Theorem sranlm 24894

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 24872	. . . . 5 ⊢ (𝑊 ∈ NrmRing → 𝑊 ∈ NrmGrp)
2	1	adantr 486	. . . 4 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑊 ∈ NrmGrp)
3		eqidd 2766	. . . . 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 2765	. . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊)
7	6	subrgss 20723	. . . . . . 7 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
8	7	adantl 487	. . . . . 6 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 ⊆ (Base‘𝑊))
9	5, 8	srabase 21350	. . . . 5 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘𝑊) = (Base‘𝐴))
10	5, 8	sraaddg 21351	. . . . . 6 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (+g‘𝑊) = (+g‘𝐴))
11	10	oveqdr 7447	. . . . 5 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g‘𝑊)𝑦) = (𝑥(+g‘𝐴)𝑦))
12	5, 8	srads 21358	. . . . . 6 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (dist‘𝑊) = (dist‘𝐴))
13	12	reseq1d 5979	. . . . 5 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) = ((dist‘𝐴) ↾ ((Base‘𝑊) × (Base‘𝑊))))
14	5, 8	sratopn 21357	. . . . 5 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (TopOpen‘𝑊) = (TopOpen‘𝐴))
15	3, 9, 11, 13, 14	ngppropd 24847	. . . 4 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊 ∈ NrmGrp ↔ 𝐴 ∈ NrmGrp))
16	2, 15	mpbid 235	. . 3 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ NrmGrp)
17	4	sralmod 21360	. . . 4 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)
18	17	adantl 487	. . 3 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ LMod)
19	5, 8	srasca 21353	. . . 4 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊 ↾s 𝑆) = (Scalar‘𝐴))
20		eqid 2765	. . . . 5 ⊢ (𝑊 ↾s 𝑆) = (𝑊 ↾s 𝑆)
21	20	subrgnrg 24883	. . . 4 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊 ↾s 𝑆) ∈ NrmRing)
22	19, 21	eqeltrrd 2866	. . 3 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Scalar‘𝐴) ∈ NrmRing)
23	16, 18, 22	3jca 1146	. 2 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝐴 ∈ NrmGrp ∧ 𝐴 ∈ LMod ∧ (Scalar‘𝐴) ∈ NrmRing))
24		eqid 2765	. . . . . . . 8 ⊢ (norm‘𝑊) = (norm‘𝑊)
25		eqid 2765	. . . . . . . 8 ⊢ (AbsVal‘𝑊) = (AbsVal‘𝑊)
26	24, 25	nrgabv 24871	. . . . . . 7 ⊢ (𝑊 ∈ NrmRing → (norm‘𝑊) ∈ (AbsVal‘𝑊))
27	26	ad2antrr 739	. . . . . 6 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (norm‘𝑊) ∈ (AbsVal‘𝑊))
28	8	adantr 486	. . . . . . 7 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑆 ⊆ (Base‘𝑊))
29		simprl 783	. . . . . . . 8 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘(Scalar‘𝐴)))
30	20	subrgbas 20732	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 = (Base‘(𝑊 ↾s 𝑆)))
31	30	adantl 487	. . . . . . . . . 10 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 = (Base‘(𝑊 ↾s 𝑆)))
32	19	fveq2d 6889	. . . . . . . . . 10 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘(𝑊 ↾s 𝑆)) = (Base‘(Scalar‘𝐴)))
33	31, 32	eqtrd 2800	. . . . . . . . 9 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 = (Base‘(Scalar‘𝐴)))
34	33	adantr 486	. . . . . . . 8 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑆 = (Base‘(Scalar‘𝐴)))
35	29, 34	eleqtrrd 2868	. . . . . . 7 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ 𝑆)
36	28, 35	sseldd 3939	. . . . . 6 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘𝑊))
37		simprr 785	. . . . . . 7 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑦 ∈ (Base‘𝐴))
38	9	adantr 486	. . . . . . 7 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (Base‘𝑊) = (Base‘𝐴))
39	37, 38	eleqtrrd 2868	. . . . . 6 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑦 ∈ (Base‘𝑊))
40		eqid 2765	. . . . . . 7 ⊢ (.r‘𝑊) = (.r‘𝑊)
41	25, 6, 40	abvmul 20976	. . . . . 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 1398	. . . . 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 24804	. . . . . . 7 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (norm‘𝑊) = (norm‘𝐴))
44	43	adantr 486	. . . . . 6 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (norm‘𝑊) = (norm‘𝐴))
45	5, 8	sravsca 21354	. . . . . . 7 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (.r‘𝑊) = ( ·𝑠 ‘𝐴))
46	45	oveqdr 7447	. . . . . 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 6892	. . . . 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 2802	. . . 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 20728	. . . . . . . 8 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ∈ (SubGrp‘𝑊))
50	49	ad2antlr 740	. . . . . . 7 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑆 ∈ (SubGrp‘𝑊))
51		eqid 2765	. . . . . . . 8 ⊢ (norm‘(𝑊 ↾s 𝑆)) = (norm‘(𝑊 ↾s 𝑆))
52	20, 24, 51	subgnm2 24844	. . . . . . 7 ⊢ ((𝑆 ∈ (SubGrp‘𝑊) ∧ 𝑥 ∈ 𝑆) → ((norm‘(𝑊 ↾s 𝑆))‘𝑥) = ((norm‘𝑊)‘𝑥))
53	50, 35, 52	syl2anc 596	. . . . . 6 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘(𝑊 ↾s 𝑆))‘𝑥) = ((norm‘𝑊)‘𝑥))
54	19	adantr 486	. . . . . . . 8 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑊 ↾s 𝑆) = (Scalar‘𝐴))
55	54	fveq2d 6889	. . . . . . 7 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (norm‘(𝑊 ↾s 𝑆)) = (norm‘(Scalar‘𝐴)))
56	55	fveq1d 6887	. . . . . 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 2802	. . . . 5 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘𝑊)‘𝑥) = ((norm‘(Scalar‘𝐴))‘𝑥))
58	44	fveq1d 6887	. . . . 5 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘𝑊)‘𝑦) = ((norm‘𝐴)‘𝑦))
59	57, 58	oveq12d 7437	. . . 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 2802	. . 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 3210	. 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 2765	. . 3 ⊢ (Base‘𝐴) = (Base‘𝐴)
63		eqid 2765	. . 3 ⊢ (norm‘𝐴) = (norm‘𝐴)
64		eqid 2765	. . 3 ⊢ ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘𝐴)
65		eqid 2765	. . 3 ⊢ (Scalar‘𝐴) = (Scalar‘𝐴)
66		eqid 2765	. . 3 ⊢ (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐴))
67		eqid 2765	. . 3 ⊢ (norm‘(Scalar‘𝐴)) = (norm‘(Scalar‘𝐴))
68	62, 63, 64, 65, 66, 67	isnlm 24885	. 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 595	1 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ NrmMod)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081   ⊆ wss 3906   × cxp 5661  ‘cfv 6540  (class class class)co 7419   · cmul 11122  Basecbs 17293   ↾_s cress 17314  +_gcplusg 17334  ._rcmulr 17335  Scalarcsca 17337   ·_𝑠 cvsca 17338  distcds 17343  SubGrpcsubg 19232  SubRingcsubrg 20720  AbsValcabv 20963  LModclmod 21033  subringAlg csra 21344  normcnm 24786  NrmGrpcngp 24787  NrmRingcnrg 24789  NrmModcnlm 24790

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11173  ax-resscn 11174  ax-1cn 11175  ax-icn 11176  ax-addcl 11177  ax-addrcl 11178  ax-mulcl 11179  ax-mulrcl 11180  ax-mulcom 11181  ax-addass 11182  ax-mulass 11183  ax-distr 11184  ax-i2m1 11185  ax-1ne0 11186  ax-1rid 11187  ax-rnegex 11188  ax-rrecex 11189  ax-cnre 11190  ax-pre-lttri 11191  ax-pre-lttrn 11192  ax-pre-ltadd 11193  ax-pre-mulgt0 11194  ax-pre-sup 11195

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-map 8832  df-en 8950  df-dom 8951  df-sdom 8952  df-sup 9409  df-inf 9410  df-pnf 11262  df-mnf 11263  df-xr 11264  df-ltxr 11265  df-le 11266  df-sub 11460  df-neg 11461  df-div 11889  df-nn 12251  df-2 12320  df-3 12321  df-4 12322  df-5 12323  df-6 12324  df-7 12325  df-8 12326  df-9 12327  df-n0 12522  df-z 12609  df-dec 12730  df-uz 12881  df-q 12991  df-rp 13035  df-xneg 13155  df-xadd 13156  df-xmul 13157  df-ico 13396  df-sets 17248  df-slot 17266  df-ndx 17278  df-base 17294  df-ress 17315  df-plusg 17347  df-mulr 17348  df-sca 17350  df-vsca 17351  df-ip 17352  df-tset 17353  df-ds 17356  df-rest 17499  df-topn 17500  df-0g 17518  df-topgen 17520  df-mgm 18722  df-sgrp 18811  df-mnd 18827  df-grp 19049  df-minusg 19050  df-sbg 19051  df-subg 19235  df-cmn 19898  df-abl 19899  df-mgp 20263  df-rng 20277  df-ur 20310  df-ring 20363  df-subrng 20697  df-subrg 20721  df-abv 20964  df-lmod 21035  df-sra 21346  df-psmet 21566  df-xmet 21567  df-met 21568  df-bl 21569  df-mopn 21570  df-top 23103  df-topon 23120  df-topsp 23142  df-bases 23155  df-xms 24530  df-ms 24531  df-nm 24792  df-ngp 24793  df-nrg 24795  df-nlm 24796

This theorem is used by:  rlmnlm  24898  srabn  25572