Theorem ftc1anclem4 33488

Description: Lemma for ftc1anc 33493. (Contributed by Brendan Leahy, 17-Jun-2018.)

Assertion

Ref	Expression
ftc1anclem4	|- ( ( F e. dom S.1 /\ G e. L^1 /\ G : RR --> RR ) -> ( S.2 ` ( t e. RR |-> ( abs ` ( ( G ` t ) - ( F ` t ) ) ) ) ) e. RR )

Distinct variable groups:   t, F   t, G

Proof of Theorem ftc1anclem4

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffvelrn 6357	. . . . . . . . . 10 |- ( ( G : RR --> RR /\ t e. RR ) -> ( G ` t ) e. RR )
2	1	recnd 10068	. . . . . . . . 9 |- ( ( G : RR --> RR /\ t e. RR ) -> ( G ` t ) e. CC )
3		i1ff 23443	. . . . . . . . . . 11 |- ( F e. dom S.1 -> F : RR --> RR )
4	3	ffvelrnda 6359	. . . . . . . . . 10 |- ( ( F e. dom S.1 /\ t e. RR ) -> ( F ` t ) e. RR )
5	4	recnd 10068	. . . . . . . . 9 |- ( ( F e. dom S.1 /\ t e. RR ) -> ( F ` t ) e. CC )
6		subcl 10280	. . . . . . . . 9 |- ( ( ( G ` t ) e. CC /\ ( F ` t ) e. CC ) -> ( ( G ` t ) - ( F ` t ) ) e. CC )
7	2, 5, 6	syl2anr 495	. . . . . . . 8 |- ( ( ( F e. dom S.1 /\ t e. RR ) /\ ( G : RR --> RR /\ t e. RR ) ) -> ( ( G ` t ) - ( F ` t ) ) e. CC )
8	7	anandirs 874	. . . . . . 7 |- ( ( ( F e. dom S.1 /\ G : RR --> RR ) /\ t e. RR ) -> ( ( G ` t ) - ( F ` t ) ) e. CC )
9	8	abscld 14175	. . . . . 6 |- ( ( ( F e. dom S.1 /\ G : RR --> RR ) /\ t e. RR ) -> ( abs ` ( ( G ` t ) - ( F ` t ) ) ) e. RR )
10	9	rexrd 10089	. . . . 5 |- ( ( ( F e. dom S.1 /\ G : RR --> RR ) /\ t e. RR ) -> ( abs ` ( ( G ` t ) - ( F ` t ) ) ) e. RR* )
11	8	absge0d 14183	. . . . 5 |- ( ( ( F e. dom S.1 /\ G : RR --> RR ) /\ t e. RR ) -> 0 <_ ( abs ` ( ( G ` t ) - ( F ` t ) ) ) )
12		elxrge0 12281	. . . . 5 |- ( ( abs ` ( ( G ` t ) - ( F ` t ) ) ) e. ( 0 [,] +oo ) <-> ( ( abs ` ( ( G ` t ) - ( F ` t ) ) ) e. RR* /\ 0 <_ ( abs ` ( ( G ` t ) - ( F ` t ) ) ) ) )
13	10, 11, 12	sylanbrc 698	. . . 4 |- ( ( ( F e. dom S.1 /\ G : RR --> RR ) /\ t e. RR ) -> ( abs ` ( ( G ` t ) - ( F ` t ) ) ) e. ( 0 [,] +oo ) )
14		eqid 2622	. . . 4 |- ( t e. RR |-> ( abs ` ( ( G ` t ) - ( F ` t ) ) ) ) = ( t e. RR |-> ( abs ` ( ( G ` t ) - ( F ` t ) ) ) )
15	13, 14	fmptd 6385	. . 3 |- ( ( F e. dom S.1 /\ G : RR --> RR ) -> ( t e. RR |-> ( abs ` ( ( G ` t ) - ( F ` t ) ) ) ) : RR --> ( 0 [,] +oo ) )
16	15	3adant2 1080	. 2 |- ( ( F e. dom S.1 /\ G e. L^1 /\ G : RR --> RR ) -> ( t e. RR |-> ( abs ` ( ( G ` t ) - ( F ` t ) ) ) ) : RR --> ( 0 [,] +oo ) )
17		reex 10027	. . . . . . 7 |- RR e. _V
18	17	a1i 11	. . . . . 6 |- ( ( F e. dom S.1 /\ G e. L^1 /\ G : RR --> RR ) -> RR e. _V )
19		fvexd 6203	. . . . . 6 |- ( ( ( F e. dom S.1 /\ G e. L^1 /\ G : RR --> RR ) /\ t e. RR ) -> ( abs ` ( G ` t ) ) e. _V )
20		fvexd 6203	. . . . . 6 |- ( ( ( F e. dom S.1 /\ G e. L^1 /\ G : RR --> RR ) /\ t e. RR ) -> ( abs ` ( F ` t ) ) e. _V )
21		eqidd 2623	. . . . . 6 |- ( ( F e. dom S.1 /\ G e. L^1 /\ G : RR --> RR ) -> ( t e. RR |-> ( abs ` ( G ` t ) ) ) = ( t e. RR |-> ( abs ` ( G ` t ) ) ) )
22		eqidd 2623	. . . . . 6 |- ( ( F e. dom S.1 /\ G e. L^1 /\ G : RR --> RR ) -> ( t e. RR |-> ( abs ` ( F ` t ) ) ) = ( t e. RR |-> ( abs ` ( F ` t ) ) ) )
23	18, 19, 20, 21, 22	offval2 6914	. . . . 5 |- ( ( F e. dom S.1 /\ G e. L^1 /\ G : RR --> RR ) -> ( ( t e. RR |-> ( abs ` ( G ` t ) ) ) oF + ( t e. RR |-> ( abs ` ( F ` t ) ) ) ) = ( t e. RR |-> ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) ) )
24	23	fveq2d 6195	. . . 4 |- ( ( F e. dom S.1 /\ G e. L^1 /\ G : RR --> RR ) -> ( S.2 ` ( ( t e. RR |-> ( abs ` ( G ` t ) ) ) oF + ( t e. RR |-> ( abs ` ( F ` t ) ) ) ) ) = ( S.2 ` ( t e. RR |-> ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) ) ) )
25		id 22	. . . . . . . . . 10 |- ( G : RR --> RR -> G : RR --> RR )
26	25	feqmptd 6249	. . . . . . . . 9 |- ( G : RR --> RR -> G = ( t e. RR |-> ( G ` t ) ) )
27		absf 14077	. . . . . . . . . . 11 |- abs : CC --> RR
28	27	a1i 11	. . . . . . . . . 10 |- ( G : RR --> RR -> abs : CC --> RR )
29	28	feqmptd 6249	. . . . . . . . 9 |- ( G : RR --> RR -> abs = ( x e. CC |-> ( abs ` x ) ) )
30		fveq2 6191	. . . . . . . . 9 |- ( x = ( G ` t ) -> ( abs ` x ) = ( abs ` ( G ` t ) ) )
31	2, 26, 29, 30	fmptco 6396	. . . . . . . 8 |- ( G : RR --> RR -> ( abs o. G ) = ( t e. RR |-> ( abs ` ( G ` t ) ) ) )
32	31	adantl 482	. . . . . . 7 |- ( ( G e. L^1 /\ G : RR --> RR ) -> ( abs o. G ) = ( t e. RR |-> ( abs ` ( G ` t ) ) ) )
33		iblmbf 23534	. . . . . . . . 9 |- ( G e. L^1 -> G e. MblFn )
34		ftc1anclem1 33485	. . . . . . . . 9 |- ( ( G : RR --> RR /\ G e. MblFn ) -> ( abs o. G ) e. MblFn )
35	33, 34	sylan2 491	. . . . . . . 8 |- ( ( G : RR --> RR /\ G e. L^1 ) -> ( abs o. G ) e. MblFn )
36	35	ancoms 469	. . . . . . 7 |- ( ( G e. L^1 /\ G : RR --> RR ) -> ( abs o. G ) e. MblFn )
37	32, 36	eqeltrrd 2702	. . . . . 6 |- ( ( G e. L^1 /\ G : RR --> RR ) -> ( t e. RR |-> ( abs ` ( G ` t ) ) ) e. MblFn )
38	37	3adant1 1079	. . . . 5 |- ( ( F e. dom S.1 /\ G e. L^1 /\ G : RR --> RR ) -> ( t e. RR |-> ( abs ` ( G ` t ) ) ) e. MblFn )
39	2	abscld 14175	. . . . . . . 8 |- ( ( G : RR --> RR /\ t e. RR ) -> ( abs ` ( G ` t ) ) e. RR )
40	2	absge0d 14183	. . . . . . . 8 |- ( ( G : RR --> RR /\ t e. RR ) -> 0 <_ ( abs ` ( G ` t ) ) )
41		elrege0 12278	. . . . . . . 8 |- ( ( abs ` ( G ` t ) ) e. ( 0 [,) +oo ) <-> ( ( abs ` ( G ` t ) ) e. RR /\ 0 <_ ( abs ` ( G ` t ) ) ) )
42	39, 40, 41	sylanbrc 698	. . . . . . 7 |- ( ( G : RR --> RR /\ t e. RR ) -> ( abs ` ( G ` t ) ) e. ( 0 [,) +oo ) )
43		eqid 2622	. . . . . . 7 |- ( t e. RR |-> ( abs ` ( G ` t ) ) ) = ( t e. RR |-> ( abs ` ( G ` t ) ) )
44	42, 43	fmptd 6385	. . . . . 6 |- ( G : RR --> RR -> ( t e. RR |-> ( abs ` ( G ` t ) ) ) : RR --> ( 0 [,) +oo ) )
45	44	3ad2ant3 1084	. . . . 5 |- ( ( F e. dom S.1 /\ G e. L^1 /\ G : RR --> RR ) -> ( t e. RR |-> ( abs ` ( G ` t ) ) ) : RR --> ( 0 [,) +oo ) )
46		iftrue 4092	. . . . . . . . 9 |- ( t e. RR -> if ( t e. RR , ( abs ` ( G ` t ) ) , 0 ) = ( abs ` ( G ` t ) ) )
47	46	mpteq2ia 4740	. . . . . . . 8 |- ( t e. RR |-> if ( t e. RR , ( abs ` ( G ` t ) ) , 0 ) ) = ( t e. RR |-> ( abs ` ( G ` t ) ) )
48	47	fveq2i 6194	. . . . . . 7 |- ( S.2 ` ( t e. RR |-> if ( t e. RR , ( abs ` ( G ` t ) ) , 0 ) ) ) = ( S.2 ` ( t e. RR |-> ( abs ` ( G ` t ) ) ) )
49	1	adantll 750	. . . . . . . . . 10 |- ( ( ( G e. L^1 /\ G : RR --> RR ) /\ t e. RR ) -> ( G ` t ) e. RR )
50		simpr 477	. . . . . . . . . . . 12 |- ( ( G e. L^1 /\ G : RR --> RR ) -> G : RR --> RR )
51	50	feqmptd 6249	. . . . . . . . . . 11 |- ( ( G e. L^1 /\ G : RR --> RR ) -> G = ( t e. RR |-> ( G ` t ) ) )
52		simpl 473	. . . . . . . . . . 11 |- ( ( G e. L^1 /\ G : RR --> RR ) -> G e. L^1 )
53	51, 52	eqeltrrd 2702	. . . . . . . . . 10 |- ( ( G e. L^1 /\ G : RR --> RR ) -> ( t e. RR |-> ( G ` t ) ) e. L^1 )
54	49, 53, 37	iblabsnc 33474	. . . . . . . . 9 |- ( ( G e. L^1 /\ G : RR --> RR ) -> ( t e. RR |-> ( abs ` ( G ` t ) ) ) e. L^1 )
55	39	adantll 750	. . . . . . . . . 10 |- ( ( ( G e. L^1 /\ G : RR --> RR ) /\ t e. RR ) -> ( abs ` ( G ` t ) ) e. RR )
56	40	adantll 750	. . . . . . . . . 10 |- ( ( ( G e. L^1 /\ G : RR --> RR ) /\ t e. RR ) -> 0 <_ ( abs ` ( G ` t ) ) )
57	55, 56	iblpos 23559	. . . . . . . . 9 |- ( ( G e. L^1 /\ G : RR --> RR ) -> ( ( t e. RR |-> ( abs ` ( G ` t ) ) ) e. L^1 <-> ( ( t e. RR |-> ( abs ` ( G ` t ) ) ) e. MblFn /\ ( S.2 ` ( t e. RR |-> if ( t e. RR , ( abs ` ( G ` t ) ) , 0 ) ) ) e. RR ) ) )
58	54, 57	mpbid 222	. . . . . . . 8 |- ( ( G e. L^1 /\ G : RR --> RR ) -> ( ( t e. RR |-> ( abs ` ( G ` t ) ) ) e. MblFn /\ ( S.2 ` ( t e. RR |-> if ( t e. RR , ( abs ` ( G ` t ) ) , 0 ) ) ) e. RR ) )
59	58	simprd 479	. . . . . . 7 |- ( ( G e. L^1 /\ G : RR --> RR ) -> ( S.2 ` ( t e. RR |-> if ( t e. RR , ( abs ` ( G ` t ) ) , 0 ) ) ) e. RR )
60	48, 59	syl5eqelr 2706	. . . . . 6 |- ( ( G e. L^1 /\ G : RR --> RR ) -> ( S.2 ` ( t e. RR |-> ( abs ` ( G ` t ) ) ) ) e. RR )
61	60	3adant1 1079	. . . . 5 |- ( ( F e. dom S.1 /\ G e. L^1 /\ G : RR --> RR ) -> ( S.2 ` ( t e. RR |-> ( abs ` ( G ` t ) ) ) ) e. RR )
62	5	abscld 14175	. . . . . . . 8 |- ( ( F e. dom S.1 /\ t e. RR ) -> ( abs ` ( F ` t ) ) e. RR )
63	5	absge0d 14183	. . . . . . . 8 |- ( ( F e. dom S.1 /\ t e. RR ) -> 0 <_ ( abs ` ( F ` t ) ) )
64		elrege0 12278	. . . . . . . 8 |- ( ( abs ` ( F ` t ) ) e. ( 0 [,) +oo ) <-> ( ( abs ` ( F ` t ) ) e. RR /\ 0 <_ ( abs ` ( F ` t ) ) ) )
65	62, 63, 64	sylanbrc 698	. . . . . . 7 |- ( ( F e. dom S.1 /\ t e. RR ) -> ( abs ` ( F ` t ) ) e. ( 0 [,) +oo ) )
66		eqid 2622	. . . . . . 7 |- ( t e. RR |-> ( abs ` ( F ` t ) ) ) = ( t e. RR |-> ( abs ` ( F ` t ) ) )
67	65, 66	fmptd 6385	. . . . . 6 |- ( F e. dom S.1 -> ( t e. RR |-> ( abs ` ( F ` t ) ) ) : RR --> ( 0 [,) +oo ) )
68	67	3ad2ant1 1082	. . . . 5 |- ( ( F e. dom S.1 /\ G e. L^1 /\ G : RR --> RR ) -> ( t e. RR |-> ( abs ` ( F ` t ) ) ) : RR --> ( 0 [,) +oo ) )
69		iftrue 4092	. . . . . . . . 9 |- ( t e. RR -> if ( t e. RR , ( abs ` ( F ` t ) ) , 0 ) = ( abs ` ( F ` t ) ) )
70	69	mpteq2ia 4740	. . . . . . . 8 |- ( t e. RR |-> if ( t e. RR , ( abs ` ( F ` t ) ) , 0 ) ) = ( t e. RR |-> ( abs ` ( F ` t ) ) )
71	70	fveq2i 6194	. . . . . . 7 |- ( S.2 ` ( t e. RR |-> if ( t e. RR , ( abs ` ( F ` t ) ) , 0 ) ) ) = ( S.2 ` ( t e. RR |-> ( abs ` ( F ` t ) ) ) )
72	3	feqmptd 6249	. . . . . . . . . . 11 |- ( F e. dom S.1 -> F = ( t e. RR |-> ( F ` t ) ) )
73		i1fibl 23574	. . . . . . . . . . 11 |- ( F e. dom S.1 -> F e. L^1 )
74	72, 73	eqeltrrd 2702	. . . . . . . . . 10 |- ( F e. dom S.1 -> ( t e. RR |-> ( F ` t ) ) e. L^1 )
75	27	a1i 11	. . . . . . . . . . . . 13 |- ( F e. dom S.1 -> abs : CC --> RR )
76	75	feqmptd 6249	. . . . . . . . . . . 12 |- ( F e. dom S.1 -> abs = ( x e. CC |-> ( abs ` x ) ) )
77		fveq2 6191	. . . . . . . . . . . 12 |- ( x = ( F ` t ) -> ( abs ` x ) = ( abs ` ( F ` t ) ) )
78	5, 72, 76, 77	fmptco 6396	. . . . . . . . . . 11 |- ( F e. dom S.1 -> ( abs o. F ) = ( t e. RR |-> ( abs ` ( F ` t ) ) ) )
79		i1fmbf 23442	. . . . . . . . . . . 12 |- ( F e. dom S.1 -> F e. MblFn )
80		ftc1anclem1 33485	. . . . . . . . . . . 12 |- ( ( F : RR --> RR /\ F e. MblFn ) -> ( abs o. F ) e. MblFn )
81	3, 79, 80	syl2anc 693	. . . . . . . . . . 11 |- ( F e. dom S.1 -> ( abs o. F ) e. MblFn )
82	78, 81	eqeltrrd 2702	. . . . . . . . . 10 |- ( F e. dom S.1 -> ( t e. RR |-> ( abs ` ( F ` t ) ) ) e. MblFn )
83	4, 74, 82	iblabsnc 33474	. . . . . . . . 9 |- ( F e. dom S.1 -> ( t e. RR |-> ( abs ` ( F ` t ) ) ) e. L^1 )
84	62, 63	iblpos 23559	. . . . . . . . 9 |- ( F e. dom S.1 -> ( ( t e. RR |-> ( abs ` ( F ` t ) ) ) e. L^1 <-> ( ( t e. RR |-> ( abs ` ( F ` t ) ) ) e. MblFn /\ ( S.2 ` ( t e. RR |-> if ( t e. RR , ( abs ` ( F ` t ) ) , 0 ) ) ) e. RR ) ) )
85	83, 84	mpbid 222	. . . . . . . 8 |- ( F e. dom S.1 -> ( ( t e. RR |-> ( abs ` ( F ` t ) ) ) e. MblFn /\ ( S.2 ` ( t e. RR |-> if ( t e. RR , ( abs ` ( F ` t ) ) , 0 ) ) ) e. RR ) )
86	85	simprd 479	. . . . . . 7 |- ( F e. dom S.1 -> ( S.2 ` ( t e. RR |-> if ( t e. RR , ( abs ` ( F ` t ) ) , 0 ) ) ) e. RR )
87	71, 86	syl5eqelr 2706	. . . . . 6 |- ( F e. dom S.1 -> ( S.2 ` ( t e. RR |-> ( abs ` ( F ` t ) ) ) ) e. RR )
88	87	3ad2ant1 1082	. . . . 5 |- ( ( F e. dom S.1 /\ G e. L^1 /\ G : RR --> RR ) -> ( S.2 ` ( t e. RR |-> ( abs ` ( F ` t ) ) ) ) e. RR )
89	38, 45, 61, 68, 88	itg2addnc 33464	. . . 4 |- ( ( F e. dom S.1 /\ G e. L^1 /\ G : RR --> RR ) -> ( S.2 ` ( ( t e. RR |-> ( abs ` ( G ` t ) ) ) oF + ( t e. RR |-> ( abs ` ( F ` t ) ) ) ) ) = ( ( S.2 ` ( t e. RR |-> ( abs ` ( G ` t ) ) ) ) + ( S.2 ` ( t e. RR |-> ( abs ` ( F ` t ) ) ) ) ) )
90	24, 89	eqtr3d 2658	. . 3 |- ( ( F e. dom S.1 /\ G e. L^1 /\ G : RR --> RR ) -> ( S.2 ` ( t e. RR |-> ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) ) ) = ( ( S.2 ` ( t e. RR |-> ( abs ` ( G ` t ) ) ) ) + ( S.2 ` ( t e. RR |-> ( abs ` ( F ` t ) ) ) ) ) )
91	61, 88	readdcld 10069	. . 3 |- ( ( F e. dom S.1 /\ G e. L^1 /\ G : RR --> RR ) -> ( ( S.2 ` ( t e. RR |-> ( abs ` ( G ` t ) ) ) ) + ( S.2 ` ( t e. RR |-> ( abs ` ( F ` t ) ) ) ) ) e. RR )
92	90, 91	eqeltrd 2701	. 2 |- ( ( F e. dom S.1 /\ G e. L^1 /\ G : RR --> RR ) -> ( S.2 ` ( t e. RR |-> ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) ) ) e. RR )
93		readdcl 10019	. . . . . . . . 9 |- ( ( ( abs ` ( G ` t ) ) e. RR /\ ( abs ` ( F ` t ) ) e. RR ) -> ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) e. RR )
94	39, 62, 93	syl2anr 495	. . . . . . . 8 |- ( ( ( F e. dom S.1 /\ t e. RR ) /\ ( G : RR --> RR /\ t e. RR ) ) -> ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) e. RR )
95	94	anandirs 874	. . . . . . 7 |- ( ( ( F e. dom S.1 /\ G : RR --> RR ) /\ t e. RR ) -> ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) e. RR )
96	95	rexrd 10089	. . . . . 6 |- ( ( ( F e. dom S.1 /\ G : RR --> RR ) /\ t e. RR ) -> ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) e. RR* )
97	39	adantll 750	. . . . . . 7 |- ( ( ( F e. dom S.1 /\ G : RR --> RR ) /\ t e. RR ) -> ( abs ` ( G ` t ) ) e. RR )
98	62	adantlr 751	. . . . . . 7 |- ( ( ( F e. dom S.1 /\ G : RR --> RR ) /\ t e. RR ) -> ( abs ` ( F ` t ) ) e. RR )
99	40	adantll 750	. . . . . . 7 |- ( ( ( F e. dom S.1 /\ G : RR --> RR ) /\ t e. RR ) -> 0 <_ ( abs ` ( G ` t ) ) )
100	63	adantlr 751	. . . . . . 7 |- ( ( ( F e. dom S.1 /\ G : RR --> RR ) /\ t e. RR ) -> 0 <_ ( abs ` ( F ` t ) ) )
101	97, 98, 99, 100	addge0d 10603	. . . . . 6 |- ( ( ( F e. dom S.1 /\ G : RR --> RR ) /\ t e. RR ) -> 0 <_ ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) )
102		elxrge0 12281	. . . . . 6 |- ( ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) e. ( 0 [,] +oo ) <-> ( ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) e. RR* /\ 0 <_ ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) ) )
103	96, 101, 102	sylanbrc 698	. . . . 5 |- ( ( ( F e. dom S.1 /\ G : RR --> RR ) /\ t e. RR ) -> ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) e. ( 0 [,] +oo ) )
104		eqid 2622	. . . . 5 |- ( t e. RR |-> ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) ) = ( t e. RR |-> ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) )
105	103, 104	fmptd 6385	. . . 4 |- ( ( F e. dom S.1 /\ G : RR --> RR ) -> ( t e. RR |-> ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) ) : RR --> ( 0 [,] +oo ) )
106	105	3adant2 1080	. . 3 |- ( ( F e. dom S.1 /\ G e. L^1 /\ G : RR --> RR ) -> ( t e. RR |-> ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) ) : RR --> ( 0 [,] +oo ) )
107		abs2dif2 14073	. . . . . . . 8 |- ( ( ( G ` t ) e. CC /\ ( F ` t ) e. CC ) -> ( abs ` ( ( G ` t ) - ( F ` t ) ) ) <_ ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) )
108	2, 5, 107	syl2anr 495	. . . . . . 7 |- ( ( ( F e. dom S.1 /\ t e. RR ) /\ ( G : RR --> RR /\ t e. RR ) ) -> ( abs ` ( ( G ` t ) - ( F ` t ) ) ) <_ ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) )
109	108	anandirs 874	. . . . . 6 |- ( ( ( F e. dom S.1 /\ G : RR --> RR ) /\ t e. RR ) -> ( abs ` ( ( G ` t ) - ( F ` t ) ) ) <_ ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) )
110	109	ralrimiva 2966	. . . . 5 |- ( ( F e. dom S.1 /\ G : RR --> RR ) -> A. t e. RR ( abs ` ( ( G ` t ) - ( F ` t ) ) ) <_ ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) )
111	17	a1i 11	. . . . . 6 |- ( ( F e. dom S.1 /\ G : RR --> RR ) -> RR e. _V )
112		eqidd 2623	. . . . . 6 |- ( ( F e. dom S.1 /\ G : RR --> RR ) -> ( t e. RR |-> ( abs ` ( ( G ` t ) - ( F ` t ) ) ) ) = ( t e. RR |-> ( abs ` ( ( G ` t ) - ( F ` t ) ) ) ) )
113		eqidd 2623	. . . . . 6 |- ( ( F e. dom S.1 /\ G : RR --> RR ) -> ( t e. RR |-> ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) ) = ( t e. RR |-> ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) ) )
114	111, 9, 95, 112, 113	ofrfval2 6915	. . . . 5 |- ( ( F e. dom S.1 /\ G : RR --> RR ) -> ( ( t e. RR |-> ( abs ` ( ( G ` t ) - ( F ` t ) ) ) ) oR <_ ( t e. RR |-> ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) ) <-> A. t e. RR ( abs ` ( ( G ` t ) - ( F ` t ) ) ) <_ ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) ) )
115	110, 114	mpbird 247	. . . 4 |- ( ( F e. dom S.1 /\ G : RR --> RR ) -> ( t e. RR |-> ( abs ` ( ( G ` t ) - ( F ` t ) ) ) ) oR <_ ( t e. RR |-> ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) ) )
116	115	3adant2 1080	. . 3 |- ( ( F e. dom S.1 /\ G e. L^1 /\ G : RR --> RR ) -> ( t e. RR |-> ( abs ` ( ( G ` t ) - ( F ` t ) ) ) ) oR <_ ( t e. RR |-> ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) ) )
117		itg2le 23506	. . 3 |- ( ( ( t e. RR |-> ( abs ` ( ( G ` t ) - ( F ` t ) ) ) ) : RR --> ( 0 [,] +oo ) /\ ( t e. RR |-> ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) ) : RR --> ( 0 [,] +oo ) /\ ( t e. RR |-> ( abs ` ( ( G ` t ) - ( F ` t ) ) ) ) oR <_ ( t e. RR |-> ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) ) ) -> ( S.2 ` ( t e. RR |-> ( abs ` ( ( G ` t ) - ( F ` t ) ) ) ) ) <_ ( S.2 ` ( t e. RR |-> ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) ) ) )
118	16, 106, 116, 117	syl3anc 1326	. 2 |- ( ( F e. dom S.1 /\ G e. L^1 /\ G : RR --> RR ) -> ( S.2 ` ( t e. RR |-> ( abs ` ( ( G ` t ) - ( F ` t ) ) ) ) ) <_ ( S.2 ` ( t e. RR |-> ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) ) ) )
119		itg2lecl 23505	. 2 |- ( ( ( t e. RR |-> ( abs ` ( ( G ` t ) - ( F ` t ) ) ) ) : RR --> ( 0 [,] +oo ) /\ ( S.2 ` ( t e. RR |-> ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) ) ) e. RR /\ ( S.2 ` ( t e. RR |-> ( abs ` ( ( G ` t ) - ( F ` t ) ) ) ) ) <_ ( S.2 ` ( t e. RR |-> ( ( abs ` ( G ` t ) ) + ( abs ` ( F ` t ) ) ) ) ) ) -> ( S.2 ` ( t e. RR |-> ( abs ` ( ( G ` t ) - ( F ` t ) ) ) ) ) e. RR )
120	16, 92, 118, 119	syl3anc 1326	1 |- ( ( F e. dom S.1 /\ G e. L^1 /\ G : RR --> RR ) -> ( S.2 ` ( t e. RR |-> ( abs ` ( ( G ` t ) - ( F ` t ) ) ) ) ) e. RR )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   oRcofr 6896   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   +oocpnf 10071   RR*cxr 10073   <_ cle 10075   - cmin 10266   [,)cico 12177   [,]cicc 12178   abscabs 13974  MblFncmbf 23383   S.1citg1 23384   S.2citg2 23385   L^1cibl 23386

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-ofr 6898  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-fi 8317  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-xneg 11946  df-xadd 11947  df-xmul 11948  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-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cmp 21190  df-ovol 23233  df-vol 23234  df-mbf 23388  df-itg1 23389  df-itg2 23390  df-ibl 23391  df-0p 23437

This theorem is referenced by:  ftc1anclem5  33489  ftc1anclem6  33490