Theorem ftc2re 30676

Description: The Fundamental Theorem of Calculus, part two, for functions continuous on D. (Contributed by Thierry Arnoux, 1-Dec-2021.)

Hypotheses

Ref	Expression
ftc2re.e	|- E = ( C (,) D )
ftc2re.a	|- ( ph -> A e. E )
ftc2re.b	|- ( ph -> B e. E )
ftc2re.le	|- ( ph -> A <_ B )
ftc2re.f	|- ( ph -> F : E --> CC )
ftc2re.1	|- ( ph -> ( RR _D F ) e. ( E -cn-> CC ) )

Assertion

Ref	Expression
ftc2re	|- ( ph -> S. ( A (,) B ) ( ( RR _D F ) ` t ) _d t = ( ( F ` B ) - ( F ` A ) ) )

Distinct variable groups:   t, A   t, B   t, F   ph, t

Allowed substitution hints:   C( t)   D( t)   E( t)

Proof of Theorem ftc2re

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ftc2re.e	. . . . . 6 |- E = ( C (,) D )
2		ioossre 12235	. . . . . 6 |- ( C (,) D ) C_ RR
3	1, 2	eqsstri 3635	. . . . 5 |- E C_ RR
4	3	a1i 11	. . . 4 |- ( ph -> E C_ RR )
5		ftc2re.a	. . . 4 |- ( ph -> A e. E )
6	4, 5	sseldd 3604	. . 3 |- ( ph -> A e. RR )
7		ftc2re.b	. . . 4 |- ( ph -> B e. E )
8	4, 7	sseldd 3604	. . 3 |- ( ph -> B e. RR )
9		ftc2re.le	. . 3 |- ( ph -> A <_ B )
10		ax-resscn 9993	. . . . . . 7 |- RR C_ CC
11	10	a1i 11	. . . . . 6 |- ( ph -> RR C_ CC )
12		ftc2re.f	. . . . . 6 |- ( ph -> F : E --> CC )
13		iccssre 12255	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
14	6, 8, 13	syl2anc 693	. . . . . 6 |- ( ph -> ( A [,] B ) C_ RR )
15		eqid 2622	. . . . . . 7 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
16	15	tgioo2 22606	. . . . . . 7 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
17	15, 16	dvres 23675	. . . . . 6 |- ( ( ( RR C_ CC /\ F : E --> CC ) /\ ( E C_ RR /\ ( A [,] B ) C_ RR ) ) -> ( RR _D ( F |` ( A [,] B ) ) ) = ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) ) )
18	11, 12, 4, 14, 17	syl22anc 1327	. . . . 5 |- ( ph -> ( RR _D ( F |` ( A [,] B ) ) ) = ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) ) )
19		iccntr 22624	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
20	6, 8, 19	syl2anc 693	. . . . . 6 |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
21	20	reseq2d 5396	. . . . 5 |- ( ph -> ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) ) = ( ( RR _D F ) |` ( A (,) B ) ) )
22	18, 21	eqtrd 2656	. . . 4 |- ( ph -> ( RR _D ( F |` ( A [,] B ) ) ) = ( ( RR _D F ) |` ( A (,) B ) ) )
23		ioossicc 12259	. . . . . . 7 |- ( A (,) B ) C_ ( A [,] B )
24	23	a1i 11	. . . . . 6 |- ( ph -> ( A (,) B ) C_ ( A [,] B ) )
25	1, 5, 7	fct2relem 30675	. . . . . 6 |- ( ph -> ( A [,] B ) C_ E )
26	24, 25	sstrd 3613	. . . . 5 |- ( ph -> ( A (,) B ) C_ E )
27		ftc2re.1	. . . . 5 |- ( ph -> ( RR _D F ) e. ( E -cn-> CC ) )
28		rescncf 22700	. . . . 5 |- ( ( A (,) B ) C_ E -> ( ( RR _D F ) e. ( E -cn-> CC ) -> ( ( RR _D F ) |` ( A (,) B ) ) e. ( ( A (,) B ) -cn-> CC ) ) )
29	26, 27, 28	sylc 65	. . . 4 |- ( ph -> ( ( RR _D F ) |` ( A (,) B ) ) e. ( ( A (,) B ) -cn-> CC ) )
30	22, 29	eqeltrd 2701	. . 3 |- ( ph -> ( RR _D ( F |` ( A [,] B ) ) ) e. ( ( A (,) B ) -cn-> CC ) )
31		ioombl 23333	. . . . . . 7 |- ( A (,) B ) e. dom vol
32	31	a1i 11	. . . . . 6 |- ( ph -> ( A (,) B ) e. dom vol )
33		cnmbf 23426	. . . . . 6 |- ( ( ( A (,) B ) e. dom vol /\ ( ( RR _D F ) |` ( A (,) B ) ) e. ( ( A (,) B ) -cn-> CC ) ) -> ( ( RR _D F ) |` ( A (,) B ) ) e. MblFn )
34	32, 29, 33	syl2anc 693	. . . . 5 |- ( ph -> ( ( RR _D F ) |` ( A (,) B ) ) e. MblFn )
35		dmres 5419	. . . . . . 7 |- dom ( ( RR _D F ) |` ( A (,) B ) ) = ( ( A (,) B ) i^i dom ( RR _D F ) )
36	35	fveq2i 6194	. . . . . 6 |- ( vol ` dom ( ( RR _D F ) |` ( A (,) B ) ) ) = ( vol ` ( ( A (,) B ) i^i dom ( RR _D F ) ) )
37		cncff 22696	. . . . . . . . . . . 12 |- ( ( RR _D F ) e. ( E -cn-> CC ) -> ( RR _D F ) : E --> CC )
38	27, 37	syl 17	. . . . . . . . . . 11 |- ( ph -> ( RR _D F ) : E --> CC )
39		fdm 6051	. . . . . . . . . . 11 |- ( ( RR _D F ) : E --> CC -> dom ( RR _D F ) = E )
40	38, 39	syl 17	. . . . . . . . . 10 |- ( ph -> dom ( RR _D F ) = E )
41	40	ineq2d 3814	. . . . . . . . 9 |- ( ph -> ( ( A (,) B ) i^i dom ( RR _D F ) ) = ( ( A (,) B ) i^i E ) )
42		df-ss 3588	. . . . . . . . . 10 |- ( ( A (,) B ) C_ E <-> ( ( A (,) B ) i^i E ) = ( A (,) B ) )
43	26, 42	sylib 208	. . . . . . . . 9 |- ( ph -> ( ( A (,) B ) i^i E ) = ( A (,) B ) )
44	41, 43	eqtrd 2656	. . . . . . . 8 |- ( ph -> ( ( A (,) B ) i^i dom ( RR _D F ) ) = ( A (,) B ) )
45	44	fveq2d 6195	. . . . . . 7 |- ( ph -> ( vol ` ( ( A (,) B ) i^i dom ( RR _D F ) ) ) = ( vol ` ( A (,) B ) ) )
46		volioo 23337	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( vol ` ( A (,) B ) ) = ( B - A ) )
47	6, 8, 9, 46	syl3anc 1326	. . . . . . . 8 |- ( ph -> ( vol ` ( A (,) B ) ) = ( B - A ) )
48	8, 6	resubcld 10458	. . . . . . . 8 |- ( ph -> ( B - A ) e. RR )
49	47, 48	eqeltrd 2701	. . . . . . 7 |- ( ph -> ( vol ` ( A (,) B ) ) e. RR )
50	45, 49	eqeltrd 2701	. . . . . 6 |- ( ph -> ( vol ` ( ( A (,) B ) i^i dom ( RR _D F ) ) ) e. RR )
51	36, 50	syl5eqel 2705	. . . . 5 |- ( ph -> ( vol ` dom ( ( RR _D F ) |` ( A (,) B ) ) ) e. RR )
52		rescncf 22700	. . . . . . . . 9 |- ( ( A [,] B ) C_ E -> ( ( RR _D F ) e. ( E -cn-> CC ) -> ( ( RR _D F ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) ) )
53	25, 52	syl 17	. . . . . . . 8 |- ( ph -> ( ( RR _D F ) e. ( E -cn-> CC ) -> ( ( RR _D F ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) ) )
54	27, 53	mpd 15	. . . . . . 7 |- ( ph -> ( ( RR _D F ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) )
55		cniccbdd 23230	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ ( ( RR _D F ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) ) -> E. x e. RR A. y e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) ) <_ x )
56	6, 8, 54, 55	syl3anc 1326	. . . . . 6 |- ( ph -> E. x e. RR A. y e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) ) <_ x )
57	35, 44	syl5eq 2668	. . . . . . . . . . 11 |- ( ph -> dom ( ( RR _D F ) |` ( A (,) B ) ) = ( A (,) B ) )
58	57, 24	eqsstrd 3639	. . . . . . . . . 10 |- ( ph -> dom ( ( RR _D F ) |` ( A (,) B ) ) C_ ( A [,] B ) )
59		ssralv 3666	. . . . . . . . . 10 |- ( dom ( ( RR _D F ) |` ( A (,) B ) ) C_ ( A [,] B ) -> ( A. y e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) ) <_ x -> A. y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) ) <_ x ) )
60	58, 59	syl 17	. . . . . . . . 9 |- ( ph -> ( A. y e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) ) <_ x -> A. y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) ) <_ x ) )
61	60	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. RR ) -> ( A. y e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) ) <_ x -> A. y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) ) <_ x ) )
62	58	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. RR ) -> dom ( ( RR _D F ) |` ( A (,) B ) ) C_ ( A [,] B ) )
63	62	sselda 3603	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. RR ) /\ y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ) -> y e. ( A [,] B ) )
64		fvres 6207	. . . . . . . . . . . . . 14 |- ( y e. ( A [,] B ) -> ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) = ( ( RR _D F ) ` y ) )
65	63, 64	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. RR ) /\ y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ) -> ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) = ( ( RR _D F ) ` y ) )
66		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. RR ) /\ y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ) -> y e. dom ( ( RR _D F ) |` ( A (,) B ) ) )
67	57	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. RR ) /\ y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ) -> dom ( ( RR _D F ) |` ( A (,) B ) ) = ( A (,) B ) )
68	66, 67	eleqtrd 2703	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. RR ) /\ y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ) -> y e. ( A (,) B ) )
69		fvres 6207	. . . . . . . . . . . . . 14 |- ( y e. ( A (,) B ) -> ( ( ( RR _D F ) |` ( A (,) B ) ) ` y ) = ( ( RR _D F ) ` y ) )
70	68, 69	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. RR ) /\ y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ) -> ( ( ( RR _D F ) |` ( A (,) B ) ) ` y ) = ( ( RR _D F ) ` y ) )
71	65, 70	eqtr4d 2659	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. RR ) /\ y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ) -> ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) = ( ( ( RR _D F ) |` ( A (,) B ) ) ` y ) )
72	71	fveq2d 6195	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR ) /\ y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ) -> ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) ) = ( abs ` ( ( ( RR _D F ) |` ( A (,) B ) ) ` y ) ) )
73	72	breq1d 4663	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR ) /\ y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ) -> ( ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) ) <_ x <-> ( abs ` ( ( ( RR _D F ) |` ( A (,) B ) ) ` y ) ) <_ x ) )
74	73	biimpd 219	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR ) /\ y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ) -> ( ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) ) <_ x -> ( abs ` ( ( ( RR _D F ) |` ( A (,) B ) ) ` y ) ) <_ x ) )
75	74	ralimdva 2962	. . . . . . . 8 |- ( ( ph /\ x e. RR ) -> ( A. y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) ) <_ x -> A. y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ( abs ` ( ( ( RR _D F ) |` ( A (,) B ) ) ` y ) ) <_ x ) )
76	61, 75	syld 47	. . . . . . 7 |- ( ( ph /\ x e. RR ) -> ( A. y e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) ) <_ x -> A. y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ( abs ` ( ( ( RR _D F ) |` ( A (,) B ) ) ` y ) ) <_ x ) )
77	76	reximdva 3017	. . . . . 6 |- ( ph -> ( E. x e. RR A. y e. ( A [,] B ) ( abs ` ( ( ( RR _D F ) |` ( A [,] B ) ) ` y ) ) <_ x -> E. x e. RR A. y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ( abs ` ( ( ( RR _D F ) |` ( A (,) B ) ) ` y ) ) <_ x ) )
78	56, 77	mpd 15	. . . . 5 |- ( ph -> E. x e. RR A. y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ( abs ` ( ( ( RR _D F ) |` ( A (,) B ) ) ` y ) ) <_ x )
79		bddibl 23606	. . . . 5 |- ( ( ( ( RR _D F ) |` ( A (,) B ) ) e. MblFn /\ ( vol ` dom ( ( RR _D F ) |` ( A (,) B ) ) ) e. RR /\ E. x e. RR A. y e. dom ( ( RR _D F ) |` ( A (,) B ) ) ( abs ` ( ( ( RR _D F ) |` ( A (,) B ) ) ` y ) ) <_ x ) -> ( ( RR _D F ) |` ( A (,) B ) ) e. L^1 )
80	34, 51, 78, 79	syl3anc 1326	. . . 4 |- ( ph -> ( ( RR _D F ) |` ( A (,) B ) ) e. L^1 )
81	22, 80	eqeltrd 2701	. . 3 |- ( ph -> ( RR _D ( F |` ( A [,] B ) ) ) e. L^1 )
82		dvcn 23684	. . . . 5 |- ( ( ( RR C_ CC /\ F : E --> CC /\ E C_ RR ) /\ dom ( RR _D F ) = E ) -> F e. ( E -cn-> CC ) )
83	11, 12, 4, 40, 82	syl31anc 1329	. . . 4 |- ( ph -> F e. ( E -cn-> CC ) )
84		rescncf 22700	. . . . 5 |- ( ( A [,] B ) C_ E -> ( F e. ( E -cn-> CC ) -> ( F |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) ) )
85	25, 84	syl 17	. . . 4 |- ( ph -> ( F e. ( E -cn-> CC ) -> ( F |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) ) )
86	83, 85	mpd 15	. . 3 |- ( ph -> ( F |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) )
87	6, 8, 9, 30, 81, 86	ftc2 23807	. 2 |- ( ph -> S. ( A (,) B ) ( ( RR _D ( F |` ( A [,] B ) ) ) ` t ) _d t = ( ( ( F |` ( A [,] B ) ) ` B ) - ( ( F |` ( A [,] B ) ) ` A ) ) )
88	22	fveq1d 6193	. . . . 5 |- ( ph -> ( ( RR _D ( F |` ( A [,] B ) ) ) ` t ) = ( ( ( RR _D F ) |` ( A (,) B ) ) ` t ) )
89		fvres 6207	. . . . 5 |- ( t e. ( A (,) B ) -> ( ( ( RR _D F ) |` ( A (,) B ) ) ` t ) = ( ( RR _D F ) ` t ) )
90	88, 89	sylan9eq 2676	. . . 4 |- ( ( ph /\ t e. ( A (,) B ) ) -> ( ( RR _D ( F |` ( A [,] B ) ) ) ` t ) = ( ( RR _D F ) ` t ) )
91	90	ralrimiva 2966	. . 3 |- ( ph -> A. t e. ( A (,) B ) ( ( RR _D ( F |` ( A [,] B ) ) ) ` t ) = ( ( RR _D F ) ` t ) )
92		itgeq2 23544	. . 3 |- ( A. t e. ( A (,) B ) ( ( RR _D ( F |` ( A [,] B ) ) ) ` t ) = ( ( RR _D F ) ` t ) -> S. ( A (,) B ) ( ( RR _D ( F |` ( A [,] B ) ) ) ` t ) _d t = S. ( A (,) B ) ( ( RR _D F ) ` t ) _d t )
93	91, 92	syl 17	. 2 |- ( ph -> S. ( A (,) B ) ( ( RR _D ( F |` ( A [,] B ) ) ) ` t ) _d t = S. ( A (,) B ) ( ( RR _D F ) ` t ) _d t )
94	6	rexrd 10089	. . . . 5 |- ( ph -> A e. RR* )
95	8	rexrd 10089	. . . . 5 |- ( ph -> B e. RR* )
96		ubicc2 12289	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
97	94, 95, 9, 96	syl3anc 1326	. . . 4 |- ( ph -> B e. ( A [,] B ) )
98	97	fvresd 6208	. . 3 |- ( ph -> ( ( F |` ( A [,] B ) ) ` B ) = ( F ` B ) )
99		lbicc2 12288	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
100	94, 95, 9, 99	syl3anc 1326	. . . 4 |- ( ph -> A e. ( A [,] B ) )
101	100	fvresd 6208	. . 3 |- ( ph -> ( ( F |` ( A [,] B ) ) ` A ) = ( F ` A ) )
102	98, 101	oveq12d 6668	. 2 |- ( ph -> ( ( ( F |` ( A [,] B ) ) ` B ) - ( ( F |` ( A [,] B ) ) ` A ) ) = ( ( F ` B ) - ( F ` A ) ) )
103	87, 93, 102	3eqtr3d 2664	1 |- ( ph -> S. ( A (,) B ) ( ( RR _D F ) ` t ) _d t = ( ( F ` B ) - ( F ` A ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   class class class wbr 4653   dom cdm 5114   ran crn 5115   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   RR*cxr 10073   <_ cle 10075   - cmin 10266   (,)cioo 12175   [,]cicc 12178   abscabs 13974   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746   intcnt 20821   -cn->ccncf 22679   volcvol 23232  MblFncmbf 23383   L^1cibl 23386   S.citg 23387   _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-limc 23630  df-dv 23631

This theorem is referenced by:  fdvposlt  30677  fdvposle  30679  itgexpif  30684