Theorem i1fadd 22228

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 9592	. . . 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 22209	. . . 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 22209	. . . 4 |- ( G e. dom S.1 -> G : RR --> RR )
8	6, 7	syl 16	. . 3 |- ( ph -> G : RR --> RR )
9		reex 9600	. . . 4 |- RR e. _V
10	9	a1i 11	. . 3 |- ( ph -> RR e. _V )
11		inidm 3703	. . 3 |- ( RR i^i RR ) = RR
12	2, 5, 8, 10, 10, 11	off 6553	. 2 |- ( ph -> ( F oF + G ) : RR --> RR )
13		i1frn 22210	. . . . . 6 |- ( F e. dom S.1 -> ran F e. Fin )
14	3, 13	syl 16	. . . . 5 |- ( ph -> ran F e. Fin )
15		i1frn 22210	. . . . . 6 |- ( G e. dom S.1 -> ran G e. Fin )
16	6, 15	syl 16	. . . . 5 |- ( ph -> ran G e. Fin )
17		xpfi 7809	. . . . 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 2457	. . . . . 6 |- ( u e. ran F , v e. ran G |-> ( u + v ) ) = ( u e. ran F , v e. ran G |-> ( u + v ) )
20		ovex 6324	. . . . . 6 |- ( u + v ) e. _V
21	19, 20	fnmpt2i 6868	. . . . 5 |- ( u e. ran F , v e. ran G |-> ( u + v ) ) Fn ( ran F X. ran G )
22		dffn4 5807	. . . . 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 7824	. . . 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 2457	. . . . . . . . 9 |- ( x + y ) = ( x + y )
27		rspceov 6335	. . . . . . . . 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 1313	. . . . . . . 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 6324	. . . . . . . . 9 |- ( x + y ) e. _V
30		eqeq1 2461	. . . . . . . . . 10 |- ( w = ( x + y ) -> ( w = ( u + v ) <-> ( x + y ) = ( u + v ) ) )
31	30	2rexbidv 2975	. . . . . . . . 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 3246	. . . . . . . 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 5737	. . . . . . . 8 |- ( F : RR --> RR -> F Fn RR )
36	5, 35	syl 16	. . . . . . 7 |- ( ph -> F Fn RR )
37		dffn3 5744	. . . . . . 7 |- ( F Fn RR <-> F : RR --> ran F )
38	36, 37	sylib 196	. . . . . 6 |- ( ph -> F : RR --> ran F )
39		ffn 5737	. . . . . . . 8 |- ( G : RR --> RR -> G Fn RR )
40	8, 39	syl 16	. . . . . . 7 |- ( ph -> G Fn RR )
41		dffn3 5744	. . . . . . 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 6553	. . . . 5 |- ( ph -> ( F oF + G ) : RR --> { w | E. u e. ran F E. v e. ran G w = ( u + v ) } )
44		frn 5743	. . . . 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 6411	. . . 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 3546	. . 3 |- ( ph -> ran ( F oF + G ) C_ ran ( u e. ran F , v e. ran G |-> ( u + v ) ) )
48		ssfi 7759	. . 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 5743	. . . . . . . 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 3638	. . . . . 6 |- ( ph -> ( ran ( F oF + G ) \ { 0 } ) C_ RR )
53	52	sselda 3499	. . . . 5 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> y e. RR )
54	53	recnd 9639	. . . 4 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> y e. CC )
55	3, 6	i1faddlem 22226	. . . 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 22208	. . . . . . . 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 3622	. . . . . . . . . 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 3500	. . . . . . . 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 5743	. . . . . . . . . 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 3499	. . . . . . . 8 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> z e. RR )
71	66, 70	resubcld 10008	. . . . . . 7 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> ( y - z ) e. RR )
72		mbfimasn 22167	. . . . . . 7 |- ( ( F e. MblFn /\ F : RR --> RR /\ ( y - z ) e. RR ) -> ( `' F " { ( y - z ) } ) e. dom vol )
73	60, 61, 71, 72	syl3anc 1228	. . . . . 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 22208	. . . . . . . 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 22167	. . . . . . 7 |- ( ( G e. MblFn /\ G : RR --> RR /\ z e. RR ) -> ( `' G " { z } ) e. dom vol )
79	76, 77, 70, 78	syl3anc 1228	. . . . . 6 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> ( `' G " { z } ) e. dom vol )
80		inmbl 22078	. . . . . 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 2871	. . . 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 22080	. . . 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 2545	. 2 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ( `' ( F oF + G ) " { y } ) e. dom vol )
86		mblvol 22067	. . . 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 22068	. . . . 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 3714	. . . . . . . . 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 22068	. . . . . . . . 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 22067	. . . . . . . . . 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 757	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> z = 0 )
98	97	oveq2d 6312	. . . . . . . . . . . . . 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 9939	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( y - 0 ) = y )
101	98, 100	eqtrd 2498	. . . . . . . . . . . . 13 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( y - z ) = y )
102	101	sneqd 4044	. . . . . . . . . . . 12 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> { ( y - z ) } = { y } )
103	102	imaeq2d 5347	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( `' F " { ( y - z ) } ) = ( `' F " { y } ) )
104	103	fveq2d 5876	. . . . . . . . . 10 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( vol ` ( `' F " { ( y - z ) } ) ) = ( vol ` ( `' F " { y } ) ) )
105		i1fima2sn 22213	. . . . . . . . . . . 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 2545	. . . . . . . . 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 2546	. . . . . . . 8 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( vol* ` ( `' F " { ( y - z ) } ) ) e. RR )
110		ovolsscl 22023	. . . . . . . 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 1228	. . . . . . 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 4157	. . . . . . . 8 |- ( z e. ( ran G \ { 0 } ) <-> ( z e. ran G /\ z =/= 0 ) )
114		inss2 3715	. . . . . . . . . 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 3622	. . . . . . . . . 10 |- ( z e. ( ran G \ { 0 } ) -> z e. ran G )
117		mblss 22068	. . . . . . . . . . 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 22211	. . . . . . . . . . . . 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 22067	. . . . . . . . . . 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 22213	. . . . . . . . . . 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 2546	. . . . . . . . 9 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ( ran G \ { 0 } ) ) -> ( vol* ` ( `' G " { z } ) ) e. RR )
129		ovolsscl 22023	. . . . . . . . 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 1228	. . . . . . . 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 2774	. . . . 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 13568	. . . 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 5876	. . . . 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 3510	. . . . . . . 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 2871	. . . . . 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 22038	. . . . . 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 4476	. . . 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 22020	. . . 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 1228	. . 3 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ( vol* ` ( `' ( F oF + G ) " { y } ) ) e. RR )
144	87, 143	eqeltrd 2545	. 2 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ( vol ` ( `' ( F oF + G ) " { y } ) ) e. RR )
145	12, 49, 85, 144	i1fd 22214	1 |- ( ph -> ( F oF + G ) e. dom S.1 )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1395   e. wcel 1819   {cab 2442   =/= wne 2652   A.wral 2807   E.wrex 2808   _Vcvv 3109   \ cdif 3468   i^i cin 3470   C_ wss 3471   {csn 4032   U_ciun 4332   class class class wbr 4456   X. cxp 5006   `'ccnv 5007   dom cdm 5008   ran crn 5009   "cima 5011   Fn wfn 5589   -->wf 5590   -onto->wfo 5592   ` cfv 5594  (class class class)co 6296   |-> cmpt2 6298   oFcof 6537   Fincfn 7535   CCcc 9507   RRcr 9508   0cc0 9509   + caddc 9512   <_ cle 9646   - cmin 9824   sum_csu 13520   vol*covol 22000   volcvol 22001  MblFncmbf 22149   S.1citg1 22150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-xadd 11344  df-ioo 11558  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-fl 11932  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323  df-sum 13521  df-xmet 18539  df-met 18540  df-ovol 22002  df-vol 22003  df-mbf 22154  df-itg1 22155

This theorem is referenced by:  itg1addlem4  22232  i1fsub  22241  itg2splitlem  22281  itg2split  22282  itg2addlem  22291  itg2addnc  30274  ftc1anclem3  30297  ftc1anclem5  30299  ftc1anclem8  30302