mbfimasn 21134	. . . . . . 7 |- ( ( G e. MblFn /\ G : RR --> RR /\ z e. RR ) -> ( `' G " { z } ) e. dom vol )
79	76, 77, 70, 78	syl3anc 1218	. . . . . 6 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> ( `' G " { z } ) e. dom vol )
80		inmbl 21045	. . . . . 6 |- ( ( ( `' F " { ( y - z ) } ) e. dom vol /\ ( `' G " { z } ) e. dom vol ) -> ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) e. dom vol )
81	73, 79, 80	syl2anc 661	. . . . 5 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) e. dom vol )
82	81	ralrimiva 2820	. . . 4 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> A. z e. ran G ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) e. dom vol )
83		finiunmbl 21047	. . . 4 |- ( ( ran G e. Fin /\ A. z e. ran G ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) e. dom vol ) -> U_ z e. ran G ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) e. dom vol )
84	57, 82, 83	syl2anc 661	. . 3 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> U_ z e. ran G ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) e. dom vol )
85	56, 84	eqeltrd 2517	. 2 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ( `' ( F oF + G ) " { y } ) e. dom vol )
86		mblvol 21035	. . . 4 |- ( ( `' ( F oF + G ) " { y } ) e. dom vol -> ( vol ` ( `' ( F oF + G ) " { y } ) ) = ( vol* ` ( `' ( F oF + G ) " { y } ) ) )
87	85, 86	syl 16	. . 3 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ( vol ` ( `' ( F oF + G ) " { y } ) ) = ( vol* ` ( `' ( F oF + G ) " { y } ) ) )
88		mblss 21036	. . . . 5 |- ( ( `' ( F oF + G ) " { y } ) e. dom vol -> ( `' ( F oF + G ) " { y } ) C_ RR )
89	85, 88	syl 16	. . . 4 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ( `' ( F oF + G ) " { y } ) C_ RR )
90		inss1 3591	. . . . . . . . 9 |- ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) C_ ( `' F " { ( y - z ) } )
91	90	a1i 11	. . . . . . . 8 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) C_ ( `' F " { ( y - z ) } ) )
92	73	adantrr 716	. . . . . . . . 9 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( `' F " { ( y - z ) } ) e. dom vol )
93		mblss 21036	. . . . . . . . 9 |- ( ( `' F " { ( y - z ) } ) e. dom vol -> ( `' F " { ( y - z ) } ) C_ RR )
94	92, 93	syl 16	. . . . . . . 8 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( `' F " { ( y - z ) } ) C_ RR )
95		mblvol 21035	. . . . . . . . . 10 |- ( ( `' F " { ( y - z ) } ) e. dom vol -> ( vol ` ( `' F " { ( y - z ) } ) ) = ( vol* ` ( `' F " { ( y - z ) } ) ) )
96	92, 95	syl 16	. . . . . . . . 9 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( vol ` ( `' F " { ( y - z ) } ) ) = ( vol* ` ( `' F " { ( y - z ) } ) ) )
97		simprr 756	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> z = 0 )
98	97	oveq2d 6128	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( y - z ) = ( y - 0 ) )
99	54	adantr 465	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> y e. CC )
100	99	subid1d 9729	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( y - 0 ) = y )
101	98, 100	eqtrd 2475	. . . . . . . . . . . . 13 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( y - z ) = y )
102	101	sneqd 3910	. . . . . . . . . . . 12 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> { ( y - z ) } = { y } )
103	102	imaeq2d 5190	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( `' F " { ( y - z ) } ) = ( `' F " { y } ) )
104	103	fveq2d 5716	. . . . . . . . . 10 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( vol ` ( `' F " { ( y - z ) } ) ) = ( vol ` ( `' F " { y } ) ) )
105		i1fima2sn 21180	. . . . . . . . . . . 12 |- ( ( F e. dom S.1 /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ( vol ` ( `' F " { y } ) ) e. RR )
106	3, 105	sylan 471	. . . . . . . . . . 11 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ( vol ` ( `' F " { y } ) ) e. RR )
107	106	adantr 465	. . . . . . . . . 10 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( vol ` ( `' F " { y } ) ) e. RR )
108	104, 107	eqeltrd 2517	. . . . . . . . 9 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( vol ` ( `' F " { ( y - z ) } ) ) e. RR )
109	96, 108	eqeltrrd 2518	. . . . . . . 8 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( vol* ` ( `' F " { ( y - z ) } ) ) e. RR )
110		ovolsscl 20991	. . . . . . . 8 |- ( ( ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) C_ ( `' F " { ( y - z ) } ) /\ ( `' F " { ( y - z ) } ) C_ RR /\ ( vol* ` ( `' F " { ( y - z ) } ) ) e. RR ) -> ( vol* ` ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) ) e. RR )
111	91, 94, 109, 110	syl3anc 1218	. . . . . . 7 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( vol* ` ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) ) e. RR )
112	111	expr 615	. . . . . 6 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> ( z = 0 -> ( vol* ` ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) ) e. RR ) )
113		eldifsn 4021	. . . . . . . 8 |- ( z e. ( ran G \ { 0 } ) <-> ( z e. ran G /\ z =/= 0 ) )
114		inss2 3592	. . . . . . . . . 10 |- ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) C_ ( `' G " { z } )
115	114	a1i 11	. . . . . . . . 9 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ( ran G \ { 0 } ) ) -> ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) C_ ( `' G " { z } ) )
116		eldifi 3499	. . . . . . . . . 10 |- ( z e. ( ran G \ { 0 } ) -> z e. ran G )
117		mblss 21036	. . . . . . . . . . 11 |- ( ( `' G " { z } ) e. dom vol -> ( `' G " { z } ) C_ RR )
118	79, 117	syl 16	. . . . . . . . . 10 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> ( `' G " { z } ) C_ RR )
119	116, 118	sylan2 474	. . . . . . . . 9 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ( ran G \ { 0 } ) ) -> ( `' G " { z } ) C_ RR )
120		i1fima 21178	. . . . . . . . . . . . 13 |- ( G e. dom S.1 -> ( `' G " { z } ) e. dom vol )
121	6, 120	syl 16	. . . . . . . . . . . 12 |- ( ph -> ( `' G " { z } ) e. dom vol )
122	121	ad2antrr 725	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ( ran G \ { 0 } ) ) -> ( `' G " { z } ) e. dom vol )
123		mblvol 21035	. . . . . . . . . . 11 |- ( ( `' G " { z } ) e. dom vol -> ( vol ` ( `' G " { z } ) ) = ( vol* ` ( `' G " { z } ) ) )
124	122, 123	syl 16	. . . . . . . . . 10 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ( ran G \ { 0 } ) ) -> ( vol ` ( `' G " { z } ) ) = ( vol* ` ( `' G " { z } ) ) )
125	6	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> G e. dom S.1 )
126		i1fima2sn 21180	. . . . . . . . . . 11 |- ( ( G e. dom S.1 /\ z e. ( ran G \ { 0 } ) ) -> ( vol ` ( `' G " { z } ) ) e. RR )
127	125, 126	sylan 471	. . . . . . . . . 10 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ( ran G \ { 0 } ) ) -> ( vol ` ( `' G " { z } ) ) e. RR )
128	124, 127	eqeltrrd 2518	. . . . . . . . 9 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ( ran G \ { 0 } ) ) -> ( vol* ` ( `' G " { z } ) ) e. RR )
129		ovolsscl 20991	. . . . . . . . 9 |- ( ( ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) C_ ( `' G " { z } ) /\ ( `' G " { z } ) C_ RR /\ ( vol* ` ( `' G " { z } ) ) e. RR ) -> ( vol* ` ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) ) e. RR )
130	115, 119, 128, 129	syl3anc 1218	. . . . . . . 8 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ( ran G \ { 0 } ) ) -> ( vol* ` ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) ) e. RR )
131	113, 130	sylan2br 476	. . . . . . 7 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z =/= 0 ) ) -> ( vol* ` ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) ) e. RR )
132	131	expr 615	. . . . . 6 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> ( z =/= 0 -> ( vol* ` ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) ) e. RR ) )
133	112, 132	pm2.61dne 2712	. . . . 5 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> ( vol* ` ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) ) e. RR )
134	57, 133	fsumrecl 13232	. . . 4 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> sum_ z e. ran G ( vol* ` ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) ) e. RR )
135	56	fveq2d 5716	. . . . 5 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ( vol* ` ( `' ( F oF + G ) " { y } ) ) = ( vol* ` U_ z e. ran G ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) ) )
136	114, 118	syl5ss 3388	. . . . . . . 8 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) C_ RR )
137	136, 133	jca 532	. . . . . . 7 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> ( ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) C_ RR /\ ( vol* ` ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) ) e. RR ) )
138	137	ralrimiva 2820	. . . . . 6 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> A. z e. ran G ( ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) C_ RR /\ ( vol* ` ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) ) e. RR ) )
139		ovolfiniun 21006	. . . . . 6 |- ( ( ran G e. Fin /\ A. z e. ran G ( ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) C_ RR /\ ( vol* ` ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) ) e. RR ) ) -> ( vol* ` U_ z e. ran G ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) ) <_ sum_ z e. ran G ( vol* ` ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) ) )
140	57, 138, 139	syl2anc 661	. . . . 5 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ( vol* ` U_ z e. ran G ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) ) <_ sum_ z e. ran G ( vol* ` ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) ) )
141	135, 140	eqbrtrd 4333	. . . 4 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ( vol* ` ( `' ( F oF + G ) " { y } ) ) <_ sum_ z e. ran G ( vol* ` ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) ) )
142		ovollecl 20988	. . . 4 |- ( ( ( `' ( F oF + G ) " { y } ) C_ RR /\ sum_ z e. ran G ( vol* ` ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) ) e. RR /\ ( vol* ` ( `' ( F oF + G ) " { y } ) ) <_ sum_ z e. ran G ( vol* ` ( ( `' F " { ( y - z ) } ) i^i ( `' G " { z } ) ) ) ) -> ( vol* ` ( `' ( F oF + G ) " { y } ) ) e. RR )
143	89, 134, 141, 142	syl3anc 1218	. . 3 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ( vol* ` ( `' ( F oF + G ) " { y } ) ) e. RR )
144	87, 143	eqeltrd 2517	. 2 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ( vol ` ( `' ( F oF + G ) " { y } ) ) e. RR )
145	12, 49, 85, 144	i1fd 21181	1 |- ( ph -> ( F oF + G ) e. dom S.1 )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1369   e. wcel 1756   {cab 2429   =/= wne 2620   A.wral 2736   E.wrex 2737   _Vcvv 2993   \ cdif 3346   i^i cin 3348   C_ wss 3349   {csn 3898   U_ciun 4192   class class class wbr 4313   X. cxp 4859   `'ccnv 4860   dom cdm 4861   ran crn 4862   "cima 4864   Fn wfn 5434   -->wf 5435   -onto->wfo 5437   ` cfv 5439  (class class class)co 6112   e. cmpt2 6114   oFcof 6339   Fincfn 7331   CCcc 9301   RRcr 9302   0cc0 9303   + caddc 9306   <_ cle 9440   - cmin 9616   sum_csu 13184   vol*covol 20968   volcvol 20969  MblFncmbf 21116   S.1citg1 21117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-of 6341  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-2o 6942  df-oadd 6945  df-er 7122  df-map 7237  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-sup 7712  df-oi 7745  df-card 8130  df-cda 8358  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-n0 10601  df-z 10668  df-uz 10883  df-q 10975  df-rp 11013  df-xadd 11111  df-ioo 11325  df-ico 11327  df-icc 11328  df-fz 11459  df-fzo 11570  df-fl 11663  df-seq 11828  df-exp 11887  df-hash 12125  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-clim 12987  df-sum 13185  df-xmet 17832  df-met 17833  df-ovol 20970  df-vol 20971  df-mbf 21121  df-itg1 21122

This theorem is referenced by:  itg1addlem4  21199  i1fsub  21208  itg2splitlem  21248  itg2split  21249  itg2addlem  21258  itg2addnc  28472  ftc1anclem3  28495  ftc1anclem5  28497  ftc1anclem8  28500