Theorem i1fadd 22732

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 9640	. . . 4 |- ( ( x e. RR /\ y e. RR ) -> ( x + y ) e. RR )
2	1	adantl 473	. . 3 |- ( ( ph /\ ( x e. RR /\ y e. RR ) ) -> ( x + y ) e. RR )
3		i1fadd.1	. . . 4 |- ( ph -> F e. dom S.1 )
4		i1ff 22713	. . . 4 |- ( F e. dom S.1 -> F : RR --> RR )
5	3, 4	syl 17	. . 3 |- ( ph -> F : RR --> RR )
6		i1fadd.2	. . . 4 |- ( ph -> G e. dom S.1 )
7		i1ff 22713	. . . 4 |- ( G e. dom S.1 -> G : RR --> RR )
8	6, 7	syl 17	. . 3 |- ( ph -> G : RR --> RR )
9		reex 9648	. . . 4 |- RR e. _V
10	9	a1i 11	. . 3 |- ( ph -> RR e. _V )
11		inidm 3632	. . 3 |- ( RR i^i RR ) = RR
12	2, 5, 8, 10, 10, 11	off 6565	. 2 |- ( ph -> ( F oF + G ) : RR --> RR )
13		i1frn 22714	. . . . . 6 |- ( F e. dom S.1 -> ran F e. Fin )
14	3, 13	syl 17	. . . . 5 |- ( ph -> ran F e. Fin )
15		i1frn 22714	. . . . . 6 |- ( G e. dom S.1 -> ran G e. Fin )
16	6, 15	syl 17	. . . . 5 |- ( ph -> ran G e. Fin )
17		xpfi 7860	. . . . 5 |- ( ( ran F e. Fin /\ ran G e. Fin ) -> ( ran F X. ran G ) e. Fin )
18	14, 16, 17	syl2anc 673	. . . 4 |- ( ph -> ( ran F X. ran G ) e. Fin )
19		eqid 2471	. . . . . 6 |- ( u e. ran F , v e. ran G |-> ( u + v ) ) = ( u e. ran F , v e. ran G |-> ( u + v ) )
20		ovex 6336	. . . . . 6 |- ( u + v ) e. _V
21	19, 20	fnmpt2i 6881	. . . . 5 |- ( u e. ran F , v e. ran G |-> ( u + v ) ) Fn ( ran F X. ran G )
22		dffn4 5812	. . . . 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 213	. . . 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 7878	. . . 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 675	. . 3 |- ( ph -> ran ( u e. ran F , v e. ran G |-> ( u + v ) ) e. Fin )
26		eqid 2471	. . . . . . . . 9 |- ( x + y ) = ( x + y )
27		rspceov 6347	. . . . . . . . 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 1379	. . . . . . . 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 6336	. . . . . . . . 9 |- ( x + y ) e. _V
30		eqeq1 2475	. . . . . . . . . 10 |- ( w = ( x + y ) -> ( w = ( u + v ) <-> ( x + y ) = ( u + v ) ) )
31	30	2rexbidv 2897	. . . . . . . . 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 3173	. . . . . . . 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 217	. . . . . . 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 473	. . . . . 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 5739	. . . . . . . 8 |- ( F : RR --> RR -> F Fn RR )
36	5, 35	syl 17	. . . . . . 7 |- ( ph -> F Fn RR )
37		dffn3 5748	. . . . . . 7 |- ( F Fn RR <-> F : RR --> ran F )
38	36, 37	sylib 201	. . . . . 6 |- ( ph -> F : RR --> ran F )
39		ffn 5739	. . . . . . . 8 |- ( G : RR --> RR -> G Fn RR )
40	8, 39	syl 17	. . . . . . 7 |- ( ph -> G Fn RR )
41		dffn3 5748	. . . . . . 7 |- ( G Fn RR <-> G : RR --> ran G )
42	40, 41	sylib 201	. . . . . 6 |- ( ph -> G : RR --> ran G )
43	34, 38, 42, 10, 10, 11	off 6565	. . . . 5 |- ( ph -> ( F oF + G ) : RR --> { w | E. u e. ran F E. v e. ran G w = ( u + v ) } )
44		frn 5747	. . . . 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 17	. . . 4 |- ( ph -> ran ( F oF + G ) C_ { w | E. u e. ran F E. v e. ran G w = ( u + v ) } )
46	19	rnmpt2 6425	. . . 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 3465	. . 3 |- ( ph -> ran ( F oF + G ) C_ ran ( u e. ran F , v e. ran G |-> ( u + v ) ) )
48		ssfi 7810	. . 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 673	. 2 |- ( ph -> ran ( F oF + G ) e. Fin )
50		frn 5747	. . . . . . . 8 |- ( ( F oF + G ) : RR --> RR -> ran ( F oF + G ) C_ RR )
51	12, 50	syl 17	. . . . . . 7 |- ( ph -> ran ( F oF + G ) C_ RR )
52	51	ssdifssd 3560	. . . . . 6 |- ( ph -> ( ran ( F oF + G ) \ { 0 } ) C_ RR )
53	52	sselda 3418	. . . . 5 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> y e. RR )
54	53	recnd 9687	. . . 4 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> y e. CC )
55	3, 6	i1faddlem 22730	. . . 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 478	. . 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 472	. . . 4 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ran G e. Fin )
58	3	ad2antrr 740	. . . . . . . 8 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> F e. dom S.1 )
59		i1fmbf 22712	. . . . . . . 8 |- ( F e. dom S.1 -> F e. MblFn )
60	58, 59	syl 17	. . . . . . 7 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> F e. MblFn )
61	5	ad2antrr 740	. . . . . . 7 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> F : RR --> RR )
62	12	ad2antrr 740	. . . . . . . . . 10 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> ( F oF + G ) : RR --> RR )
63	62, 50	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> ran ( F oF + G ) C_ RR )
64		eldifi 3544	. . . . . . . . . 10 |- ( y e. ( ran ( F oF + G ) \ { 0 } ) -> y e. ran ( F oF + G ) )
65	64	ad2antlr 741	. . . . . . . . 9 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> y e. ran ( F oF + G ) )
66	63, 65	sseldd 3419	. . . . . . . 8 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> y e. RR )
67	8	adantr 472	. . . . . . . . . 10 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> G : RR --> RR )
68		frn 5747	. . . . . . . . . 10 |- ( G : RR --> RR -> ran G C_ RR )
69	67, 68	syl 17	. . . . . . . . 9 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ran G C_ RR )
70	69	sselda 3418	. . . . . . . 8 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> z e. RR )
71	66, 70	resubcld 10068	. . . . . . 7 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> ( y - z ) e. RR )
72		mbfimasn 22669	. . . . . . 7 |- ( ( F e. MblFn /\ F : RR --> RR /\ ( y - z ) e. RR ) -> ( `' F " { ( y - z ) } ) e. dom vol )
73	60, 61, 71, 72	syl3anc 1292	. . . . . 6 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> ( `' F " { ( y - z ) } ) e. dom vol )
74	6	ad2antrr 740	. . . . . . . 8 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> G e. dom S.1 )
75		i1fmbf 22712	. . . . . . . 8 |- ( G e. dom S.1 -> G e. MblFn )
76	74, 75	syl 17	. . . . . . 7 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> G e. MblFn )
77	8	ad2antrr 740	. . . . . . 7 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> G : RR --> RR )
78		mbfimasn 22669	. . . . . . 7 |- ( ( G e. MblFn /\ G : RR --> RR /\ z e. RR ) -> ( `' G " { z } ) e. dom vol )
79	76, 77, 70, 78	syl3anc 1292	. . . . . 6 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> ( `' G " { z } ) e. dom vol )
80		inmbl 22574	. . . . . 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 673	. . . . 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 2809	. . . 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 22576	. . . 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 673	. . 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 2549	. 2 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ( `' ( F oF + G ) " { y } ) e. dom vol )
86		mblvol 22562	. . . 4 |- ( ( `' ( F oF + G ) " { y } ) e. dom vol -> ( vol ` ( `' ( F oF + G ) " { y } ) ) = ( vol* ` ( `' ( F oF + G ) " { y } ) ) )
87	85, 86	syl 17	. . 3 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ( vol ` ( `' ( F oF + G ) " { y } ) ) = ( vol* ` ( `' ( F oF + G ) " { y } ) ) )
88		mblss 22563	. . . . 5 |- ( ( `' ( F oF + G ) " { y } ) e. dom vol -> ( `' ( F oF + G ) " { y } ) C_ RR )
89	85, 88	syl 17	. . . 4 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ( `' ( F oF + G ) " { y } ) C_ RR )
90		inss1 3643	. . . . . . . . 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 731	. . . . . . . . 9 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( `' F " { ( y - z ) } ) e. dom vol )
93		mblss 22563	. . . . . . . . 9 |- ( ( `' F " { ( y - z ) } ) e. dom vol -> ( `' F " { ( y - z ) } ) C_ RR )
94	92, 93	syl 17	. . . . . . . 8 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( `' F " { ( y - z ) } ) C_ RR )
95		mblvol 22562	. . . . . . . . . 10 |- ( ( `' F " { ( y - z ) } ) e. dom vol -> ( vol ` ( `' F " { ( y - z ) } ) ) = ( vol* ` ( `' F " { ( y - z ) } ) ) )
96	92, 95	syl 17	. . . . . . . . 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 774	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> z = 0 )
98	97	oveq2d 6324	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( y - z ) = ( y - 0 ) )
99	54	adantr 472	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> y e. CC )
100	99	subid1d 9994	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( y - 0 ) = y )
101	98, 100	eqtrd 2505	. . . . . . . . . . . . 13 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( y - z ) = y )
102	101	sneqd 3971	. . . . . . . . . . . 12 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> { ( y - z ) } = { y } )
103	102	imaeq2d 5174	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( `' F " { ( y - z ) } ) = ( `' F " { y } ) )
104	103	fveq2d 5883	. . . . . . . . . 10 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( vol ` ( `' F " { ( y - z ) } ) ) = ( vol ` ( `' F " { y } ) ) )
105		i1fima2sn 22717	. . . . . . . . . . . 12 |- ( ( F e. dom S.1 /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ( vol ` ( `' F " { y } ) ) e. RR )
106	3, 105	sylan 479	. . . . . . . . . . 11 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ( vol ` ( `' F " { y } ) ) e. RR )
107	106	adantr 472	. . . . . . . . . 10 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( vol ` ( `' F " { y } ) ) e. RR )
108	104, 107	eqeltrd 2549	. . . . . . . . 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 2550	. . . . . . . 8 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( vol* ` ( `' F " { ( y - z ) } ) ) e. RR )
110		ovolsscl 22517	. . . . . . . 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 1292	. . . . . . 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 626	. . . . . 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 4088	. . . . . . . 8 |- ( z e. ( ran G \ { 0 } ) <-> ( z e. ran G /\ z =/= 0 ) )
114		inss2 3644	. . . . . . . . . 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 3544	. . . . . . . . . 10 |- ( z e. ( ran G \ { 0 } ) -> z e. ran G )
117		mblss 22563	. . . . . . . . . . 11 |- ( ( `' G " { z } ) e. dom vol -> ( `' G " { z } ) C_ RR )
118	79, 117	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> ( `' G " { z } ) C_ RR )
119	116, 118	sylan2 482	. . . . . . . . 9 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ( ran G \ { 0 } ) ) -> ( `' G " { z } ) C_ RR )
120		i1fima 22715	. . . . . . . . . . . . 13 |- ( G e. dom S.1 -> ( `' G " { z } ) e. dom vol )
121	6, 120	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( `' G " { z } ) e. dom vol )
122	121	ad2antrr 740	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ( ran G \ { 0 } ) ) -> ( `' G " { z } ) e. dom vol )
123		mblvol 22562	. . . . . . . . . . 11 |- ( ( `' G " { z } ) e. dom vol -> ( vol ` ( `' G " { z } ) ) = ( vol* ` ( `' G " { z } ) ) )
124	122, 123	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ( ran G \ { 0 } ) ) -> ( vol ` ( `' G " { z } ) ) = ( vol* ` ( `' G " { z } ) ) )
125	6	adantr 472	. . . . . . . . . . 11 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> G e. dom S.1 )
126		i1fima2sn 22717	. . . . . . . . . . 11 |- ( ( G e. dom S.1 /\ z e. ( ran G \ { 0 } ) ) -> ( vol ` ( `' G " { z } ) ) e. RR )
127	125, 126	sylan 479	. . . . . . . . . 10 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ( ran G \ { 0 } ) ) -> ( vol ` ( `' G " { z } ) ) e. RR )
128	124, 127	eqeltrrd 2550	. . . . . . . . 9 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ( ran G \ { 0 } ) ) -> ( vol* ` ( `' G " { z } ) ) e. RR )
129		ovolsscl 22517	. . . . . . . . 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 1292	. . . . . . . 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 484	. . . . . . 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 626	. . . . . 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 2729	. . . . 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 13877	. . . 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 5883	. . . . 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 3429	. . . . . . . 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 541	. . . . . . 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 2809	. . . . . 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 22532	. . . . . 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 673	. . . . 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 4416	. . . 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 22514	. . . 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 1292	. . 3 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ( vol* ` ( `' ( F oF + G ) " { y } ) ) e. RR )
144	87, 143	eqeltrd 2549	. 2 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ( vol ` ( `' ( F oF + G ) " { y } ) ) e. RR )
145	12, 49, 85, 144	i1fd 22718	1 |- ( ph -> ( F oF + G ) e. dom S.1 )

Syntax hints:   -> wi 4   /\ wa 376   = wceq 1452   e. wcel 1904   {cab 2457   =/= wne 2641   A.wral 2756   E.wrex 2757   _Vcvv 3031   \ cdif 3387   i^i cin 3389   C_ wss 3390   {csn 3959   U_ciun 4269   class class class wbr 4395   X. cxp 4837   `'ccnv 4838   dom cdm 4839   ran crn 4840   "cima 4842   Fn wfn 5584   -->wf 5585   -onto->wfo 5587   ` cfv 5589  (class class class)co 6308   |-> cmpt2 6310   oFcof 6548   Fincfn 7587   CCcc 9555   RRcr 9556   0cc0 9557   + caddc 9560   <_ cle 9694   - cmin 9880   sum_csu 13829   vol*covol 22491   volcvol 22493  MblFncmbf 22651   S.1citg1 22652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-xadd 11433  df-ioo 11664  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-xmet 19040  df-met 19041  df-ovol 22494  df-vol 22496  df-mbf 22656  df-itg1 22657

This theorem is referenced by:  itg1addlem4  22736  i1fsub  22745  itg2splitlem  22785  itg2split  22786  itg2addlem  22795  itg2addnc  32060  ftc1anclem3  32083  ftc1anclem5  32085  ftc1anclem8  32088