Theorem sranlm 24201

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 24179	. . . . 5 ⊢ (𝑊 ∈ NrmRing → 𝑊 ∈ NrmGrp)
2	1	adantr 482	. . . 4 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑊 ∈ NrmGrp)
3		eqidd 2734	. . . . 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 2733	. . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊)
7	6	subrgss 20320	. . . . . . 7 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
8	7	adantl 483	. . . . . 6 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 ⊆ (Base‘𝑊))
9	5, 8	srabase 20792	. . . . 5 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘𝑊) = (Base‘𝐴))
10	5, 8	sraaddg 20794	. . . . . 6 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (+g‘𝑊) = (+g‘𝐴))
11	10	oveqdr 7437	. . . . 5 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g‘𝑊)𝑦) = (𝑥(+g‘𝐴)𝑦))
12	5, 8	srads 20806	. . . . . 6 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (dist‘𝑊) = (dist‘𝐴))
13	12	reseq1d 5981	. . . . 5 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → ((dist‘𝑊) ↾ ((Base‘𝑊) × (Base‘𝑊))) = ((dist‘𝐴) ↾ ((Base‘𝑊) × (Base‘𝑊))))
14	5, 8	sratopn 20805	. . . . 5 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (TopOpen‘𝑊) = (TopOpen‘𝐴))
15	3, 9, 11, 13, 14	ngppropd 24146	. . . 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 20809	. . . 4 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)
18	17	adantl 483	. . 3 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ LMod)
19	5, 8	srasca 20798	. . . 4 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊 ↾s 𝑆) = (Scalar‘𝐴))
20		eqid 2733	. . . . 5 ⊢ (𝑊 ↾s 𝑆) = (𝑊 ↾s 𝑆)
21	20	subrgnrg 24190	. . . 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 1129	. 2 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝐴 ∈ NrmGrp ∧ 𝐴 ∈ LMod ∧ (Scalar‘𝐴) ∈ NrmRing))
24		eqid 2733	. . . . . . . 8 ⊢ (norm‘𝑊) = (norm‘𝑊)
25		eqid 2733	. . . . . . . 8 ⊢ (AbsVal‘𝑊) = (AbsVal‘𝑊)
26	24, 25	nrgabv 24178	. . . . . . 7 ⊢ (𝑊 ∈ NrmRing → (norm‘𝑊) ∈ (AbsVal‘𝑊))
27	26	ad2antrr 725	. . . . . 6 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (norm‘𝑊) ∈ (AbsVal‘𝑊))
28	8	adantr 482	. . . . . . 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 20328	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 = (Base‘(𝑊 ↾s 𝑆)))
31	30	adantl 483	. . . . . . . . . 10 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 = (Base‘(𝑊 ↾s 𝑆)))
32	19	fveq2d 6896	. . . . . . . . . 10 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘(𝑊 ↾s 𝑆)) = (Base‘(Scalar‘𝐴)))
33	31, 32	eqtrd 2773	. . . . . . . . 9 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 = (Base‘(Scalar‘𝐴)))
34	33	adantr 482	. . . . . . . 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 3984	. . . . . 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 482	. . . . . . 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 2733	. . . . . . 7 ⊢ (.r‘𝑊) = (.r‘𝑊)
41	25, 6, 40	abvmul 20437	. . . . . 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 1372	. . . . 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 24103	. . . . . . 7 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (norm‘𝑊) = (norm‘𝐴))
44	43	adantr 482	. . . . . 6 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (norm‘𝑊) = (norm‘𝐴))
45	5, 8	sravsca 20800	. . . . . . 7 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (.r‘𝑊) = ( ·𝑠 ‘𝐴))
46	45	oveqdr 7437	. . . . . 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 6899	. . . . 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 2775	. . . 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 20325	. . . . . . . 8 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ∈ (SubGrp‘𝑊))
50	49	ad2antlr 726	. . . . . . 7 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑆 ∈ (SubGrp‘𝑊))
51		eqid 2733	. . . . . . . 8 ⊢ (norm‘(𝑊 ↾s 𝑆)) = (norm‘(𝑊 ↾s 𝑆))
52	20, 24, 51	subgnm2 24143	. . . . . . 7 ⊢ ((𝑆 ∈ (SubGrp‘𝑊) ∧ 𝑥 ∈ 𝑆) → ((norm‘(𝑊 ↾s 𝑆))‘𝑥) = ((norm‘𝑊)‘𝑥))
53	50, 35, 52	syl2anc 585	. . . . . 6 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘(𝑊 ↾s 𝑆))‘𝑥) = ((norm‘𝑊)‘𝑥))
54	19	adantr 482	. . . . . . . 8 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑊 ↾s 𝑆) = (Scalar‘𝐴))
55	54	fveq2d 6896	. . . . . . 7 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → (norm‘(𝑊 ↾s 𝑆)) = (norm‘(Scalar‘𝐴)))
56	55	fveq1d 6894	. . . . . 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 2775	. . . . 5 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘𝑊)‘𝑥) = ((norm‘(Scalar‘𝐴))‘𝑥))
58	44	fveq1d 6894	. . . . 5 ⊢ (((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴))) → ((norm‘𝑊)‘𝑦) = ((norm‘𝐴)‘𝑦))
59	57, 58	oveq12d 7427	. . . 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 2775	. . 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 3201	. 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 2733	. . 3 ⊢ (Base‘𝐴) = (Base‘𝐴)
63		eqid 2733	. . 3 ⊢ (norm‘𝐴) = (norm‘𝐴)
64		eqid 2733	. . 3 ⊢ ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘𝐴)
65		eqid 2733	. . 3 ⊢ (Scalar‘𝐴) = (Scalar‘𝐴)
66		eqid 2733	. . 3 ⊢ (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐴))
67		eqid 2733	. . 3 ⊢ (norm‘(Scalar‘𝐴)) = (norm‘(Scalar‘𝐴))
68	62, 63, 64, 65, 66, 67	isnlm 24192	. 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 584	1 ⊢ ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ NrmMod)

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∀wral 3062   ⊆ wss 3949   × cxp 5675  ‘cfv 6544  (class class class)co 7409   · cmul 11115  Basecbs 17144   ↾_s cress 17173  +_gcplusg 17197  ._rcmulr 17198  Scalarcsca 17200   ·_𝑠 cvsca 17201  distcds 17206  SubGrpcsubg 19000  SubRingcsubrg 20315  AbsValcabv 20424  LModclmod 20471  subringAlg csra 20781  normcnm 24085  NrmGrpcngp 24086  NrmRingcnrg 24088  NrmModcnlm 24089

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-sup 9437  df-inf 9438  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-q 12933  df-rp 12975  df-xneg 13092  df-xadd 13093  df-xmul 13094  df-ico 13330  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mulr 17211  df-sca 17213  df-vsca 17214  df-ip 17215  df-tset 17216  df-ds 17219  df-rest 17368  df-topn 17369  df-0g 17387  df-topgen 17389  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-sbg 18824  df-subg 19003  df-mgp 19988  df-ur 20005  df-ring 20058  df-subrg 20317  df-abv 20425  df-lmod 20473  df-sra 20785  df-psmet 20936  df-xmet 20937  df-met 20938  df-bl 20939  df-mopn 20940  df-top 22396  df-topon 22413  df-topsp 22435  df-bases 22449  df-xms 23826  df-ms 23827  df-nm 24091  df-ngp 24092  df-nrg 24094  df-nlm 24095

This theorem is referenced by:  rlmnlm  24205  srabn  24877