Theorem sitgaddlemb 30410

Description: Lemma for * sitgadd . (Contributed by Thierry Arnoux, 10-Mar-2019.)

Hypotheses

Ref	Expression
sitgval.b	|- B = ( Base ` W )
sitgval.j	|- J = ( TopOpen ` W )
sitgval.s	|- S = (sigaGen ` J )
sitgval.0	|- .0. = ( 0g ` W )
sitgval.x	|- .x. = ( .s ` W )
sitgval.h	|- H = (RRHom ` (Scalar ` W ) )
sitgval.1	|- ( ph -> W e. V )
sitgval.2	|- ( ph -> M e. U. ran measures )
sitgadd.1	|- ( ph -> W e. TopSp )
sitgadd.2	|- ( ph -> ( W ↾v ( H " ( 0 [,) +oo ) ) ) e. SLMod )
sitgadd.3	|- ( ph -> J e. Fre )
sitgadd.4	|- ( ph -> F e. dom ( Wsitg M ) )
sitgadd.5	|- ( ph -> G e. dom ( Wsitg M ) )
sitgadd.6	|- ( ph -> (Scalar ` W ) e. ℝExt )
sitgadd.7	|- .+ = ( +g ` W )

Assertion

Ref	Expression
sitgaddlemb	|- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> ( ( H ` ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) ) .x. ( 2nd ` p ) ) e. B )

Proof of Theorem sitgaddlemb

Step	Hyp	Ref	Expression
1		sitgadd.2	. . 3 |- ( ph -> ( W ↾v ( H " ( 0 [,) +oo ) ) ) e. SLMod )
2	1	adantr 481	. 2 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> ( W ↾v ( H " ( 0 [,) +oo ) ) ) e. SLMod )
3		simpl 473	. . . . 5 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> ph )
4		sitgadd.6	. . . . . . . 8 |- ( ph -> (Scalar ` W ) e. ℝExt )
5		eqid 2622	. . . . . . . . 9 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
6	5	rrhfe 30056	. . . . . . . 8 |- ( (Scalar ` W ) e. ℝExt -> (RRHom ` (Scalar ` W ) ) : RR --> ( Base ` (Scalar ` W ) ) )
7	4, 6	syl 17	. . . . . . 7 |- ( ph -> (RRHom ` (Scalar ` W ) ) : RR --> ( Base ` (Scalar ` W ) ) )
8		sitgval.h	. . . . . . . 8 |- H = (RRHom ` (Scalar ` W ) )
9	8	feq1i 6036	. . . . . . 7 |- ( H : RR --> ( Base ` (Scalar ` W ) ) <-> (RRHom ` (Scalar ` W ) ) : RR --> ( Base ` (Scalar ` W ) ) )
10	7, 9	sylibr 224	. . . . . 6 |- ( ph -> H : RR --> ( Base ` (Scalar ` W ) ) )
11		ffn 6045	. . . . . 6 |- ( H : RR --> ( Base ` (Scalar ` W ) ) -> H Fn RR )
12	10, 11	syl 17	. . . . 5 |- ( ph -> H Fn RR )
13	3, 12	syl 17	. . . 4 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> H Fn RR )
14		rge0ssre 12280	. . . . 5 |- ( 0 [,) +oo ) C_ RR
15	14	a1i 11	. . . 4 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> ( 0 [,) +oo ) C_ RR )
16		simpr 477	. . . . . . 7 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) )
17	16	eldifad 3586	. . . . . 6 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> p e. ( ran F X. ran G ) )
18		xp1st 7198	. . . . . 6 |- ( p e. ( ran F X. ran G ) -> ( 1st ` p ) e. ran F )
19	17, 18	syl 17	. . . . 5 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> ( 1st ` p ) e. ran F )
20		xp2nd 7199	. . . . . 6 |- ( p e. ( ran F X. ran G ) -> ( 2nd ` p ) e. ran G )
21	17, 20	syl 17	. . . . 5 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> ( 2nd ` p ) e. ran G )
22	16	eldifbd 3587	. . . . . . . 8 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> -. p e. { <. .0. , .0. >. } )
23		velsn 4193	. . . . . . . . 9 |- ( p e. { <. .0. , .0. >. } <-> p = <. .0. , .0. >. )
24	23	notbii 310	. . . . . . . 8 |- ( -. p e. { <. .0. , .0. >. } <-> -. p = <. .0. , .0. >. )
25	22, 24	sylib 208	. . . . . . 7 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> -. p = <. .0. , .0. >. )
26		eqopi 7202	. . . . . . . . . 10 |- ( ( p e. ( ran F X. ran G ) /\ ( ( 1st ` p ) = .0. /\ ( 2nd ` p ) = .0. ) ) -> p = <. .0. , .0. >. )
27	26	ex 450	. . . . . . . . 9 |- ( p e. ( ran F X. ran G ) -> ( ( ( 1st ` p ) = .0. /\ ( 2nd ` p ) = .0. ) -> p = <. .0. , .0. >. ) )
28	27	con3d 148	. . . . . . . 8 |- ( p e. ( ran F X. ran G ) -> ( -. p = <. .0. , .0. >. -> -. ( ( 1st ` p ) = .0. /\ ( 2nd ` p ) = .0. ) ) )
29	28	imp 445	. . . . . . 7 |- ( ( p e. ( ran F X. ran G ) /\ -. p = <. .0. , .0. >. ) -> -. ( ( 1st ` p ) = .0. /\ ( 2nd ` p ) = .0. ) )
30	17, 25, 29	syl2anc 693	. . . . . 6 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> -. ( ( 1st ` p ) = .0. /\ ( 2nd ` p ) = .0. ) )
31		ianor 509	. . . . . . 7 |- ( -. ( ( 1st ` p ) = .0. /\ ( 2nd ` p ) = .0. ) <-> ( -. ( 1st ` p ) = .0. \/ -. ( 2nd ` p ) = .0. ) )
32		df-ne 2795	. . . . . . . 8 |- ( ( 1st ` p ) =/= .0. <-> -. ( 1st ` p ) = .0. )
33		df-ne 2795	. . . . . . . 8 |- ( ( 2nd ` p ) =/= .0. <-> -. ( 2nd ` p ) = .0. )
34	32, 33	orbi12i 543	. . . . . . 7 |- ( ( ( 1st ` p ) =/= .0. \/ ( 2nd ` p ) =/= .0. ) <-> ( -. ( 1st ` p ) = .0. \/ -. ( 2nd ` p ) = .0. ) )
35	31, 34	bitr4i 267	. . . . . 6 |- ( -. ( ( 1st ` p ) = .0. /\ ( 2nd ` p ) = .0. ) <-> ( ( 1st ` p ) =/= .0. \/ ( 2nd ` p ) =/= .0. ) )
36	30, 35	sylib 208	. . . . 5 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> ( ( 1st ` p ) =/= .0. \/ ( 2nd ` p ) =/= .0. ) )
37		sitgval.b	. . . . . 6 |- B = ( Base ` W )
38		sitgval.j	. . . . . 6 |- J = ( TopOpen ` W )
39		sitgval.s	. . . . . 6 |- S = (sigaGen ` J )
40		sitgval.0	. . . . . 6 |- .0. = ( 0g ` W )
41		sitgval.x	. . . . . 6 |- .x. = ( .s ` W )
42		sitgval.1	. . . . . 6 |- ( ph -> W e. V )
43		sitgval.2	. . . . . 6 |- ( ph -> M e. U. ran measures )
44		sitgadd.4	. . . . . 6 |- ( ph -> F e. dom ( Wsitg M ) )
45		sitgadd.5	. . . . . 6 |- ( ph -> G e. dom ( Wsitg M ) )
46		sitgadd.1	. . . . . 6 |- ( ph -> W e. TopSp )
47		sitgadd.3	. . . . . 6 |- ( ph -> J e. Fre )
48	37, 38, 39, 40, 41, 8, 42, 43, 44, 45, 46, 47	sibfinima 30401	. . . . 5 |- ( ( ( ph /\ ( 1st ` p ) e. ran F /\ ( 2nd ` p ) e. ran G ) /\ ( ( 1st ` p ) =/= .0. \/ ( 2nd ` p ) =/= .0. ) ) -> ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) e. ( 0 [,) +oo ) )
49	3, 19, 21, 36, 48	syl31anc 1329	. . . 4 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) e. ( 0 [,) +oo ) )
50		fnfvima 6496	. . . 4 |- ( ( H Fn RR /\ ( 0 [,) +oo ) C_ RR /\ ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) e. ( 0 [,) +oo ) ) -> ( H ` ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) ) e. ( H " ( 0 [,) +oo ) ) )
51	13, 15, 49, 50	syl3anc 1326	. . 3 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> ( H ` ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) ) e. ( H " ( 0 [,) +oo ) ) )
52		imassrn 5477	. . . . . 6 |- ( H " ( 0 [,) +oo ) ) C_ ran H
53		frn 6053	. . . . . . 7 |- ( H : RR --> ( Base ` (Scalar ` W ) ) -> ran H C_ ( Base ` (Scalar ` W ) ) )
54	10, 53	syl 17	. . . . . 6 |- ( ph -> ran H C_ ( Base ` (Scalar ` W ) ) )
55	52, 54	syl5ss 3614	. . . . 5 |- ( ph -> ( H " ( 0 [,) +oo ) ) C_ ( Base ` (Scalar ` W ) ) )
56		eqid 2622	. . . . . 6 |- ( (Scalar ` W ) ↾s ( H " ( 0 [,) +oo ) ) ) = ( (Scalar ` W ) ↾s ( H " ( 0 [,) +oo ) ) )
57	56, 5	ressbas2 15931	. . . . 5 |- ( ( H " ( 0 [,) +oo ) ) C_ ( Base ` (Scalar ` W ) ) -> ( H " ( 0 [,) +oo ) ) = ( Base ` ( (Scalar ` W ) ↾s ( H " ( 0 [,) +oo ) ) ) ) )
58	55, 57	syl 17	. . . 4 |- ( ph -> ( H " ( 0 [,) +oo ) ) = ( Base ` ( (Scalar ` W ) ↾s ( H " ( 0 [,) +oo ) ) ) ) )
59	3, 58	syl 17	. . 3 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> ( H " ( 0 [,) +oo ) ) = ( Base ` ( (Scalar ` W ) ↾s ( H " ( 0 [,) +oo ) ) ) ) )
60	51, 59	eleqtrd 2703	. 2 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> ( H ` ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) ) e. ( Base ` ( (Scalar ` W ) ↾s ( H " ( 0 [,) +oo ) ) ) ) )
61	37, 38, 39, 40, 41, 8, 42, 43, 45	sibff 30398	. . . . . 6 |- ( ph -> G : U. dom M --> U. J )
62	37, 38	tpsuni 20740	. . . . . . 7 |- ( W e. TopSp -> B = U. J )
63		feq3 6028	. . . . . . 7 |- ( B = U. J -> ( G : U. dom M --> B <-> G : U. dom M --> U. J ) )
64	46, 62, 63	3syl 18	. . . . . 6 |- ( ph -> ( G : U. dom M --> B <-> G : U. dom M --> U. J ) )
65	61, 64	mpbird 247	. . . . 5 |- ( ph -> G : U. dom M --> B )
66		frn 6053	. . . . 5 |- ( G : U. dom M --> B -> ran G C_ B )
67	65, 66	syl 17	. . . 4 |- ( ph -> ran G C_ B )
68	67	adantr 481	. . 3 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> ran G C_ B )
69	68, 21	sseldd 3604	. 2 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> ( 2nd ` p ) e. B )
70		fvex 6201	. . . . 5 |- (RRHom ` (Scalar ` W ) ) e. _V
71	8, 70	eqeltri 2697	. . . 4 |- H e. _V
72		imaexg 7103	. . . 4 |- ( H e. _V -> ( H " ( 0 [,) +oo ) ) e. _V )
73		eqid 2622	. . . . 5 |- ( W ↾v ( H " ( 0 [,) +oo ) ) ) = ( W ↾v ( H " ( 0 [,) +oo ) ) )
74	73, 37	resvbas 29832	. . . 4 |- ( ( H " ( 0 [,) +oo ) ) e. _V -> B = ( Base ` ( W ↾v ( H " ( 0 [,) +oo ) ) ) ) )
75	71, 72, 74	mp2b 10	. . 3 |- B = ( Base ` ( W ↾v ( H " ( 0 [,) +oo ) ) ) )
76		eqid 2622	. . . . 5 |- (Scalar ` W ) = (Scalar ` W )
77	73, 76, 5	resvsca 29830	. . . 4 |- ( ( H " ( 0 [,) +oo ) ) e. _V -> ( (Scalar ` W ) ↾s ( H " ( 0 [,) +oo ) ) ) = (Scalar ` ( W ↾v ( H " ( 0 [,) +oo ) ) ) ) )
78	71, 72, 77	mp2b 10	. . 3 |- ( (Scalar ` W ) ↾s ( H " ( 0 [,) +oo ) ) ) = (Scalar ` ( W ↾v ( H " ( 0 [,) +oo ) ) ) )
79	73, 41	resvvsca 29834	. . . 4 |- ( ( H " ( 0 [,) +oo ) ) e. _V -> .x. = ( .s ` ( W ↾v ( H " ( 0 [,) +oo ) ) ) ) )
80	71, 72, 79	mp2b 10	. . 3 |- .x. = ( .s ` ( W ↾v ( H " ( 0 [,) +oo ) ) ) )
81		eqid 2622	. . 3 |- ( Base ` ( (Scalar ` W ) ↾s ( H " ( 0 [,) +oo ) ) ) ) = ( Base ` ( (Scalar ` W ) ↾s ( H " ( 0 [,) +oo ) ) ) )
82	75, 78, 80, 81	slmdvscl 29767	. 2 |- ( ( ( W ↾v ( H " ( 0 [,) +oo ) ) ) e. SLMod /\ ( H ` ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) ) e. ( Base ` ( (Scalar ` W ) ↾s ( H " ( 0 [,) +oo ) ) ) ) /\ ( 2nd ` p ) e. B ) -> ( ( H ` ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) ) .x. ( 2nd ` p ) ) e. B )
83	2, 60, 69, 82	syl3anc 1326	1 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> ( ( H ` ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) ) .x. ( 2nd ` p ) ) e. B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   {csn 4177   <.cop 4183   U.cuni 4436   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   RRcr 9935   0cc0 9936   +oocpnf 10071   [,)cico 12177   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941  Scalarcsca 15944   .scvsca 15945   TopOpenctopn 16082   0gc0g 16100   TopSpctps 20736   Frect1 21111  SLModcslmd 29753   ↾_v cresv 29824  RRHomcrrh 30037   ℝExt crrext 30038  sigaGencsigagen 30201  measurescmeas 30258  sitgcsitg 30391

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-ac2 9285  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-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  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-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-ac 8939  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-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  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-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-dvds 14984  df-gcd 15217  df-numer 15443  df-denom 15444  df-gz 15634  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-ordt 16161  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-ps 17200  df-tsr 17201  df-plusf 17241  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-od 17948  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-rnghom 18715  df-drng 18749  df-subrg 18778  df-abv 18817  df-lmod 18865  df-scaf 18866  df-sra 19172  df-rgmod 19173  df-nzr 19258  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-metu 19745  df-cnfld 19747  df-zring 19819  df-zrh 19852  df-zlm 19853  df-chr 19854  df-refld 19951  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-t1 21118  df-haus 21119  df-reg 21120  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-fcls 21745  df-cnext 21864  df-tmd 21876  df-tgp 21877  df-tsms 21930  df-trg 21963  df-ust 22004  df-utop 22035  df-uss 22060  df-usp 22061  df-ucn 22080  df-cfilu 22091  df-cusp 22102  df-xms 22125  df-ms 22126  df-tms 22127  df-nm 22387  df-ngp 22388  df-nrg 22390  df-nlm 22391  df-ii 22680  df-cncf 22681  df-cfil 23053  df-cmet 23055  df-cms 23132  df-limc 23630  df-dv 23631  df-log 24303  df-slmd 29754  df-resv 29825  df-qqh 30017  df-rrh 30039  df-rrext 30043  df-esum 30090  df-siga 30171  df-sigagen 30202  df-meas 30259  df-mbfm 30313  df-sitg 30392

This theorem is referenced by: (None)