Theorem itgsubsticclem 40191

Description: lemma for itgsubsticc 40192. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
itgsubsticclem.1	|- F = ( u e. ( K [,] L ) |-> C )
itgsubsticclem.2	|- G = ( u e. RR |-> if ( u e. ( K [,] L ) , ( F ` u ) , if ( u < K , ( F ` K ) , ( F ` L ) ) ) )
itgsubsticclem.3	|- ( ph -> X e. RR )
itgsubsticclem.4	|- ( ph -> Y e. RR )
itgsubsticclem.5	|- ( ph -> X <_ Y )
itgsubsticclem.6	|- ( ph -> ( x e. ( X [,] Y ) |-> A ) e. ( ( X [,] Y ) -cn-> ( K [,] L ) ) )
itgsubsticclem.7	|- ( ph -> ( x e. ( X (,) Y ) |-> B ) e. ( ( ( X (,) Y ) -cn-> CC ) i^i L^1 ) )
itgsubsticclem.8	|- ( ph -> F e. ( ( K [,] L ) -cn-> CC ) )
itgsubsticclem.9	|- ( ph -> K e. RR )
itgsubsticclem.10	|- ( ph -> L e. RR )
itgsubsticclem.11	|- ( ph -> K <_ L )
itgsubsticclem.12	|- ( ph -> ( RR _D ( x e. ( X [,] Y ) |-> A ) ) = ( x e. ( X (,) Y ) |-> B ) )
itgsubsticclem.13	|- ( u = A -> C = E )
itgsubsticclem.14	|- ( x = X -> A = K )
itgsubsticclem.15	|- ( x = Y -> A = L )

Assertion

Ref	Expression
itgsubsticclem	|- ( ph -> S__ [ K -> L ] C _d u = S__ [ X -> Y ] ( E x. B ) _d x )

Distinct variable groups:   u, A   u, E   x, G   u, K, x   u, L, x   u, X, x   u, Y, x   ph, u, x

Allowed substitution hints:   A( x)   B( x, u)   C( x, u)   E( x)   F( x, u)   G( u)

Proof of Theorem itgsubsticclem

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 |- ( u = w -> ( G ` u ) = ( G ` w ) )
2		nfcv 2764	. . . 4 |- F/_ w ( G ` u )
3		itgsubsticclem.2	. . . . . 6 |- G = ( u e. RR |-> if ( u e. ( K [,] L ) , ( F ` u ) , if ( u < K , ( F ` K ) , ( F ` L ) ) ) )
4		nfmpt1 4747	. . . . . 6 |- F/_ u ( u e. RR |-> if ( u e. ( K [,] L ) , ( F ` u ) , if ( u < K , ( F ` K ) , ( F ` L ) ) ) )
5	3, 4	nfcxfr 2762	. . . . 5 |- F/_ u G
6		nfcv 2764	. . . . 5 |- F/_ u w
7	5, 6	nffv 6198	. . . 4 |- F/_ u ( G ` w )
8	1, 2, 7	cbvditg 23618	. . 3 |- S__ [ K -> L ] ( G ` u ) _d u = S__ [ K -> L ] ( G ` w ) _d w
9		itgsubsticclem.11	. . . 4 |- ( ph -> K <_ L )
10		itgsubsticclem.9	. . . . . . . . 9 |- ( ph -> K e. RR )
11		itgsubsticclem.10	. . . . . . . . 9 |- ( ph -> L e. RR )
12	10, 11	iccssred 39727	. . . . . . . 8 |- ( ph -> ( K [,] L ) C_ RR )
13	12	adantr 481	. . . . . . 7 |- ( ( ph /\ u e. ( K (,) L ) ) -> ( K [,] L ) C_ RR )
14		ioossicc 12259	. . . . . . . . 9 |- ( K (,) L ) C_ ( K [,] L )
15	14	sseli 3599	. . . . . . . 8 |- ( u e. ( K (,) L ) -> u e. ( K [,] L ) )
16	15	adantl 482	. . . . . . 7 |- ( ( ph /\ u e. ( K (,) L ) ) -> u e. ( K [,] L ) )
17	13, 16	sseldd 3604	. . . . . 6 |- ( ( ph /\ u e. ( K (,) L ) ) -> u e. RR )
18	16	iftrued 4094	. . . . . . 7 |- ( ( ph /\ u e. ( K (,) L ) ) -> if ( u e. ( K [,] L ) , ( F ` u ) , if ( u < K , ( F ` K ) , ( F ` L ) ) ) = ( F ` u ) )
19		itgsubsticclem.1	. . . . . . . . . . . . 13 |- F = ( u e. ( K [,] L ) |-> C )
20	19	a1i 11	. . . . . . . . . . . 12 |- ( ph -> F = ( u e. ( K [,] L ) |-> C ) )
21		itgsubsticclem.8	. . . . . . . . . . . . 13 |- ( ph -> F e. ( ( K [,] L ) -cn-> CC ) )
22		cncff 22696	. . . . . . . . . . . . 13 |- ( F e. ( ( K [,] L ) -cn-> CC ) -> F : ( K [,] L ) --> CC )
23	21, 22	syl 17	. . . . . . . . . . . 12 |- ( ph -> F : ( K [,] L ) --> CC )
24	20, 23	feq1dd 39347	. . . . . . . . . . 11 |- ( ph -> ( u e. ( K [,] L ) |-> C ) : ( K [,] L ) --> CC )
25	24	mptex2 6384	. . . . . . . . . 10 |- ( ( ph /\ u e. ( K [,] L ) ) -> C e. CC )
26	16, 25	syldan 487	. . . . . . . . 9 |- ( ( ph /\ u e. ( K (,) L ) ) -> C e. CC )
27	19	fvmpt2 6291	. . . . . . . . 9 |- ( ( u e. ( K [,] L ) /\ C e. CC ) -> ( F ` u ) = C )
28	16, 26, 27	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ u e. ( K (,) L ) ) -> ( F ` u ) = C )
29	28, 26	eqeltrd 2701	. . . . . . 7 |- ( ( ph /\ u e. ( K (,) L ) ) -> ( F ` u ) e. CC )
30	18, 29	eqeltrd 2701	. . . . . 6 |- ( ( ph /\ u e. ( K (,) L ) ) -> if ( u e. ( K [,] L ) , ( F ` u ) , if ( u < K , ( F ` K ) , ( F ` L ) ) ) e. CC )
31	3	fvmpt2 6291	. . . . . 6 |- ( ( u e. RR /\ if ( u e. ( K [,] L ) , ( F ` u ) , if ( u < K , ( F ` K ) , ( F ` L ) ) ) e. CC ) -> ( G ` u ) = if ( u e. ( K [,] L ) , ( F ` u ) , if ( u < K , ( F ` K ) , ( F ` L ) ) ) )
32	17, 30, 31	syl2anc 693	. . . . 5 |- ( ( ph /\ u e. ( K (,) L ) ) -> ( G ` u ) = if ( u e. ( K [,] L ) , ( F ` u ) , if ( u < K , ( F ` K ) , ( F ` L ) ) ) )
33	32, 18, 28	3eqtrd 2660	. . . 4 |- ( ( ph /\ u e. ( K (,) L ) ) -> ( G ` u ) = C )
34	9, 33	ditgeq3d 40180	. . 3 |- ( ph -> S__ [ K -> L ] ( G ` u ) _d u = S__ [ K -> L ] C _d u )
35		itgsubsticclem.3	. . . 4 |- ( ph -> X e. RR )
36		itgsubsticclem.4	. . . 4 |- ( ph -> Y e. RR )
37		itgsubsticclem.5	. . . 4 |- ( ph -> X <_ Y )
38		mnfxr 10096	. . . . 5 |- -oo e. RR*
39	38	a1i 11	. . . 4 |- ( ph -> -oo e. RR* )
40		pnfxr 10092	. . . . 5 |- +oo e. RR*
41	40	a1i 11	. . . 4 |- ( ph -> +oo e. RR* )
42		ioomax 12248	. . . . . . . . 9 |- ( -oo (,) +oo ) = RR
43	42	eqcomi 2631	. . . . . . . 8 |- RR = ( -oo (,) +oo )
44	43	a1i 11	. . . . . . 7 |- ( ph -> RR = ( -oo (,) +oo ) )
45	12, 44	sseqtrd 3641	. . . . . 6 |- ( ph -> ( K [,] L ) C_ ( -oo (,) +oo ) )
46		ax-resscn 9993	. . . . . . 7 |- RR C_ CC
47	44, 46	syl6eqssr 3656	. . . . . 6 |- ( ph -> ( -oo (,) +oo ) C_ CC )
48		cncfss 22702	. . . . . 6 |- ( ( ( K [,] L ) C_ ( -oo (,) +oo ) /\ ( -oo (,) +oo ) C_ CC ) -> ( ( X [,] Y ) -cn-> ( K [,] L ) ) C_ ( ( X [,] Y ) -cn-> ( -oo (,) +oo ) ) )
49	45, 47, 48	syl2anc 693	. . . . 5 |- ( ph -> ( ( X [,] Y ) -cn-> ( K [,] L ) ) C_ ( ( X [,] Y ) -cn-> ( -oo (,) +oo ) ) )
50		itgsubsticclem.6	. . . . 5 |- ( ph -> ( x e. ( X [,] Y ) |-> A ) e. ( ( X [,] Y ) -cn-> ( K [,] L ) ) )
51	49, 50	sseldd 3604	. . . 4 |- ( ph -> ( x e. ( X [,] Y ) |-> A ) e. ( ( X [,] Y ) -cn-> ( -oo (,) +oo ) ) )
52		itgsubsticclem.7	. . . 4 |- ( ph -> ( x e. ( X (,) Y ) |-> B ) e. ( ( ( X (,) Y ) -cn-> CC ) i^i L^1 ) )
53		nfmpt1 4747	. . . . . . . . . . 11 |- F/_ u ( u e. ( K [,] L ) |-> C )
54	19, 53	nfcxfr 2762	. . . . . . . . . 10 |- F/_ u F
55		eqid 2622	. . . . . . . . . 10 |- ( topGen ` ran (,) ) = ( topGen ` ran (,) )
56		eqid 2622	. . . . . . . . . 10 |- U. ( TopOpen ` ℂfld ) = U. ( TopOpen ` ℂfld )
57		eqid 2622	. . . . . . . . . . . 12 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
58	57	cnfldtop 22587	. . . . . . . . . . 11 |- ( TopOpen ` ℂfld ) e. Top
59	58	a1i 11	. . . . . . . . . 10 |- ( ph -> ( TopOpen ` ℂfld ) e. Top )
60	12, 46	syl6ss 3615	. . . . . . . . . . . . 13 |- ( ph -> ( K [,] L ) C_ CC )
61		ssid 3624	. . . . . . . . . . . . 13 |- CC C_ CC
62		eqid 2622	. . . . . . . . . . . . . 14 |- ( ( TopOpen ` ℂfld ) ↾t ( K [,] L ) ) = ( ( TopOpen ` ℂfld ) ↾t ( K [,] L ) )
63		unicntop 22589	. . . . . . . . . . . . . . . . 17 |- CC = U. ( TopOpen ` ℂfld )
64	63	restid 16094	. . . . . . . . . . . . . . . 16 |- ( ( TopOpen ` ℂfld ) e. Top -> ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld ) )
65	58, 64	ax-mp 5	. . . . . . . . . . . . . . 15 |- ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld )
66	65	eqcomi 2631	. . . . . . . . . . . . . 14 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
67	57, 62, 66	cncfcn 22712	. . . . . . . . . . . . 13 |- ( ( ( K [,] L ) C_ CC /\ CC C_ CC ) -> ( ( K [,] L ) -cn-> CC ) = ( ( ( TopOpen ` ℂfld ) ↾t ( K [,] L ) ) Cn ( TopOpen ` ℂfld ) ) )
68	60, 61, 67	sylancl 694	. . . . . . . . . . . 12 |- ( ph -> ( ( K [,] L ) -cn-> CC ) = ( ( ( TopOpen ` ℂfld ) ↾t ( K [,] L ) ) Cn ( TopOpen ` ℂfld ) ) )
69		reex 10027	. . . . . . . . . . . . . . . 16 |- RR e. _V
70	69	a1i 11	. . . . . . . . . . . . . . 15 |- ( ph -> RR e. _V )
71		restabs 20969	. . . . . . . . . . . . . . 15 |- ( ( ( TopOpen ` ℂfld ) e. Top /\ ( K [,] L ) C_ RR /\ RR e. _V ) -> ( ( ( TopOpen ` ℂfld ) ↾t RR ) ↾t ( K [,] L ) ) = ( ( TopOpen ` ℂfld ) ↾t ( K [,] L ) ) )
72	59, 12, 70, 71	syl3anc 1326	. . . . . . . . . . . . . 14 |- ( ph -> ( ( ( TopOpen ` ℂfld ) ↾t RR ) ↾t ( K [,] L ) ) = ( ( TopOpen ` ℂfld ) ↾t ( K [,] L ) ) )
73	57	tgioo2 22606	. . . . . . . . . . . . . . . . 17 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
74	73	eqcomi 2631	. . . . . . . . . . . . . . . 16 |- ( ( TopOpen ` ℂfld ) ↾t RR ) = ( topGen ` ran (,) )
75	74	a1i 11	. . . . . . . . . . . . . . 15 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t RR ) = ( topGen ` ran (,) ) )
76	75	oveq1d 6665	. . . . . . . . . . . . . 14 |- ( ph -> ( ( ( TopOpen ` ℂfld ) ↾t RR ) ↾t ( K [,] L ) ) = ( ( topGen ` ran (,) ) ↾t ( K [,] L ) ) )
77	72, 76	eqtr3d 2658	. . . . . . . . . . . . 13 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t ( K [,] L ) ) = ( ( topGen ` ran (,) ) ↾t ( K [,] L ) ) )
78	77	oveq1d 6665	. . . . . . . . . . . 12 |- ( ph -> ( ( ( TopOpen ` ℂfld ) ↾t ( K [,] L ) ) Cn ( TopOpen ` ℂfld ) ) = ( ( ( topGen ` ran (,) ) ↾t ( K [,] L ) ) Cn ( TopOpen ` ℂfld ) ) )
79	68, 78	eqtrd 2656	. . . . . . . . . . 11 |- ( ph -> ( ( K [,] L ) -cn-> CC ) = ( ( ( topGen ` ran (,) ) ↾t ( K [,] L ) ) Cn ( TopOpen ` ℂfld ) ) )
80	21, 79	eleqtrd 2703	. . . . . . . . . 10 |- ( ph -> F e. ( ( ( topGen ` ran (,) ) ↾t ( K [,] L ) ) Cn ( TopOpen ` ℂfld ) ) )
81	54, 55, 56, 3, 10, 11, 9, 59, 80	icccncfext 40100	. . . . . . . . 9 |- ( ph -> ( G e. ( ( topGen ` ran (,) ) Cn ( ( TopOpen ` ℂfld ) ↾t ran F ) ) /\ ( G |` ( K [,] L ) ) = F ) )
82	81	simpld 475	. . . . . . . 8 |- ( ph -> G e. ( ( topGen ` ran (,) ) Cn ( ( TopOpen ` ℂfld ) ↾t ran F ) ) )
83		uniretop 22566	. . . . . . . . 9 |- RR = U. ( topGen ` ran (,) )
84		eqid 2622	. . . . . . . . 9 |- U. ( ( TopOpen ` ℂfld ) ↾t ran F ) = U. ( ( TopOpen ` ℂfld ) ↾t ran F )
85	83, 84	cnf 21050	. . . . . . . 8 |- ( G e. ( ( topGen ` ran (,) ) Cn ( ( TopOpen ` ℂfld ) ↾t ran F ) ) -> G : RR --> U. ( ( TopOpen ` ℂfld ) ↾t ran F ) )
86	82, 85	syl 17	. . . . . . 7 |- ( ph -> G : RR --> U. ( ( TopOpen ` ℂfld ) ↾t ran F ) )
87	44	feq2d 6031	. . . . . . 7 |- ( ph -> ( G : RR --> U. ( ( TopOpen ` ℂfld ) ↾t ran F ) <-> G : ( -oo (,) +oo ) --> U. ( ( TopOpen ` ℂfld ) ↾t ran F ) ) )
88	86, 87	mpbid 222	. . . . . 6 |- ( ph -> G : ( -oo (,) +oo ) --> U. ( ( TopOpen ` ℂfld ) ↾t ran F ) )
89	88	feqmptd 6249	. . . . 5 |- ( ph -> G = ( w e. ( -oo (,) +oo ) |-> ( G ` w ) ) )
90		frn 6053	. . . . . . . 8 |- ( F : ( K [,] L ) --> CC -> ran F C_ CC )
91	23, 90	syl 17	. . . . . . 7 |- ( ph -> ran F C_ CC )
92		cncfss 22702	. . . . . . 7 |- ( ( ran F C_ CC /\ CC C_ CC ) -> ( ( -oo (,) +oo ) -cn-> ran F ) C_ ( ( -oo (,) +oo ) -cn-> CC ) )
93	91, 61, 92	sylancl 694	. . . . . 6 |- ( ph -> ( ( -oo (,) +oo ) -cn-> ran F ) C_ ( ( -oo (,) +oo ) -cn-> CC ) )
94	43	oveq2i 6661	. . . . . . . . . . 11 |- ( ( TopOpen ` ℂfld ) ↾t RR ) = ( ( TopOpen ` ℂfld ) ↾t ( -oo (,) +oo ) )
95	73, 94	eqtri 2644	. . . . . . . . . 10 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t ( -oo (,) +oo ) )
96		eqid 2622	. . . . . . . . . 10 |- ( ( TopOpen ` ℂfld ) ↾t ran F ) = ( ( TopOpen ` ℂfld ) ↾t ran F )
97	57, 95, 96	cncfcn 22712	. . . . . . . . 9 |- ( ( ( -oo (,) +oo ) C_ CC /\ ran F C_ CC ) -> ( ( -oo (,) +oo ) -cn-> ran F ) = ( ( topGen ` ran (,) ) Cn ( ( TopOpen ` ℂfld ) ↾t ran F ) ) )
98	47, 91, 97	syl2anc 693	. . . . . . . 8 |- ( ph -> ( ( -oo (,) +oo ) -cn-> ran F ) = ( ( topGen ` ran (,) ) Cn ( ( TopOpen ` ℂfld ) ↾t ran F ) ) )
99	98	eqcomd 2628	. . . . . . 7 |- ( ph -> ( ( topGen ` ran (,) ) Cn ( ( TopOpen ` ℂfld ) ↾t ran F ) ) = ( ( -oo (,) +oo ) -cn-> ran F ) )
100	82, 99	eleqtrd 2703	. . . . . 6 |- ( ph -> G e. ( ( -oo (,) +oo ) -cn-> ran F ) )
101	93, 100	sseldd 3604	. . . . 5 |- ( ph -> G e. ( ( -oo (,) +oo ) -cn-> CC ) )
102	89, 101	eqeltrrd 2702	. . . 4 |- ( ph -> ( w e. ( -oo (,) +oo ) |-> ( G ` w ) ) e. ( ( -oo (,) +oo ) -cn-> CC ) )
103		itgsubsticclem.12	. . . 4 |- ( ph -> ( RR _D ( x e. ( X [,] Y ) |-> A ) ) = ( x e. ( X (,) Y ) |-> B ) )
104		fveq2 6191	. . . 4 |- ( w = A -> ( G ` w ) = ( G ` A ) )
105		itgsubsticclem.14	. . . 4 |- ( x = X -> A = K )
106		itgsubsticclem.15	. . . 4 |- ( x = Y -> A = L )
107	35, 36, 37, 39, 41, 51, 52, 102, 103, 104, 105, 106	itgsubst 23812	. . 3 |- ( ph -> S__ [ K -> L ] ( G ` w ) _d w = S__ [ X -> Y ] ( ( G ` A ) x. B ) _d x )
108	8, 34, 107	3eqtr3a 2680	. 2 |- ( ph -> S__ [ K -> L ] C _d u = S__ [ X -> Y ] ( ( G ` A ) x. B ) _d x )
109	3	a1i 11	. . . . 5 |- ( ( ph /\ x e. ( X (,) Y ) ) -> G = ( u e. RR |-> if ( u e. ( K [,] L ) , ( F ` u ) , if ( u < K , ( F ` K ) , ( F ` L ) ) ) ) )
110		simpr 477	. . . . . . . 8 |- ( ( ( ph /\ x e. ( X (,) Y ) ) /\ u = A ) -> u = A )
111	57	cnfldtopon 22586	. . . . . . . . . . . . . 14 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
112	35, 36	iccssred 39727	. . . . . . . . . . . . . . 15 |- ( ph -> ( X [,] Y ) C_ RR )
113	112, 46	syl6ss 3615	. . . . . . . . . . . . . 14 |- ( ph -> ( X [,] Y ) C_ CC )
114		resttopon 20965	. . . . . . . . . . . . . 14 |- ( ( ( TopOpen ` ℂfld ) e. (TopOn ` CC ) /\ ( X [,] Y ) C_ CC ) -> ( ( TopOpen ` ℂfld ) ↾t ( X [,] Y ) ) e. (TopOn ` ( X [,] Y ) ) )
115	111, 113, 114	sylancr 695	. . . . . . . . . . . . 13 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t ( X [,] Y ) ) e. (TopOn ` ( X [,] Y ) ) )
116		resttopon 20965	. . . . . . . . . . . . . 14 |- ( ( ( TopOpen ` ℂfld ) e. (TopOn ` CC ) /\ ( K [,] L ) C_ CC ) -> ( ( TopOpen ` ℂfld ) ↾t ( K [,] L ) ) e. (TopOn ` ( K [,] L ) ) )
117	111, 60, 116	sylancr 695	. . . . . . . . . . . . 13 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t ( K [,] L ) ) e. (TopOn ` ( K [,] L ) ) )
118		eqid 2622	. . . . . . . . . . . . . . . 16 |- ( ( TopOpen ` ℂfld ) ↾t ( X [,] Y ) ) = ( ( TopOpen ` ℂfld ) ↾t ( X [,] Y ) )
119	57, 118, 62	cncfcn 22712	. . . . . . . . . . . . . . 15 |- ( ( ( X [,] Y ) C_ CC /\ ( K [,] L ) C_ CC ) -> ( ( X [,] Y ) -cn-> ( K [,] L ) ) = ( ( ( TopOpen ` ℂfld ) ↾t ( X [,] Y ) ) Cn ( ( TopOpen ` ℂfld ) ↾t ( K [,] L ) ) ) )
120	113, 60, 119	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ph -> ( ( X [,] Y ) -cn-> ( K [,] L ) ) = ( ( ( TopOpen ` ℂfld ) ↾t ( X [,] Y ) ) Cn ( ( TopOpen ` ℂfld ) ↾t ( K [,] L ) ) ) )
121	50, 120	eleqtrd 2703	. . . . . . . . . . . . 13 |- ( ph -> ( x e. ( X [,] Y ) |-> A ) e. ( ( ( TopOpen ` ℂfld ) ↾t ( X [,] Y ) ) Cn ( ( TopOpen ` ℂfld ) ↾t ( K [,] L ) ) ) )
122		cnf2 21053	. . . . . . . . . . . . 13 |- ( ( ( ( TopOpen ` ℂfld ) ↾t ( X [,] Y ) ) e. (TopOn ` ( X [,] Y ) ) /\ ( ( TopOpen ` ℂfld ) ↾t ( K [,] L ) ) e. (TopOn ` ( K [,] L ) ) /\ ( x e. ( X [,] Y ) |-> A ) e. ( ( ( TopOpen ` ℂfld ) ↾t ( X [,] Y ) ) Cn ( ( TopOpen ` ℂfld ) ↾t ( K [,] L ) ) ) ) -> ( x e. ( X [,] Y ) |-> A ) : ( X [,] Y ) --> ( K [,] L ) )
123	115, 117, 121, 122	syl3anc 1326	. . . . . . . . . . . 12 |- ( ph -> ( x e. ( X [,] Y ) |-> A ) : ( X [,] Y ) --> ( K [,] L ) )
124	123	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( X (,) Y ) ) -> ( x e. ( X [,] Y ) |-> A ) : ( X [,] Y ) --> ( K [,] L ) )
125		eqid 2622	. . . . . . . . . . . 12 |- ( x e. ( X [,] Y ) |-> A ) = ( x e. ( X [,] Y ) |-> A )
126	125	fmpt 6381	. . . . . . . . . . 11 |- ( A. x e. ( X [,] Y ) A e. ( K [,] L ) <-> ( x e. ( X [,] Y ) |-> A ) : ( X [,] Y ) --> ( K [,] L ) )
127	124, 126	sylibr 224	. . . . . . . . . 10 |- ( ( ph /\ x e. ( X (,) Y ) ) -> A. x e. ( X [,] Y ) A e. ( K [,] L ) )
128		ioossicc 12259	. . . . . . . . . . . 12 |- ( X (,) Y ) C_ ( X [,] Y )
129	128	sseli 3599	. . . . . . . . . . 11 |- ( x e. ( X (,) Y ) -> x e. ( X [,] Y ) )
130	129	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ x e. ( X (,) Y ) ) -> x e. ( X [,] Y ) )
131		rsp 2929	. . . . . . . . . 10 |- ( A. x e. ( X [,] Y ) A e. ( K [,] L ) -> ( x e. ( X [,] Y ) -> A e. ( K [,] L ) ) )
132	127, 130, 131	sylc 65	. . . . . . . . 9 |- ( ( ph /\ x e. ( X (,) Y ) ) -> A e. ( K [,] L ) )
133	132	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ x e. ( X (,) Y ) ) /\ u = A ) -> A e. ( K [,] L ) )
134	110, 133	eqeltrd 2701	. . . . . . 7 |- ( ( ( ph /\ x e. ( X (,) Y ) ) /\ u = A ) -> u e. ( K [,] L ) )
135	134	iftrued 4094	. . . . . 6 |- ( ( ( ph /\ x e. ( X (,) Y ) ) /\ u = A ) -> if ( u e. ( K [,] L ) , ( F ` u ) , if ( u < K , ( F ` K ) , ( F ` L ) ) ) = ( F ` u ) )
136		simpll 790	. . . . . . . 8 |- ( ( ( ph /\ x e. ( X (,) Y ) ) /\ u = A ) -> ph )
137	136, 134, 25	syl2anc 693	. . . . . . 7 |- ( ( ( ph /\ x e. ( X (,) Y ) ) /\ u = A ) -> C e. CC )
138	134, 137, 27	syl2anc 693	. . . . . 6 |- ( ( ( ph /\ x e. ( X (,) Y ) ) /\ u = A ) -> ( F ` u ) = C )
139		itgsubsticclem.13	. . . . . . 7 |- ( u = A -> C = E )
140	139	adantl 482	. . . . . 6 |- ( ( ( ph /\ x e. ( X (,) Y ) ) /\ u = A ) -> C = E )
141	135, 138, 140	3eqtrd 2660	. . . . 5 |- ( ( ( ph /\ x e. ( X (,) Y ) ) /\ u = A ) -> if ( u e. ( K [,] L ) , ( F ` u ) , if ( u < K , ( F ` K ) , ( F ` L ) ) ) = E )
142	12	adantr 481	. . . . . 6 |- ( ( ph /\ x e. ( X (,) Y ) ) -> ( K [,] L ) C_ RR )
143	142, 132	sseldd 3604	. . . . 5 |- ( ( ph /\ x e. ( X (,) Y ) ) -> A e. RR )
144		elex 3212	. . . . . . . 8 |- ( A e. ( K [,] L ) -> A e. _V )
145	132, 144	syl 17	. . . . . . 7 |- ( ( ph /\ x e. ( X (,) Y ) ) -> A e. _V )
146		isset 3207	. . . . . . 7 |- ( A e. _V <-> E. u u = A )
147	145, 146	sylib 208	. . . . . 6 |- ( ( ph /\ x e. ( X (,) Y ) ) -> E. u u = A )
148	140, 137	eqeltrrd 2702	. . . . . 6 |- ( ( ( ph /\ x e. ( X (,) Y ) ) /\ u = A ) -> E e. CC )
149	147, 148	exlimddv 1863	. . . . 5 |- ( ( ph /\ x e. ( X (,) Y ) ) -> E e. CC )
150	109, 141, 143, 149	fvmptd 6288	. . . 4 |- ( ( ph /\ x e. ( X (,) Y ) ) -> ( G ` A ) = E )
151	150	oveq1d 6665	. . 3 |- ( ( ph /\ x e. ( X (,) Y ) ) -> ( ( G ` A ) x. B ) = ( E x. B ) )
152	37, 151	ditgeq3d 40180	. 2 |- ( ph -> S__ [ X -> Y ] ( ( G ` A ) x. B ) _d x = S__ [ X -> Y ] ( E x. B ) _d x )
153	108, 152	eqtrd 2656	1 |- ( ph -> S__ [ K -> L ] C _d u = S__ [ X -> Y ] ( E x. B ) _d x )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ifcif 4086   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   x. cmul 9941   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   (,)cioo 12175   [,]cicc 12178   ↾_t crest 16081   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746   Topctop 20698  TopOnctopon 20715   Cn ccn 21028   -cn->ccncf 22679   L^1cibl 23386   S__cdit 23610   _D cdv 23627

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-cc 9257  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  ax-mulf 10016

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-iin 4523  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-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  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-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  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-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-ovol 23233  df-vol 23234  df-mbf 23388  df-itg1 23389  df-itg2 23390  df-ibl 23391  df-itg 23392  df-0p 23437  df-ditg 23611  df-limc 23630  df-dv 23631

This theorem is referenced by:  itgsubsticc  40192