Theorem sranlm 23848

Description: The subring algebra over a normed ring is a normed left module. (Contributed by Mario Carneiro, 4-Oct-2015.)

Hypothesis

Ref	Expression
sranlm.a	⊢ 𝐴 = ((subringAlg ‘𝑊)‘𝑆)

Assertion

Ref	Expression
sranlm	⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ NrmMod)

Proof of Theorem sranlm

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

Step	Hyp	Ref	Expression
1		nrgngp 23826	. . . . 5 ⊢ (𝑊 ∈ NrmRing → 𝑊 ∈ NrmGrp)
2	1	adantr 481	. . . 4 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑊 ∈ NrmGrp)
3		eqidd 2739	. . . . 5 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘𝑊) = (Base‘𝑊))
4		sranlm.a	. . . . . . 7 ⊢ 𝐴 = ((subringAlg ‘𝑊)‘𝑆)
5	4	a1i 11	. . . . . 6 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))
6		eqid 2738	. . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊)
7	6	subrgss 20025	. . . . . . 7 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
8	7	adantl 482	. . . . . 6 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 ⊆ (Base‘𝑊))
9	5, 8	srabase 20441	. . . . 5 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘𝑊) = (Base‘𝐴))
10	5, 8	sraaddg 20443	. . . . . 6 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (+g‘𝑊) = (+g‘𝐴))
11	10	oveqdr 7303	. . . . 5 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g‘𝑊)𝑦) = (𝑥(+g‘𝐴)𝑦))
12	5, 8	srads 20455	. . . . . 6 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (dist‘𝑊) = (dist‘𝐴))
13	12	reseq1d 5890	. . . . 5 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) = ((dist‘𝐴) ↾ ((Base‘𝑊) × (Base‘𝑊))))
14	5, 8	sratopn 20454	. . . . 5 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (TopOpen‘𝑊) = (TopOpen‘𝐴))
15	3, 9, 11, 13, 14	ngppropd 23793	. . . 4 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊 ∈ NrmGrp ↔ 𝐴 ∈ NrmGrp))
16	2, 15	mpbid 231	. . 3 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ NrmGrp)
17	4	sralmod 20457	. . . 4 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)
18	17	adantl 482	. . 3 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ LMod)
19	5, 8	srasca 20447	. . . 4 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊 ↾s 𝑆) = (Scalar‘𝐴))
20		eqid 2738	. . . . 5 ⊢ (𝑊 ↾s 𝑆) = (𝑊 ↾s 𝑆)
21	20	subrgnrg 23837	. . . 4 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊 ↾s 𝑆) ∈ NrmRing)
22	19, 21	eqeltrrd 2840	. . 3 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Scalar‘𝐴) ∈ NrmRing)
23	16, 18, 22	3jca 1127	. 2 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝐴 ∈ NrmGrp ∧ 𝐴 ∈ LMod ∧ (Scalar‘𝐴) ∈ NrmRing))
24		eqid 2738	. . . . . . . 8 ⊢ (norm‘𝑊) = (norm‘𝑊)
25		eqid 2738	. . . . . . . 8 ⊢ (AbsVal‘𝑊) = (AbsVal‘𝑊)
26	24, 25	nrgabv 23825	. . . . . . 7 ⊢ (𝑊 ∈ NrmRing → (norm‘𝑊) ∈ (AbsVal‘𝑊))
27	26	ad2antrr 723	. . . . . 6 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (norm‘𝑊) ∈ (AbsVal‘𝑊))
28	8	adantr 481	. . . . . . 7 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑆 ⊆ (Base‘𝑊))
29		simprl 768	. . . . . . . 8 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘(Scalar‘𝐴)))
30	20	subrgbas 20033	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 = (Base‘(𝑊 ↾s 𝑆)))
31	30	adantl 482	. . . . . . . . . 10 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 = (Base‘(𝑊 ↾s 𝑆)))
32	19	fveq2d 6778	. . . . . . . . . 10 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘(𝑊 ↾s 𝑆)) = (Base‘(Scalar‘𝐴)))
33	31, 32	eqtrd 2778	. . . . . . . . 9 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 = (Base‘(Scalar‘𝐴)))
34	33	adantr 481	. . . . . . . 8 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑆 = (Base‘(Scalar‘𝐴)))
35	29, 34	eleqtrrd 2842	. . . . . . 7 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ 𝑆)
36	28, 35	sseldd 3922	. . . . . 6 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘𝑊))
37		simprr 770	. . . . . . 7 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑦 ∈ (Base‘𝐴))
38	9	adantr 481	. . . . . . 7 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (Base‘𝑊) = (Base‘𝐴))
39	37, 38	eleqtrrd 2842	. . . . . 6 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑦 ∈ (Base‘𝑊))
40		eqid 2738	. . . . . . 7 ⊢ (.r‘𝑊) = (.r‘𝑊)
41	25, 6, 40	abvmul 20089	. . . . . 6 ⊢ (((norm‘𝑊) ∈ (AbsVal‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → ((norm‘𝑊)‘(𝑥(.r‘𝑊)𝑦)) = (((norm‘𝑊)‘𝑥) · ((norm‘𝑊)‘𝑦)))
42	27, 36, 39, 41	syl3anc 1370	. . . . 5 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘𝑊)‘(𝑥(.r‘𝑊)𝑦)) = (((norm‘𝑊)‘𝑥) · ((norm‘𝑊)‘𝑦)))
43	9, 10, 12	nmpropd 23750	. . . . . . 7 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (norm‘𝑊) = (norm‘𝐴))
44	43	adantr 481	. . . . . 6 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (norm‘𝑊) = (norm‘𝐴))
45	5, 8	sravsca 20449	. . . . . . 7 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (.r‘𝑊) = ( ·𝑠 ‘𝐴))
46	45	oveqdr 7303	. . . . . 6 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥(.r‘𝑊)𝑦) = (𝑥( ·𝑠 ‘𝐴)𝑦))
47	44, 46	fveq12d 6781	. . . . 5 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘𝑊)‘(𝑥(.r‘𝑊)𝑦)) = ((norm‘𝐴)‘(𝑥( ·𝑠 ‘𝐴)𝑦)))
48	42, 47	eqtr3d 2780	. . . 4 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (((norm‘𝑊)‘𝑥) · ((norm‘𝑊)‘𝑦)) = ((norm‘𝐴)‘(𝑥( ·𝑠 ‘𝐴)𝑦)))
49		subrgsubg 20030	. . . . . . . 8 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ∈ (SubGrp‘𝑊))
50	49	ad2antlr 724	. . . . . . 7 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑆 ∈ (SubGrp‘𝑊))
51		eqid 2738	. . . . . . . 8 ⊢ (norm‘(𝑊 ↾s 𝑆)) = (norm‘(𝑊 ↾s 𝑆))
52	20, 24, 51	subgnm2 23790	. . . . . . 7 ⊢ ((𝑆 ∈ (SubGrp‘𝑊) ∧ 𝑥 ∈ 𝑆) → ((norm‘(𝑊 ↾s 𝑆))‘𝑥) = ((norm‘𝑊)‘𝑥))
53	50, 35, 52	syl2anc 584	. . . . . 6 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘(𝑊 ↾s 𝑆))‘𝑥) = ((norm‘𝑊)‘𝑥))
54	19	adantr 481	. . . . . . . 8 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑊 ↾s 𝑆) = (Scalar‘𝐴))
55	54	fveq2d 6778	. . . . . . 7 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (norm‘(𝑊 ↾s 𝑆)) = (norm‘(Scalar‘𝐴)))
56	55	fveq1d 6776	. . . . . 6 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘(𝑊 ↾s 𝑆))‘𝑥) = ((norm‘(Scalar‘𝐴))‘𝑥))
57	53, 56	eqtr3d 2780	. . . . 5 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘𝑊)‘𝑥) = ((norm‘(Scalar‘𝐴))‘𝑥))
58	44	fveq1d 6776	. . . . 5 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘𝑊)‘𝑦) = ((norm‘𝐴)‘𝑦))
59	57, 58	oveq12d 7293	. . . 4 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (((norm‘𝑊)‘𝑥) · ((norm‘𝑊)‘𝑦)) = (((norm‘(Scalar‘𝐴))‘𝑥) · ((norm‘𝐴)‘𝑦)))
60	48, 59	eqtr3d 2780	. . 3 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘𝐴)‘(𝑥( ·𝑠 ‘𝐴)𝑦)) = (((norm‘(Scalar‘𝐴))‘𝑥) · ((norm‘𝐴)‘𝑦)))
61	60	ralrimivva 3123	. 2 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → ∀𝑥 ∈ (Base‘(Scalar‘𝐴))∀𝑦 ∈ (Base‘𝐴)((norm‘𝐴)‘(𝑥( ·𝑠 ‘𝐴)𝑦)) = (((norm‘(Scalar‘𝐴))‘𝑥) · ((norm‘𝐴)‘𝑦)))
62		eqid 2738	. . 3 ⊢ (Base‘𝐴) = (Base‘𝐴)
63		eqid 2738	. . 3 ⊢ (norm‘𝐴) = (norm‘𝐴)
64		eqid 2738	. . 3 ⊢ ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘𝐴)
65		eqid 2738	. . 3 ⊢ (Scalar‘𝐴) = (Scalar‘𝐴)
66		eqid 2738	. . 3 ⊢ (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐴))
67		eqid 2738	. . 3 ⊢ (norm‘(Scalar‘𝐴)) = (norm‘(Scalar‘𝐴))
68	62, 63, 64, 65, 66, 67	isnlm 23839	. 2 ⊢ (𝐴 ∈ NrmMod ↔ ((𝐴 ∈ NrmGrp ∧ 𝐴 ∈ LMod ∧ (Scalar‘𝐴) ∈ NrmRing) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝐴))∀𝑦 ∈ (Base‘𝐴)((norm‘𝐴)‘(𝑥( ·𝑠 ‘𝐴)𝑦)) = (((norm‘(Scalar‘𝐴))‘𝑥) · ((norm‘𝐴)‘𝑦))))
69	23, 61, 68	sylanbrc 583	1 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ NrmMod)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064   ⊆ wss 3887   × cxp 5587  ‘cfv 6433  (class class class)co 7275   · cmul 10876  Basecbs 16912   ↾_s cress 16941  +_gcplusg 16962  ._rcmulr 16963  Scalarcsca 16965   ·_𝑠 cvsca 16966  distcds 16971  SubGrpcsubg 18749  SubRingcsubrg 20020  AbsValcabv 20076  LModclmod 20123  subringAlg csra 20430  normcnm 23732  NrmGrpcngp 23733  NrmRingcnrg 23735  NrmModcnlm 23736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-sup 9201  df-inf 9202  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-ico 13085  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-sca 16978  df-vsca 16979  df-ip 16980  df-tset 16981  df-ds 16984  df-rest 17133  df-topn 17134  df-0g 17152  df-topgen 17154  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-grp 18580  df-minusg 18581  df-sbg 18582  df-subg 18752  df-mgp 19721  df-ur 19738  df-ring 19785  df-subrg 20022  df-abv 20077  df-lmod 20125  df-sra 20434  df-psmet 20589  df-xmet 20590  df-met 20591  df-bl 20592  df-mopn 20593  df-top 22043  df-topon 22060  df-topsp 22082  df-bases 22096  df-xms 23473  df-ms 23474  df-nm 23738  df-ngp 23739  df-nrg 23741  df-nlm 23742

This theorem is referenced by:  rlmnlm  23852  srabn  24524