Theorem i1fmul 23463

Description: The pointwise product of two simple functions is a simple function. (Contributed by Mario Carneiro, 5-Sep-2014.)

Hypotheses

Ref	Expression
i1fadd.1	|- ( ph -> F e. dom S.1 )
i1fadd.2	|- ( ph -> G e. dom S.1 )

Assertion

Ref	Expression
i1fmul	|- ( ph -> ( F oF x. G ) e. dom S.1 )

Proof of Theorem i1fmul

Dummy variables y z w v x u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		remulcl 10021	. . . 4 |- ( ( x e. RR /\ y e. RR ) -> ( x x. y ) e. RR )
2	1	adantl 482	. . 3 |- ( ( ph /\ ( x e. RR /\ y e. RR ) ) -> ( x x. y ) e. RR )
3		i1fadd.1	. . . 4 |- ( ph -> F e. dom S.1 )
4		i1ff 23443	. . . 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 23443	. . . 4 |- ( G e. dom S.1 -> G : RR --> RR )
8	6, 7	syl 17	. . 3 |- ( ph -> G : RR --> RR )
9		reex 10027	. . . 4 |- RR e. _V
10	9	a1i 11	. . 3 |- ( ph -> RR e. _V )
11		inidm 3822	. . 3 |- ( RR i^i RR ) = RR
12	2, 5, 8, 10, 10, 11	off 6912	. 2 |- ( ph -> ( F oF x. G ) : RR --> RR )
13		i1frn 23444	. . . . . 6 |- ( F e. dom S.1 -> ran F e. Fin )
14	3, 13	syl 17	. . . . 5 |- ( ph -> ran F e. Fin )
15		i1frn 23444	. . . . . 6 |- ( G e. dom S.1 -> ran G e. Fin )
16	6, 15	syl 17	. . . . 5 |- ( ph -> ran G e. Fin )
17		xpfi 8231	. . . . 5 |- ( ( ran F e. Fin /\ ran G e. Fin ) -> ( ran F X. ran G ) e. Fin )
18	14, 16, 17	syl2anc 693	. . . 4 |- ( ph -> ( ran F X. ran G ) e. Fin )
19		eqid 2622	. . . . . 6 |- ( u e. ran F , v e. ran G |-> ( u x. v ) ) = ( u e. ran F , v e. ran G |-> ( u x. v ) )
20		ovex 6678	. . . . . 6 |- ( u x. v ) e. _V
21	19, 20	fnmpt2i 7239	. . . . 5 |- ( u e. ran F , v e. ran G |-> ( u x. v ) ) Fn ( ran F X. ran G )
22		dffn4 6121	. . . . 5 |- ( ( u e. ran F , v e. ran G |-> ( u x. v ) ) Fn ( ran F X. ran G ) <-> ( u e. ran F , v e. ran G |-> ( u x. v ) ) : ( ran F X. ran G ) -onto-> ran ( u e. ran F , v e. ran G |-> ( u x. v ) ) )
23	21, 22	mpbi 220	. . . 4 |- ( u e. ran F , v e. ran G |-> ( u x. v ) ) : ( ran F X. ran G ) -onto-> ran ( u e. ran F , v e. ran G |-> ( u x. v ) )
24		fofi 8252	. . . 4 |- ( ( ( ran F X. ran G ) e. Fin /\ ( u e. ran F , v e. ran G |-> ( u x. v ) ) : ( ran F X. ran G ) -onto-> ran ( u e. ran F , v e. ran G |-> ( u x. v ) ) ) -> ran ( u e. ran F , v e. ran G |-> ( u x. v ) ) e. Fin )
25	18, 23, 24	sylancl 694	. . 3 |- ( ph -> ran ( u e. ran F , v e. ran G |-> ( u x. v ) ) e. Fin )
26		eqid 2622	. . . . . . . . 9 |- ( x x. y ) = ( x x. y )
27		rspceov 6692	. . . . . . . . 9 |- ( ( x e. ran F /\ y e. ran G /\ ( x x. y ) = ( x x. y ) ) -> E. u e. ran F E. v e. ran G ( x x. y ) = ( u x. v ) )
28	26, 27	mp3an3 1413	. . . . . . . 8 |- ( ( x e. ran F /\ y e. ran G ) -> E. u e. ran F E. v e. ran G ( x x. y ) = ( u x. v ) )
29		ovex 6678	. . . . . . . . 9 |- ( x x. y ) e. _V
30		eqeq1 2626	. . . . . . . . . 10 |- ( w = ( x x. y ) -> ( w = ( u x. v ) <-> ( x x. y ) = ( u x. v ) ) )
31	30	2rexbidv 3057	. . . . . . . . 9 |- ( w = ( x x. y ) -> ( E. u e. ran F E. v e. ran G w = ( u x. v ) <-> E. u e. ran F E. v e. ran G ( x x. y ) = ( u x. v ) ) )
32	29, 31	elab 3350	. . . . . . . 8 |- ( ( x x. y ) e. { w | E. u e. ran F E. v e. ran G w = ( u x. v ) } <-> E. u e. ran F E. v e. ran G ( x x. y ) = ( u x. v ) )
33	28, 32	sylibr 224	. . . . . . 7 |- ( ( x e. ran F /\ y e. ran G ) -> ( x x. y ) e. { w | E. u e. ran F E. v e. ran G w = ( u x. v ) } )
34	33	adantl 482	. . . . . 6 |- ( ( ph /\ ( x e. ran F /\ y e. ran G ) ) -> ( x x. y ) e. { w | E. u e. ran F E. v e. ran G w = ( u x. v ) } )
35		ffn 6045	. . . . . . . 8 |- ( F : RR --> RR -> F Fn RR )
36	5, 35	syl 17	. . . . . . 7 |- ( ph -> F Fn RR )
37		dffn3 6054	. . . . . . 7 |- ( F Fn RR <-> F : RR --> ran F )
38	36, 37	sylib 208	. . . . . 6 |- ( ph -> F : RR --> ran F )
39		ffn 6045	. . . . . . . 8 |- ( G : RR --> RR -> G Fn RR )
40	8, 39	syl 17	. . . . . . 7 |- ( ph -> G Fn RR )
41		dffn3 6054	. . . . . . 7 |- ( G Fn RR <-> G : RR --> ran G )
42	40, 41	sylib 208	. . . . . 6 |- ( ph -> G : RR --> ran G )
43	34, 38, 42, 10, 10, 11	off 6912	. . . . 5 |- ( ph -> ( F oF x. G ) : RR --> { w | E. u e. ran F E. v e. ran G w = ( u x. v ) } )
44		frn 6053	. . . . 5 |- ( ( F oF x. G ) : RR --> { w | E. u e. ran F E. v e. ran G w = ( u x. v ) } -> ran ( F oF x. G ) C_ { w | E. u e. ran F E. v e. ran G w = ( u x. v ) } )
45	43, 44	syl 17	. . . 4 |- ( ph -> ran ( F oF x. G ) C_ { w | E. u e. ran F E. v e. ran G w = ( u x. v ) } )
46	19	rnmpt2 6770	. . . 4 |- ran ( u e. ran F , v e. ran G |-> ( u x. v ) ) = { w | E. u e. ran F E. v e. ran G w = ( u x. v ) }
47	45, 46	syl6sseqr 3652	. . 3 |- ( ph -> ran ( F oF x. G ) C_ ran ( u e. ran F , v e. ran G |-> ( u x. v ) ) )
48		ssfi 8180	. . 3 |- ( ( ran ( u e. ran F , v e. ran G |-> ( u x. v ) ) e. Fin /\ ran ( F oF x. G ) C_ ran ( u e. ran F , v e. ran G |-> ( u x. v ) ) ) -> ran ( F oF x. G ) e. Fin )
49	25, 47, 48	syl2anc 693	. 2 |- ( ph -> ran ( F oF x. G ) e. Fin )
50		frn 6053	. . . . . . . 8 |- ( ( F oF x. G ) : RR --> RR -> ran ( F oF x. G ) C_ RR )
51	12, 50	syl 17	. . . . . . 7 |- ( ph -> ran ( F oF x. G ) C_ RR )
52		ax-resscn 9993	. . . . . . 7 |- RR C_ CC
53	51, 52	syl6ss 3615	. . . . . 6 |- ( ph -> ran ( F oF x. G ) C_ CC )
54	53	ssdifd 3746	. . . . 5 |- ( ph -> ( ran ( F oF x. G ) \ { 0 } ) C_ ( CC \ { 0 } ) )
55	54	sselda 3603	. . . 4 |- ( ( ph /\ y e. ( ran ( F oF x. G ) \ { 0 } ) ) -> y e. ( CC \ { 0 } ) )
56	3, 6	i1fmullem 23461	. . . 4 |- ( ( ph /\ y e. ( CC \ { 0 } ) ) -> ( `' ( F oF x. G ) " { y } ) = U_ z e. ( ran G \ { 0 } ) ( ( `' F " { ( y / z ) } ) i^i ( `' G " { z } ) ) )
57	55, 56	syldan 487	. . 3 |- ( ( ph /\ y e. ( ran ( F oF x. G ) \ { 0 } ) ) -> ( `' ( F oF x. G ) " { y } ) = U_ z e. ( ran G \ { 0 } ) ( ( `' F " { ( y / z ) } ) i^i ( `' G " { z } ) ) )
58		difss 3737	. . . . . 6 |- ( ran G \ { 0 } ) C_ ran G
59		ssfi 8180	. . . . . 6 |- ( ( ran G e. Fin /\ ( ran G \ { 0 } ) C_ ran G ) -> ( ran G \ { 0 } ) e. Fin )
60	16, 58, 59	sylancl 694	. . . . 5 |- ( ph -> ( ran G \ { 0 } ) e. Fin )
61		i1fima 23445	. . . . . . . 8 |- ( F e. dom S.1 -> ( `' F " { ( y / z ) } ) e. dom vol )
62	3, 61	syl 17	. . . . . . 7 |- ( ph -> ( `' F " { ( y / z ) } ) e. dom vol )
63		i1fima 23445	. . . . . . . 8 |- ( G e. dom S.1 -> ( `' G " { z } ) e. dom vol )
64	6, 63	syl 17	. . . . . . 7 |- ( ph -> ( `' G " { z } ) e. dom vol )
65		inmbl 23310	. . . . . . 7 |- ( ( ( `' F " { ( y / z ) } ) e. dom vol /\ ( `' G " { z } ) e. dom vol ) -> ( ( `' F " { ( y / z ) } ) i^i ( `' G " { z } ) ) e. dom vol )
66	62, 64, 65	syl2anc 693	. . . . . 6 |- ( ph -> ( ( `' F " { ( y / z ) } ) i^i ( `' G " { z } ) ) e. dom vol )
67	66	ralrimivw 2967	. . . . 5 |- ( ph -> A. z e. ( ran G \ { 0 } ) ( ( `' F " { ( y / z ) } ) i^i ( `' G " { z } ) ) e. dom vol )
68		finiunmbl 23312	. . . . 5 |- ( ( ( ran G \ { 0 } ) e. Fin /\ A. z e. ( ran G \ { 0 } ) ( ( `' F " { ( y / z ) } ) i^i ( `' G " { z } ) ) e. dom vol ) -> U_ z e. ( ran G \ { 0 } ) ( ( `' F " { ( y / z ) } ) i^i ( `' G " { z } ) ) e. dom vol )
69	60, 67, 68	syl2anc 693	. . . 4 |- ( ph -> U_ z e. ( ran G \ { 0 } ) ( ( `' F " { ( y / z ) } ) i^i ( `' G " { z } ) ) e. dom vol )
70	69	adantr 481	. . 3 |- ( ( ph /\ y e. ( ran ( F oF x. G ) \ { 0 } ) ) -> U_ z e. ( ran G \ { 0 } ) ( ( `' F " { ( y / z ) } ) i^i ( `' G " { z } ) ) e. dom vol )
71	57, 70	eqeltrd 2701	. 2 |- ( ( ph /\ y e. ( ran ( F oF x. G ) \ { 0 } ) ) -> ( `' ( F oF x. G ) " { y } ) e. dom vol )
72		mblvol 23298	. . . 4 |- ( ( `' ( F oF x. G ) " { y } ) e. dom vol -> ( vol ` ( `' ( F oF x. G ) " { y } ) ) = ( vol* ` ( `' ( F oF x. G ) " { y } ) ) )
73	71, 72	syl 17	. . 3 |- ( ( ph /\ y e. ( ran ( F oF x. G ) \ { 0 } ) ) -> ( vol ` ( `' ( F oF x. G ) " { y } ) ) = ( vol* ` ( `' ( F oF x. G ) " { y } ) ) )
74		mblss 23299	. . . . 5 |- ( ( `' ( F oF x. G ) " { y } ) e. dom vol -> ( `' ( F oF x. G ) " { y } ) C_ RR )
75	71, 74	syl 17	. . . 4 |- ( ( ph /\ y e. ( ran ( F oF x. G ) \ { 0 } ) ) -> ( `' ( F oF x. G ) " { y } ) C_ RR )
76	60	adantr 481	. . . . 5 |- ( ( ph /\ y e. ( ran ( F oF x. G ) \ { 0 } ) ) -> ( ran G \ { 0 } ) e. Fin )
77		inss2 3834	. . . . . . 7 |- ( ( `' F " { ( y / z ) } ) i^i ( `' G " { z } ) ) C_ ( `' G " { z } )
78	77	a1i 11	. . . . . 6 |- ( ( ( ph /\ y e. ( ran ( F oF x. G ) \ { 0 } ) ) /\ z e. ( ran G \ { 0 } ) ) -> ( ( `' F " { ( y / z ) } ) i^i ( `' G " { z } ) ) C_ ( `' G " { z } ) )
79	64	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ y e. ( ran ( F oF x. G ) \ { 0 } ) ) /\ z e. ( ran G \ { 0 } ) ) -> ( `' G " { z } ) e. dom vol )
80		mblss 23299	. . . . . . 7 |- ( ( `' G " { z } ) e. dom vol -> ( `' G " { z } ) C_ RR )
81	79, 80	syl 17	. . . . . 6 |- ( ( ( ph /\ y e. ( ran ( F oF x. G ) \ { 0 } ) ) /\ z e. ( ran G \ { 0 } ) ) -> ( `' G " { z } ) C_ RR )
82		mblvol 23298	. . . . . . . 8 |- ( ( `' G " { z } ) e. dom vol -> ( vol ` ( `' G " { z } ) ) = ( vol* ` ( `' G " { z } ) ) )
83	79, 82	syl 17	. . . . . . 7 |- ( ( ( ph /\ y e. ( ran ( F oF x. G ) \ { 0 } ) ) /\ z e. ( ran G \ { 0 } ) ) -> ( vol ` ( `' G " { z } ) ) = ( vol* ` ( `' G " { z } ) ) )
84	6	adantr 481	. . . . . . . 8 |- ( ( ph /\ y e. ( ran ( F oF x. G ) \ { 0 } ) ) -> G e. dom S.1 )
85		i1fima2sn 23447	. . . . . . . 8 |- ( ( G e. dom S.1 /\ z e. ( ran G \ { 0 } ) ) -> ( vol ` ( `' G " { z } ) ) e. RR )
86	84, 85	sylan 488	. . . . . . 7 |- ( ( ( ph /\ y e. ( ran ( F oF x. G ) \ { 0 } ) ) /\ z e. ( ran G \ { 0 } ) ) -> ( vol ` ( `' G " { z } ) ) e. RR )
87	83, 86	eqeltrrd 2702	. . . . . 6 |- ( ( ( ph /\ y e. ( ran ( F oF x. G ) \ { 0 } ) ) /\ z e. ( ran G \ { 0 } ) ) -> ( vol* ` ( `' G " { z } ) ) e. RR )
88		ovolsscl 23254	. . . . . 6 |- ( ( ( ( `' 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 )
89	78, 81, 87, 88	syl3anc 1326	. . . . 5 |- ( ( ( ph /\ y e. ( ran ( F oF x. G ) \ { 0 } ) ) /\ z e. ( ran G \ { 0 } ) ) -> ( vol* ` ( ( `' F " { ( y / z ) } ) i^i ( `' G " { z } ) ) ) e. RR )
90	76, 89	fsumrecl 14465	. . . 4 |- ( ( ph /\ y e. ( ran ( F oF x. G ) \ { 0 } ) ) -> sum_ z e. ( ran G \ { 0 } ) ( vol* ` ( ( `' F " { ( y / z ) } ) i^i ( `' G " { z } ) ) ) e. RR )
91	57	fveq2d 6195	. . . . 5 |- ( ( ph /\ y e. ( ran ( F oF x. G ) \ { 0 } ) ) -> ( vol* ` ( `' ( F oF x. G ) " { y } ) ) = ( vol* ` U_ z e. ( ran G \ { 0 } ) ( ( `' F " { ( y / z ) } ) i^i ( `' G " { z } ) ) ) )
92		mblss 23299	. . . . . . . . . 10 |- ( ( ( `' F " { ( y / z ) } ) i^i ( `' G " { z } ) ) e. dom vol -> ( ( `' F " { ( y / z ) } ) i^i ( `' G " { z } ) ) C_ RR )
93	66, 92	syl 17	. . . . . . . . 9 |- ( ph -> ( ( `' F " { ( y / z ) } ) i^i ( `' G " { z } ) ) C_ RR )
94	93	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ y e. ( ran ( F oF x. G ) \ { 0 } ) ) /\ z e. ( ran G \ { 0 } ) ) -> ( ( `' F " { ( y / z ) } ) i^i ( `' G " { z } ) ) C_ RR )
95	94, 89	jca 554	. . . . . . 7 |- ( ( ( ph /\ y e. ( ran ( F oF x. G ) \ { 0 } ) ) /\ z e. ( ran G \ { 0 } ) ) -> ( ( ( `' F " { ( y / z ) } ) i^i ( `' G " { z } ) ) C_ RR /\ ( vol* ` ( ( `' F " { ( y / z ) } ) i^i ( `' G " { z } ) ) ) e. RR ) )
96	95	ralrimiva 2966	. . . . . 6 |- ( ( ph /\ y e. ( ran ( F oF x. G ) \ { 0 } ) ) -> A. z e. ( ran G \ { 0 } ) ( ( ( `' F " { ( y / z ) } ) i^i ( `' G " { z } ) ) C_ RR /\ ( vol* ` ( ( `' F " { ( y / z ) } ) i^i ( `' G " { z } ) ) ) e. RR ) )
97		ovolfiniun 23269	. . . . . 6 |- ( ( ( ran G \ { 0 } ) e. Fin /\ A. z e. ( ran G \ { 0 } ) ( ( ( `' 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 \ { 0 } ) ( ( `' F " { ( y / z ) } ) i^i ( `' G " { z } ) ) ) <_ sum_ z e. ( ran G \ { 0 } ) ( vol* ` ( ( `' F " { ( y / z ) } ) i^i ( `' G " { z } ) ) ) )
98	76, 96, 97	syl2anc 693	. . . . 5 |- ( ( ph /\ y e. ( ran ( F oF x. G ) \ { 0 } ) ) -> ( vol* ` U_ z e. ( ran G \ { 0 } ) ( ( `' F " { ( y / z ) } ) i^i ( `' G " { z } ) ) ) <_ sum_ z e. ( ran G \ { 0 } ) ( vol* ` ( ( `' F " { ( y / z ) } ) i^i ( `' G " { z } ) ) ) )
99	91, 98	eqbrtrd 4675	. . . 4 |- ( ( ph /\ y e. ( ran ( F oF x. G ) \ { 0 } ) ) -> ( vol* ` ( `' ( F oF x. G ) " { y } ) ) <_ sum_ z e. ( ran G \ { 0 } ) ( vol* ` ( ( `' F " { ( y / z ) } ) i^i ( `' G " { z } ) ) ) )
100		ovollecl 23251	. . . 4 |- ( ( ( `' ( F oF x. G ) " { y } ) C_ RR /\ sum_ z e. ( ran G \ { 0 } ) ( vol* ` ( ( `' F " { ( y / z ) } ) i^i ( `' G " { z } ) ) ) e. RR /\ ( vol* ` ( `' ( F oF x. G ) " { y } ) ) <_ sum_ z e. ( ran G \ { 0 } ) ( vol* ` ( ( `' F " { ( y / z ) } ) i^i ( `' G " { z } ) ) ) ) -> ( vol* ` ( `' ( F oF x. G ) " { y } ) ) e. RR )
101	75, 90, 99, 100	syl3anc 1326	. . 3 |- ( ( ph /\ y e. ( ran ( F oF x. G ) \ { 0 } ) ) -> ( vol* ` ( `' ( F oF x. G ) " { y } ) ) e. RR )
102	73, 101	eqeltrd 2701	. 2 |- ( ( ph /\ y e. ( ran ( F oF x. G ) \ { 0 } ) ) -> ( vol ` ( `' ( F oF x. G ) " { y } ) ) e. RR )
103	12, 49, 71, 102	i1fd 23448	1 |- ( ph -> ( F oF x. G ) e. dom S.1 )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   {csn 4177   U_ciun 4520   class class class wbr 4653   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   oFcof 6895   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   x. cmul 9941   <_ cle 10075   / cdiv 10684   sum_csu 14416   vol*covol 23231   volcvol 23232   S.1citg1 23384

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

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-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-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-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-xadd 11947  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-xmet 19739  df-met 19740  df-ovol 23233  df-vol 23234  df-mbf 23388  df-itg1 23389

This theorem is referenced by:  mbfmullem2  23491  ftc1anclem3  33487