Theorem sranlm 24626

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 24604	. . . . 5 ⊢ (𝑊 ∈ NrmRing → 𝑊 ∈ NrmGrp)
2	1	adantr 480	. . . 4 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑊 ∈ NrmGrp)
3		eqidd 2735	. . . . 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 2734	. . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊)
7	6	subrgss 20503	. . . . . . 7 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
8	7	adantl 481	. . . . . 6 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 ⊆ (Base‘𝑊))
9	5, 8	srabase 21127	. . . . 5 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘𝑊) = (Base‘𝐴))
10	5, 8	sraaddg 21128	. . . . . 6 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (+g‘𝑊) = (+g‘𝐴))
11	10	oveqdr 7384	. . . . 5 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g‘𝑊)𝑦) = (𝑥(+g‘𝐴)𝑦))
12	5, 8	srads 21135	. . . . . 6 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (dist‘𝑊) = (dist‘𝐴))
13	12	reseq1d 5935	. . . . 5 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) = ((dist‘𝐴) ↾ ((Base‘𝑊) × (Base‘𝑊))))
14	5, 8	sratopn 21134	. . . . 5 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (TopOpen‘𝑊) = (TopOpen‘𝐴))
15	3, 9, 11, 13, 14	ngppropd 24579	. . . 4 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊 ∈ NrmGrp ↔ 𝐴 ∈ NrmGrp))
16	2, 15	mpbid 232	. . 3 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ NrmGrp)
17	4	sralmod 21137	. . . 4 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)
18	17	adantl 481	. . 3 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ LMod)
19	5, 8	srasca 21130	. . . 4 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊 ↾s 𝑆) = (Scalar‘𝐴))
20		eqid 2734	. . . . 5 ⊢ (𝑊 ↾s 𝑆) = (𝑊 ↾s 𝑆)
21	20	subrgnrg 24615	. . . 4 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊 ↾s 𝑆) ∈ NrmRing)
22	19, 21	eqeltrrd 2835	. . 3 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Scalar‘𝐴) ∈ NrmRing)
23	16, 18, 22	3jca 1128	. 2 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝐴 ∈ NrmGrp ∧ 𝐴 ∈ LMod ∧ (Scalar‘𝐴) ∈ NrmRing))
24		eqid 2734	. . . . . . . 8 ⊢ (norm‘𝑊) = (norm‘𝑊)
25		eqid 2734	. . . . . . . 8 ⊢ (AbsVal‘𝑊) = (AbsVal‘𝑊)
26	24, 25	nrgabv 24603	. . . . . . 7 ⊢ (𝑊 ∈ NrmRing → (norm‘𝑊) ∈ (AbsVal‘𝑊))
27	26	ad2antrr 726	. . . . . 6 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (norm‘𝑊) ∈ (AbsVal‘𝑊))
28	8	adantr 480	. . . . . . 7 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑆 ⊆ (Base‘𝑊))
29		simprl 770	. . . . . . . 8 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘(Scalar‘𝐴)))
30	20	subrgbas 20512	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 = (Base‘(𝑊 ↾s 𝑆)))
31	30	adantl 481	. . . . . . . . . 10 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 = (Base‘(𝑊 ↾s 𝑆)))
32	19	fveq2d 6836	. . . . . . . . . 10 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘(𝑊 ↾s 𝑆)) = (Base‘(Scalar‘𝐴)))
33	31, 32	eqtrd 2769	. . . . . . . . 9 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 = (Base‘(Scalar‘𝐴)))
34	33	adantr 480	. . . . . . . 8 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑆 = (Base‘(Scalar‘𝐴)))
35	29, 34	eleqtrrd 2837	. . . . . . 7 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ 𝑆)
36	28, 35	sseldd 3932	. . . . . 6 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘𝑊))
37		simprr 772	. . . . . . 7 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑦 ∈ (Base‘𝐴))
38	9	adantr 480	. . . . . . 7 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (Base‘𝑊) = (Base‘𝐴))
39	37, 38	eleqtrrd 2837	. . . . . 6 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑦 ∈ (Base‘𝑊))
40		eqid 2734	. . . . . . 7 ⊢ (.r‘𝑊) = (.r‘𝑊)
41	25, 6, 40	abvmul 20752	. . . . . 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 1373	. . . . 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 24536	. . . . . . 7 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (norm‘𝑊) = (norm‘𝐴))
44	43	adantr 480	. . . . . 6 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (norm‘𝑊) = (norm‘𝐴))
45	5, 8	sravsca 21131	. . . . . . 7 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (.r‘𝑊) = ( ·𝑠 ‘𝐴))
46	45	oveqdr 7384	. . . . . 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 6839	. . . . 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 2771	. . . 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 20508	. . . . . . . 8 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ∈ (SubGrp‘𝑊))
50	49	ad2antlr 727	. . . . . . 7 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑆 ∈ (SubGrp‘𝑊))
51		eqid 2734	. . . . . . . 8 ⊢ (norm‘(𝑊 ↾s 𝑆)) = (norm‘(𝑊 ↾s 𝑆))
52	20, 24, 51	subgnm2 24576	. . . . . . 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 480	. . . . . . . 8 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑊 ↾s 𝑆) = (Scalar‘𝐴))
55	54	fveq2d 6836	. . . . . . 7 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (norm‘(𝑊 ↾s 𝑆)) = (norm‘(Scalar‘𝐴)))
56	55	fveq1d 6834	. . . . . 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 2771	. . . . 5 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘𝑊)‘𝑥) = ((norm‘(Scalar‘𝐴))‘𝑥))
58	44	fveq1d 6834	. . . . 5 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘𝑊)‘𝑦) = ((norm‘𝐴)‘𝑦))
59	57, 58	oveq12d 7374	. . . 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 2771	. . 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 3177	. 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 2734	. . 3 ⊢ (Base‘𝐴) = (Base‘𝐴)
63		eqid 2734	. . 3 ⊢ (norm‘𝐴) = (norm‘𝐴)
64		eqid 2734	. . 3 ⊢ ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘𝐴)
65		eqid 2734	. . 3 ⊢ (Scalar‘𝐴) = (Scalar‘𝐴)
66		eqid 2734	. . 3 ⊢ (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐴))
67		eqid 2734	. . 3 ⊢ (norm‘(Scalar‘𝐴)) = (norm‘(Scalar‘𝐴))
68	62, 63, 64, 65, 66, 67	isnlm 24617	. 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 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3049   ⊆ wss 3899   × cxp 5620  ‘cfv 6490  (class class class)co 7356   · cmul 11029  Basecbs 17134   ↾_s cress 17155  +_gcplusg 17175  ._rcmulr 17176  Scalarcsca 17178   ·_𝑠 cvsca 17179  distcds 17184  SubGrpcsubg 19048  SubRingcsubrg 20500  AbsValcabv 20739  LModclmod 20809  subringAlg csra 21121  normcnm 24518  NrmGrpcngp 24519  NrmRingcnrg 24521  NrmModcnlm 24522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101  ax-pre-sup 11102

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8633  df-map 8763  df-en 8882  df-dom 8883  df-sdom 8884  df-sup 9343  df-inf 9344  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-div 11793  df-nn 12144  df-2 12206  df-3 12207  df-4 12208  df-5 12209  df-6 12210  df-7 12211  df-8 12212  df-9 12213  df-n0 12400  df-z 12487  df-dec 12606  df-uz 12750  df-q 12860  df-rp 12904  df-xneg 13024  df-xadd 13025  df-xmul 13026  df-ico 13265  df-sets 17089  df-slot 17107  df-ndx 17119  df-base 17135  df-ress 17156  df-plusg 17188  df-mulr 17189  df-sca 17191  df-vsca 17192  df-ip 17193  df-tset 17194  df-ds 17197  df-rest 17340  df-topn 17341  df-0g 17359  df-topgen 17361  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-grp 18864  df-minusg 18865  df-sbg 18866  df-subg 19051  df-cmn 19709  df-abl 19710  df-mgp 20074  df-rng 20086  df-ur 20115  df-ring 20168  df-subrng 20477  df-subrg 20501  df-abv 20740  df-lmod 20811  df-sra 21123  df-psmet 21299  df-xmet 21300  df-met 21301  df-bl 21302  df-mopn 21303  df-top 22836  df-topon 22853  df-topsp 22875  df-bases 22888  df-xms 24262  df-ms 24263  df-nm 24524  df-ngp 24525  df-nrg 24527  df-nlm 24528

This theorem is referenced by:  rlmnlm  24630  srabn  25314