Theorem i1fadd 22646

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 9619	. . . 4 |- ( ( x e. RR /\ y e. RR ) -> ( x + y ) e. RR )
2	1	adantl 468	. . 3 |- ( ( ph /\ ( x e. RR /\ y e. RR ) ) -> ( x + y ) e. RR )
3		i1fadd.1	. . . 4 |- ( ph -> F e. dom S.1 )
4		i1ff 22627	. . . 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 22627	. . . 4 |- ( G e. dom S.1 -> G : RR --> RR )
8	6, 7	syl 17	. . 3 |- ( ph -> G : RR --> RR )
9		reex 9627	. . . 4 |- RR e. _V
10	9	a1i 11	. . 3 |- ( ph -> RR e. _V )
11		inidm 3640	. . 3 |- ( RR i^i RR ) = RR
12	2, 5, 8, 10, 10, 11	off 6543	. 2 |- ( ph -> ( F oF + G ) : RR --> RR )
13		i1frn 22628	. . . . . 6 |- ( F e. dom S.1 -> ran F e. Fin )
14	3, 13	syl 17	. . . . 5 |- ( ph -> ran F e. Fin )
15		i1frn 22628	. . . . . 6 |- ( G e. dom S.1 -> ran G e. Fin )
16	6, 15	syl 17	. . . . 5 |- ( ph -> ran G e. Fin )
17		xpfi 7839	. . . . 5 |- ( ( ran F e. Fin /\ ran G e. Fin ) -> ( ran F X. ran G ) e. Fin )
18	14, 16, 17	syl2anc 666	. . . 4 |- ( ph -> ( ran F X. ran G ) e. Fin )
19		eqid 2450	. . . . . 6 |- ( u e. ran F , v e. ran G |-> ( u + v ) ) = ( u e. ran F , v e. ran G |-> ( u + v ) )
20		ovex 6316	. . . . . 6 |- ( u + v ) e. _V
21	19, 20	fnmpt2i 6859	. . . . 5 |- ( u e. ran F , v e. ran G |-> ( u + v ) ) Fn ( ran F X. ran G )
22		dffn4 5797	. . . . 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 212	. . . 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 7857	. . . 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 667	. . 3 |- ( ph -> ran ( u e. ran F , v e. ran G |-> ( u + v ) ) e. Fin )
26		eqid 2450	. . . . . . . . 9 |- ( x + y ) = ( x + y )
27		rspceov 6327	. . . . . . . . 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 1352	. . . . . . . 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 6316	. . . . . . . . 9 |- ( x + y ) e. _V
30		eqeq1 2454	. . . . . . . . . 10 |- ( w = ( x + y ) -> ( w = ( u + v ) <-> ( x + y ) = ( u + v ) ) )
31	30	2rexbidv 2907	. . . . . . . . 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 3184	. . . . . . . 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 216	. . . . . . 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 468	. . . . . 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 5726	. . . . . . . 8 |- ( F : RR --> RR -> F Fn RR )
36	5, 35	syl 17	. . . . . . 7 |- ( ph -> F Fn RR )
37		dffn3 5734	. . . . . . 7 |- ( F Fn RR <-> F : RR --> ran F )
38	36, 37	sylib 200	. . . . . 6 |- ( ph -> F : RR --> ran F )
39		ffn 5726	. . . . . . . 8 |- ( G : RR --> RR -> G Fn RR )
40	8, 39	syl 17	. . . . . . 7 |- ( ph -> G Fn RR )
41		dffn3 5734	. . . . . . 7 |- ( G Fn RR <-> G : RR --> ran G )
42	40, 41	sylib 200	. . . . . 6 |- ( ph -> G : RR --> ran G )
43	34, 38, 42, 10, 10, 11	off 6543	. . . . 5 |- ( ph -> ( F oF + G ) : RR --> { w | E. u e. ran F E. v e. ran G w = ( u + v ) } )
44		frn 5733	. . . . 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 6403	. . . 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 3478	. . 3 |- ( ph -> ran ( F oF + G ) C_ ran ( u e. ran F , v e. ran G |-> ( u + v ) ) )
48		ssfi 7789	. . 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 666	. 2 |- ( ph -> ran ( F oF + G ) e. Fin )
50		frn 5733	. . . . . . . 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 3570	. . . . . 6 |- ( ph -> ( ran ( F oF + G ) \ { 0 } ) C_ RR )
53	52	sselda 3431	. . . . 5 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> y e. RR )
54	53	recnd 9666	. . . 4 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> y e. CC )
55	3, 6	i1faddlem 22644	. . . 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 473	. . 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 467	. . . 4 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ran G e. Fin )
58	3	ad2antrr 731	. . . . . . . 8 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> F e. dom S.1 )
59		i1fmbf 22626	. . . . . . . 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 731	. . . . . . 7 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> F : RR --> RR )
62	12	ad2antrr 731	. . . . . . . . . 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 3554	. . . . . . . . . 10 |- ( y e. ( ran ( F oF + G ) \ { 0 } ) -> y e. ran ( F oF + G ) )
65	64	ad2antlr 732	. . . . . . . . 9 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> y e. ran ( F oF + G ) )
66	63, 65	sseldd 3432	. . . . . . . 8 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> y e. RR )
67	8	adantr 467	. . . . . . . . . 10 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> G : RR --> RR )
68		frn 5733	. . . . . . . . . 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 3431	. . . . . . . 8 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> z e. RR )
71	66, 70	resubcld 10044	. . . . . . 7 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> ( y - z ) e. RR )
72		mbfimasn 22583	. . . . . . 7 |- ( ( F e. MblFn /\ F : RR --> RR /\ ( y - z ) e. RR ) -> ( `' F " { ( y - z ) } ) e. dom vol )
73	60, 61, 71, 72	syl3anc 1267	. . . . . 6 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> ( `' F " { ( y - z ) } ) e. dom vol )
74	6	ad2antrr 731	. . . . . . . 8 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> G e. dom S.1 )
75		i1fmbf 22626	. . . . . . . 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 731	. . . . . . 7 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> G : RR --> RR )
78		mbfimasn 22583	. . . . . . 7 |- ( ( G e. MblFn /\ G : RR --> RR /\ z e. RR ) -> ( `' G " { z } ) e. dom vol )
79	76, 77, 70, 78	syl3anc 1267	. . . . . 6 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ran G ) -> ( `' G " { z } ) e. dom vol )
80		inmbl 22488	. . . . . 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 666	. . . . 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 2801	. . . 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 22490	. . . 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 666	. . 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 2528	. 2 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ( `' ( F oF + G ) " { y } ) e. dom vol )
86		mblvol 22477	. . . 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 22478	. . . . 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 3651	. . . . . . . . 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 722	. . . . . . . . 9 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( `' F " { ( y - z ) } ) e. dom vol )
93		mblss 22478	. . . . . . . . 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 22477	. . . . . . . . . 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 765	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> z = 0 )
98	97	oveq2d 6304	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( y - z ) = ( y - 0 ) )
99	54	adantr 467	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> y e. CC )
100	99	subid1d 9972	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( y - 0 ) = y )
101	98, 100	eqtrd 2484	. . . . . . . . . . . . 13 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( y - z ) = y )
102	101	sneqd 3979	. . . . . . . . . . . 12 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> { ( y - z ) } = { y } )
103	102	imaeq2d 5167	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( `' F " { ( y - z ) } ) = ( `' F " { y } ) )
104	103	fveq2d 5867	. . . . . . . . . 10 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( vol ` ( `' F " { ( y - z ) } ) ) = ( vol ` ( `' F " { y } ) ) )
105		i1fima2sn 22631	. . . . . . . . . . . 12 |- ( ( F e. dom S.1 /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ( vol ` ( `' F " { y } ) ) e. RR )
106	3, 105	sylan 474	. . . . . . . . . . 11 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ( vol ` ( `' F " { y } ) ) e. RR )
107	106	adantr 467	. . . . . . . . . 10 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( vol ` ( `' F " { y } ) ) e. RR )
108	104, 107	eqeltrd 2528	. . . . . . . . 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 2529	. . . . . . . 8 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ ( z e. ran G /\ z = 0 ) ) -> ( vol* ` ( `' F " { ( y - z ) } ) ) e. RR )
110		ovolsscl 22432	. . . . . . . 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 1267	. . . . . . 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 619	. . . . . 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 4096	. . . . . . . 8 |- ( z e. ( ran G \ { 0 } ) <-> ( z e. ran G /\ z =/= 0 ) )
114		inss2 3652	. . . . . . . . . 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 3554	. . . . . . . . . 10 |- ( z e. ( ran G \ { 0 } ) -> z e. ran G )
117		mblss 22478	. . . . . . . . . . 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 477	. . . . . . . . 9 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ( ran G \ { 0 } ) ) -> ( `' G " { z } ) C_ RR )
120		i1fima 22629	. . . . . . . . . . . . 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 731	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ( ran G \ { 0 } ) ) -> ( `' G " { z } ) e. dom vol )
123		mblvol 22477	. . . . . . . . . . 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 467	. . . . . . . . . . 11 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> G e. dom S.1 )
126		i1fima2sn 22631	. . . . . . . . . . 11 |- ( ( G e. dom S.1 /\ z e. ( ran G \ { 0 } ) ) -> ( vol ` ( `' G " { z } ) ) e. RR )
127	125, 126	sylan 474	. . . . . . . . . 10 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ( ran G \ { 0 } ) ) -> ( vol ` ( `' G " { z } ) ) e. RR )
128	124, 127	eqeltrrd 2529	. . . . . . . . 9 |- ( ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) /\ z e. ( ran G \ { 0 } ) ) -> ( vol* ` ( `' G " { z } ) ) e. RR )
129		ovolsscl 22432	. . . . . . . . 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 1267	. . . . . . . 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 479	. . . . . . 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 619	. . . . . 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 2709	. . . . 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 13793	. . . 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 5867	. . . . 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 3442	. . . . . . . 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 535	. . . . . . 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 2801	. . . . . 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 22447	. . . . . 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 666	. . . . 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 4422	. . . 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 22429	. . . 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 1267	. . 3 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ( vol* ` ( `' ( F oF + G ) " { y } ) ) e. RR )
144	87, 143	eqeltrd 2528	. 2 |- ( ( ph /\ y e. ( ran ( F oF + G ) \ { 0 } ) ) -> ( vol ` ( `' ( F oF + G ) " { y } ) ) e. RR )
145	12, 49, 85, 144	i1fd 22632	1 |- ( ph -> ( F oF + G ) e. dom S.1 )

Syntax hints:   -> wi 4   /\ wa 371   = wceq 1443   e. wcel 1886   {cab 2436   =/= wne 2621   A.wral 2736   E.wrex 2737   _Vcvv 3044   \ cdif 3400   i^i cin 3402   C_ wss 3403   {csn 3967   U_ciun 4277   class class class wbr 4401   X. cxp 4831   `'ccnv 4832   dom cdm 4833   ran crn 4834   "cima 4836   Fn wfn 5576   -->wf 5577   -onto->wfo 5579   ` cfv 5581  (class class class)co 6288   |-> cmpt2 6290   oFcof 6526   Fincfn 7566   CCcc 9534   RRcr 9535   0cc0 9536   + caddc 9539   <_ cle 9673   - cmin 9857   sum_csu 13745   vol*covol 22406   volcvol 22408  MblFncmbf 22565   S.1citg1 22566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-n0 10867  df-z 10935  df-uz 11157  df-q 11262  df-rp 11300  df-xadd 11407  df-ioo 11636  df-ico 11638  df-icc 11639  df-fz 11782  df-fzo 11913  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-clim 13545  df-sum 13746  df-xmet 18956  df-met 18957  df-ovol 22409  df-vol 22411  df-mbf 22570  df-itg1 22571

This theorem is referenced by:  itg1addlem4  22650  i1fsub  22659  itg2splitlem  22699  itg2split  22700  itg2addlem  22709  itg2addnc  31989  ftc1anclem3  32012  ftc1anclem5  32014  ftc1anclem8  32017