Theorem sitgaddlemb 29230

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 471	. 2 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> ( W ↾v ( H " ( 0 [,) +oo ) ) ) e. SLMod )
3		simpl 463	. . . . 5 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> ph )
4		sitgadd.6	. . . . . . . 8 |- ( ph -> (Scalar ` W ) e. ℝExt )
5		eqid 2462	. . . . . . . . 9 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
6	5	rrhfe 28865	. . . . . . . 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 5742	. . . . . . 7 |- ( H : RR --> ( Base ` (Scalar ` W ) ) <-> (RRHom ` (Scalar ` W ) ) : RR --> ( Base ` (Scalar ` W ) ) )
10	7, 9	sylibr 217	. . . . . 6 |- ( ph -> H : RR --> ( Base ` (Scalar ` W ) ) )
11		ffn 5751	. . . . . 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 11769	. . . . 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 467	. . . . . . 7 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) )
17	16	eldifad 3428	. . . . . 6 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> p e. ( ran F X. ran G ) )
18		xp1st 6850	. . . . . 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 6851	. . . . . 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 3429	. . . . . . . 8 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> -. p e. { <. .0. , .0. >. } )
23		vex 3060	. . . . . . . . . 10 |- p e. _V
24	23	elsnc 4004	. . . . . . . . 9 |- ( p e. { <. .0. , .0. >. } <-> p = <. .0. , .0. >. )
25	24	notbii 302	. . . . . . . 8 |- ( -. p e. { <. .0. , .0. >. } <-> -. p = <. .0. , .0. >. )
26	22, 25	sylib 201	. . . . . . 7 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> -. p = <. .0. , .0. >. )
27		eqopi 6854	. . . . . . . . . 10 |- ( ( p e. ( ran F X. ran G ) /\ ( ( 1st ` p ) = .0. /\ ( 2nd ` p ) = .0. ) ) -> p = <. .0. , .0. >. )
28	27	ex 440	. . . . . . . . 9 |- ( p e. ( ran F X. ran G ) -> ( ( ( 1st ` p ) = .0. /\ ( 2nd ` p ) = .0. ) -> p = <. .0. , .0. >. ) )
29	28	con3d 140	. . . . . . . 8 |- ( p e. ( ran F X. ran G ) -> ( -. p = <. .0. , .0. >. -> -. ( ( 1st ` p ) = .0. /\ ( 2nd ` p ) = .0. ) ) )
30	29	imp 435	. . . . . . 7 |- ( ( p e. ( ran F X. ran G ) /\ -. p = <. .0. , .0. >. ) -> -. ( ( 1st ` p ) = .0. /\ ( 2nd ` p ) = .0. ) )
31	17, 26, 30	syl2anc 671	. . . . . 6 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> -. ( ( 1st ` p ) = .0. /\ ( 2nd ` p ) = .0. ) )
32		ianor 495	. . . . . . 7 |- ( -. ( ( 1st ` p ) = .0. /\ ( 2nd ` p ) = .0. ) <-> ( -. ( 1st ` p ) = .0. \/ -. ( 2nd ` p ) = .0. ) )
33		df-ne 2635	. . . . . . . 8 |- ( ( 1st ` p ) =/= .0. <-> -. ( 1st ` p ) = .0. )
34		df-ne 2635	. . . . . . . 8 |- ( ( 2nd ` p ) =/= .0. <-> -. ( 2nd ` p ) = .0. )
35	33, 34	orbi12i 528	. . . . . . 7 |- ( ( ( 1st ` p ) =/= .0. \/ ( 2nd ` p ) =/= .0. ) <-> ( -. ( 1st ` p ) = .0. \/ -. ( 2nd ` p ) = .0. ) )
36	32, 35	bitr4i 260	. . . . . 6 |- ( -. ( ( 1st ` p ) = .0. /\ ( 2nd ` p ) = .0. ) <-> ( ( 1st ` p ) =/= .0. \/ ( 2nd ` p ) =/= .0. ) )
37	31, 36	sylib 201	. . . . 5 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> ( ( 1st ` p ) =/= .0. \/ ( 2nd ` p ) =/= .0. ) )
38		sitgval.b	. . . . . 6 |- B = ( Base ` W )
39		sitgval.j	. . . . . 6 |- J = ( TopOpen ` W )
40		sitgval.s	. . . . . 6 |- S = (sigaGen ` J )
41		sitgval.0	. . . . . 6 |- .0. = ( 0g ` W )
42		sitgval.x	. . . . . 6 |- .x. = ( .s ` W )
43		sitgval.1	. . . . . 6 |- ( ph -> W e. V )
44		sitgval.2	. . . . . 6 |- ( ph -> M e. U. ran measures )
45		sitgadd.4	. . . . . 6 |- ( ph -> F e. dom ( Wsitg M ) )
46		sitgadd.5	. . . . . 6 |- ( ph -> G e. dom ( Wsitg M ) )
47		sitgadd.1	. . . . . 6 |- ( ph -> W e. TopSp )
48		sitgadd.3	. . . . . 6 |- ( ph -> J e. Fre )
49	38, 39, 40, 41, 42, 8, 43, 44, 45, 46, 47, 48	sibfinima 29221	. . . . 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 ) )
50	3, 19, 21, 37, 49	syl31anc 1279	. . . 4 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) e. ( 0 [,) +oo ) )
51		fnfvima 6168	. . . 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 ) ) )
52	13, 15, 50, 51	syl3anc 1276	. . 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 ) ) )
53		imassrn 5198	. . . . . 6 |- ( H " ( 0 [,) +oo ) ) C_ ran H
54		frn 5758	. . . . . . 7 |- ( H : RR --> ( Base ` (Scalar ` W ) ) -> ran H C_ ( Base ` (Scalar ` W ) ) )
55	10, 54	syl 17	. . . . . 6 |- ( ph -> ran H C_ ( Base ` (Scalar ` W ) ) )
56	53, 55	syl5ss 3455	. . . . 5 |- ( ph -> ( H " ( 0 [,) +oo ) ) C_ ( Base ` (Scalar ` W ) ) )
57		eqid 2462	. . . . . 6 |- ( (Scalar ` W ) ↾s ( H " ( 0 [,) +oo ) ) ) = ( (Scalar ` W ) ↾s ( H " ( 0 [,) +oo ) ) )
58	57, 5	ressbas2 15229	. . . . 5 |- ( ( H " ( 0 [,) +oo ) ) C_ ( Base ` (Scalar ` W ) ) -> ( H " ( 0 [,) +oo ) ) = ( Base ` ( (Scalar ` W ) ↾s ( H " ( 0 [,) +oo ) ) ) ) )
59	56, 58	syl 17	. . . 4 |- ( ph -> ( H " ( 0 [,) +oo ) ) = ( Base ` ( (Scalar ` W ) ↾s ( H " ( 0 [,) +oo ) ) ) ) )
60	3, 59	syl 17	. . 3 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> ( H " ( 0 [,) +oo ) ) = ( Base ` ( (Scalar ` W ) ↾s ( H " ( 0 [,) +oo ) ) ) ) )
61	52, 60	eleqtrd 2542	. 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 ) ) ) ) )
62	38, 39, 40, 41, 42, 8, 43, 44, 46	sibff 29218	. . . . . 6 |- ( ph -> G : U. dom M --> U. J )
63	38, 39	tpsuni 20002	. . . . . . 7 |- ( W e. TopSp -> B = U. J )
64		feq3 5734	. . . . . . 7 |- ( B = U. J -> ( G : U. dom M --> B <-> G : U. dom M --> U. J ) )
65	47, 63, 64	3syl 18	. . . . . 6 |- ( ph -> ( G : U. dom M --> B <-> G : U. dom M --> U. J ) )
66	62, 65	mpbird 240	. . . . 5 |- ( ph -> G : U. dom M --> B )
67		frn 5758	. . . . 5 |- ( G : U. dom M --> B -> ran G C_ B )
68	66, 67	syl 17	. . . 4 |- ( ph -> ran G C_ B )
69	68	adantr 471	. . 3 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> ran G C_ B )
70	69, 21	sseldd 3445	. 2 |- ( ( ph /\ p e. ( ( ran F X. ran G ) \ { <. .0. , .0. >. } ) ) -> ( 2nd ` p ) e. B )
71		fvex 5898	. . . . 5 |- (RRHom ` (Scalar ` W ) ) e. _V
72	8, 71	eqeltri 2536	. . . 4 |- H e. _V
73		imaexg 6757	. . . 4 |- ( H e. _V -> ( H " ( 0 [,) +oo ) ) e. _V )
74		eqid 2462	. . . . 5 |- ( W ↾v ( H " ( 0 [,) +oo ) ) ) = ( W ↾v ( H " ( 0 [,) +oo ) ) )
75	74, 38	resvbas 28644	. . . 4 |- ( ( H " ( 0 [,) +oo ) ) e. _V -> B = ( Base ` ( W ↾v ( H " ( 0 [,) +oo ) ) ) ) )
76	72, 73, 75	mp2b 10	. . 3 |- B = ( Base ` ( W ↾v ( H " ( 0 [,) +oo ) ) ) )
77		eqid 2462	. . . . 5 |- (Scalar ` W ) = (Scalar ` W )
78	74, 77, 5	resvsca 28642	. . . 4 |- ( ( H " ( 0 [,) +oo ) ) e. _V -> ( (Scalar ` W ) ↾s ( H " ( 0 [,) +oo ) ) ) = (Scalar ` ( W ↾v ( H " ( 0 [,) +oo ) ) ) ) )
79	72, 73, 78	mp2b 10	. . 3 |- ( (Scalar ` W ) ↾s ( H " ( 0 [,) +oo ) ) ) = (Scalar ` ( W ↾v ( H " ( 0 [,) +oo ) ) ) )
80	74, 42	resvvsca 28646	. . . 4 |- ( ( H " ( 0 [,) +oo ) ) e. _V -> .x. = ( .s ` ( W ↾v ( H " ( 0 [,) +oo ) ) ) ) )
81	72, 73, 80	mp2b 10	. . 3 |- .x. = ( .s ` ( W ↾v ( H " ( 0 [,) +oo ) ) ) )
82		eqid 2462	. . 3 |- ( Base ` ( (Scalar ` W ) ↾s ( H " ( 0 [,) +oo ) ) ) ) = ( Base ` ( (Scalar ` W ) ↾s ( H " ( 0 [,) +oo ) ) ) )
83	76, 79, 81, 82	slmdvscl 28579	. 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 )
84	2, 61, 70, 83	syl3anc 1276	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 189   \/ wo 374   /\ wa 375   = wceq 1455   e. wcel 1898   =/= wne 2633   _Vcvv 3057   \ cdif 3413   i^i cin 3415   C_ wss 3416   {csn 3980   <.cop 3986   U.cuni 4212   X. cxp 4851   `'ccnv 4852   dom cdm 4853   ran crn 4854   "cima 4856   Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6315   1stc1st 6818   2ndc2nd 6819   RRcr 9564   0cc0 9565   +oocpnf 9698   [,)cico 11666   Basecbs 15170   ↾_s cress 15171   +g cplusg 15239  Scalarcsca 15242   .scvsca 15243   TopOpenctopn 15369   0gc0g 15387   TopSpctps 19968   Frect1 20372  SLModcslmd 28565   ↾_v cresv 28636  RRHomcrrh 28846   ℝExt crrext 28847  sigaGencsigagen 29009  measurescmeas 29066  sitgcsitg 29211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-inf2 8172  ax-ac2 8919  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642  ax-pre-sup 9643  ax-addf 9644  ax-mulf 9645

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-iin 4295  df-disj 4388  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-of 6558  df-om 6720  df-1st 6820  df-2nd 6821  df-supp 6942  df-tpos 6999  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-2o 7209  df-oadd 7212  df-er 7389  df-map 7500  df-pm 7501  df-ixp 7549  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-fsupp 7910  df-fi 7951  df-sup 7982  df-inf 7983  df-oi 8051  df-card 8399  df-acn 8402  df-ac 8573  df-cda 8624  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-div 10298  df-nn 10638  df-2 10696  df-3 10697  df-4 10698  df-5 10699  df-6 10700  df-7 10701  df-8 10702  df-9 10703  df-10 10704  df-n0 10899  df-z 10967  df-dec 11081  df-uz 11189  df-q 11294  df-rp 11332  df-xneg 11438  df-xadd 11439  df-xmul 11440  df-ioo 11668  df-ioc 11669  df-ico 11670  df-icc 11671  df-fz 11814  df-fzo 11947  df-fl 12060  df-mod 12129  df-seq 12246  df-exp 12305  df-fac 12492  df-bc 12520  df-hash 12548  df-shft 13179  df-cj 13211  df-re 13212  df-im 13213  df-sqrt 13347  df-abs 13348  df-limsup 13575  df-clim 13601  df-rlim 13602  df-sum 13802  df-ef 14170  df-sin 14172  df-cos 14173  df-pi 14175  df-dvds 14355  df-gcd 14518  df-numer 14733  df-denom 14734  df-gz 14923  df-struct 15172  df-ndx 15173  df-slot 15174  df-base 15175  df-sets 15176  df-ress 15177  df-plusg 15252  df-mulr 15253  df-starv 15254  df-sca 15255  df-vsca 15256  df-ip 15257  df-tset 15258  df-ple 15259  df-ds 15261  df-unif 15262  df-hom 15263  df-cco 15264  df-rest 15370  df-topn 15371  df-0g 15389  df-gsum 15390  df-topgen 15391  df-pt 15392  df-prds 15395  df-ordt 15448  df-xrs 15449  df-qtop 15455  df-imas 15456  df-xps 15459  df-mre 15541  df-mrc 15542  df-acs 15544  df-ps 16495  df-tsr 16496  df-plusf 16536  df-mgm 16537  df-sgrp 16576  df-mnd 16586  df-mhm 16631  df-submnd 16632  df-grp 16722  df-minusg 16723  df-sbg 16724  df-mulg 16725  df-subg 16863  df-ghm 16930  df-cntz 17020  df-od 17221  df-cmn 17481  df-abl 17482  df-mgp 17773  df-ur 17785  df-ring 17831  df-cring 17832  df-oppr 17900  df-dvdsr 17918  df-unit 17919  df-invr 17949  df-dvr 17960  df-rnghom 17992  df-drng 18026  df-subrg 18055  df-abv 18094  df-lmod 18142  df-scaf 18143  df-sra 18444  df-rgmod 18445  df-nzr 18531  df-psmet 19011  df-xmet 19012  df-met 19013  df-bl 19014  df-mopn 19015  df-fbas 19016  df-fg 19017  df-metu 19018  df-cnfld 19020  df-zring 19089  df-zrh 19124  df-zlm 19125  df-chr 19126  df-refld 19222  df-top 19970  df-bases 19971  df-topon 19972  df-topsp 19973  df-cld 20083  df-ntr 20084  df-cls 20085  df-nei 20163  df-lp 20201  df-perf 20202  df-cn 20292  df-cnp 20293  df-t1 20379  df-haus 20380  df-reg 20381  df-cmp 20451  df-tx 20626  df-hmeo 20819  df-fil 20910  df-fm 21002  df-flim 21003  df-flf 21004  df-fcls 21005  df-cnext 21124  df-tmd 21136  df-tgp 21137  df-tsms 21190  df-trg 21223  df-ust 21264  df-utop 21295  df-uss 21320  df-usp 21321  df-ucn 21340  df-cfilu 21351  df-cusp 21362  df-xms 21384  df-ms 21385  df-tms 21386  df-nm 21646  df-ngp 21647  df-nrg 21649  df-nlm 21650  df-ii 21958  df-cncf 21959  df-cfil 22274  df-cmet 22276  df-cms 22352  df-limc 22870  df-dv 22871  df-log 23555  df-slmd 28566  df-resv 28637  df-qqh 28826  df-rrh 28848  df-rrext 28852  df-esum 28898  df-siga 28979  df-sigagen 29010  df-meas 29067  df-mbfm 29122  df-sitg 29212

This theorem is referenced by: (None)