Theorem sranlm 24705

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 24683	. . . . 5 ⊢ (𝑊 ∈ NrmRing → 𝑊 ∈ NrmGrp)
2	1	adantr 480	. . . 4 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑊 ∈ NrmGrp)
3		eqidd 2738	. . . . 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 2737	. . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊)
7	6	subrgss 20572	. . . . . . 7 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
8	7	adantl 481	. . . . . 6 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 ⊆ (Base‘𝑊))
9	5, 8	srabase 21177	. . . . 5 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘𝑊) = (Base‘𝐴))
10	5, 8	sraaddg 21179	. . . . . 6 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (+g‘𝑊) = (+g‘𝐴))
11	10	oveqdr 7459	. . . . 5 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g‘𝑊)𝑦) = (𝑥(+g‘𝐴)𝑦))
12	5, 8	srads 21191	. . . . . 6 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (dist‘𝑊) = (dist‘𝐴))
13	12	reseq1d 5996	. . . . 5 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) = ((dist‘𝐴) ↾ ((Base‘𝑊) × (Base‘𝑊))))
14	5, 8	sratopn 21190	. . . . 5 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (TopOpen‘𝑊) = (TopOpen‘𝐴))
15	3, 9, 11, 13, 14	ngppropd 24650	. . . 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 21194	. . . 4 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)
18	17	adantl 481	. . 3 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ LMod)
19	5, 8	srasca 21183	. . . 4 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊 ↾s 𝑆) = (Scalar‘𝐴))
20		eqid 2737	. . . . 5 ⊢ (𝑊 ↾s 𝑆) = (𝑊 ↾s 𝑆)
21	20	subrgnrg 24694	. . . 4 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊 ↾s 𝑆) ∈ NrmRing)
22	19, 21	eqeltrrd 2842	. . 3 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Scalar‘𝐴) ∈ NrmRing)
23	16, 18, 22	3jca 1129	. 2 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝐴 ∈ NrmGrp ∧ 𝐴 ∈ LMod ∧ (Scalar‘𝐴) ∈ NrmRing))
24		eqid 2737	. . . . . . . 8 ⊢ (norm‘𝑊) = (norm‘𝑊)
25		eqid 2737	. . . . . . . 8 ⊢ (AbsVal‘𝑊) = (AbsVal‘𝑊)
26	24, 25	nrgabv 24682	. . . . . . 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 771	. . . . . . . 8 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘(Scalar‘𝐴)))
30	20	subrgbas 20581	. . . . . . . . . . 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 6910	. . . . . . . . . 10 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘(𝑊 ↾s 𝑆)) = (Base‘(Scalar‘𝐴)))
33	31, 32	eqtrd 2777	. . . . . . . . 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 2844	. . . . . . 7 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ 𝑆)
36	28, 35	sseldd 3984	. . . . . 6 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘𝑊))
37		simprr 773	. . . . . . 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 2844	. . . . . 6 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑦 ∈ (Base‘𝑊))
40		eqid 2737	. . . . . . 7 ⊢ (.r‘𝑊) = (.r‘𝑊)
41	25, 6, 40	abvmul 20822	. . . . . 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 24607	. . . . . . 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 21185	. . . . . . 7 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (.r‘𝑊) = ( ·𝑠 ‘𝐴))
46	45	oveqdr 7459	. . . . . 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 6913	. . . . 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 2779	. . . 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 20577	. . . . . . . 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 2737	. . . . . . . 8 ⊢ (norm‘(𝑊 ↾s 𝑆)) = (norm‘(𝑊 ↾s 𝑆))
52	20, 24, 51	subgnm2 24647	. . . . . . 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 6910	. . . . . . 7 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (norm‘(𝑊 ↾s 𝑆)) = (norm‘(Scalar‘𝐴)))
56	55	fveq1d 6908	. . . . . 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 2779	. . . . 5 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘𝑊)‘𝑥) = ((norm‘(Scalar‘𝐴))‘𝑥))
58	44	fveq1d 6908	. . . . 5 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘𝑊)‘𝑦) = ((norm‘𝐴)‘𝑦))
59	57, 58	oveq12d 7449	. . . 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 2779	. . 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 3202	. 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 2737	. . 3 ⊢ (Base‘𝐴) = (Base‘𝐴)
63		eqid 2737	. . 3 ⊢ (norm‘𝐴) = (norm‘𝐴)
64		eqid 2737	. . 3 ⊢ ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘𝐴)
65		eqid 2737	. . 3 ⊢ (Scalar‘𝐴) = (Scalar‘𝐴)
66		eqid 2737	. . 3 ⊢ (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐴))
67		eqid 2737	. . 3 ⊢ (norm‘(Scalar‘𝐴)) = (norm‘(Scalar‘𝐴))
68	62, 63, 64, 65, 66, 67	isnlm 24696	. 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 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061   ⊆ wss 3951   × cxp 5683  ‘cfv 6561  (class class class)co 7431   · cmul 11160  Basecbs 17247   ↾_s cress 17274  +_gcplusg 17297  ._rcmulr 17298  Scalarcsca 17300   ·_𝑠 cvsca 17301  distcds 17306  SubGrpcsubg 19138  SubRingcsubrg 20569  AbsValcabv 20809  LModclmod 20858  subringAlg csra 21170  normcnm 24589  NrmGrpcngp 24590  NrmRingcnrg 24592  NrmModcnlm 24593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ico 13393  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ds 17319  df-rest 17467  df-topn 17468  df-0g 17486  df-topgen 17488  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-sbg 18956  df-subg 19141  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-subrng 20546  df-subrg 20570  df-abv 20810  df-lmod 20860  df-sra 21172  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-xms 24330  df-ms 24331  df-nm 24595  df-ngp 24596  df-nrg 24598  df-nlm 24599

This theorem is referenced by:  rlmnlm  24709  srabn  25394