Theorem sranlm 22488

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 e. NrmRing /\ S e. (SubRing ` W ) ) -> A e. NrmMod )

Proof of Theorem sranlm

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

Step	Hyp	Ref	Expression
1		nrgngp 22466	. . . . 5 |- ( W e. NrmRing -> W e. NrmGrp )
2	1	adantr 481	. . . 4 |- ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) -> W e. NrmGrp )
3		eqidd 2623	. . . . 5 |- ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) -> ( Base ` W ) = ( Base ` W ) )
4		sranlm.a	. . . . . . 7 |- A = ( (subringAlg ` W ) ` S )
5	4	a1i 11	. . . . . 6 |- ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) -> A = ( (subringAlg ` W ) ` S ) )
6		eqid 2622	. . . . . . . 8 |- ( Base ` W ) = ( Base ` W )
7	6	subrgss 18781	. . . . . . 7 |- ( S e. (SubRing ` W ) -> S C_ ( Base ` W ) )
8	7	adantl 482	. . . . . 6 |- ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) -> S C_ ( Base ` W ) )
9	5, 8	srabase 19178	. . . . 5 |- ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) -> ( Base ` W ) = ( Base ` A ) )
10	5, 8	sraaddg 19179	. . . . . 6 |- ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) -> ( +g ` W ) = ( +g ` A ) )
11	10	oveqdr 6674	. . . . 5 |- ( ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) /\ ( x e. ( Base ` W ) /\ y e. ( Base ` W ) ) ) -> ( x ( +g ` W ) y ) = ( x ( +g ` A ) y ) )
12	5, 8	srads 19186	. . . . . 6 |- ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) -> ( dist ` W ) = ( dist ` A ) )
13	12	reseq1d 5395	. . . . 5 |- ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) -> ( ( dist ` W ) |` ( ( Base ` W ) X. ( Base ` W ) ) ) = ( ( dist ` A ) |` ( ( Base ` W ) X. ( Base ` W ) ) ) )
14	5, 8	sratopn 19185	. . . . 5 |- ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) -> ( TopOpen ` W ) = ( TopOpen ` A ) )
15	3, 9, 11, 13, 14	ngppropd 22441	. . . 4 |- ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) -> ( W e. NrmGrp <-> A e. NrmGrp ) )
16	2, 15	mpbid 222	. . 3 |- ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) -> A e. NrmGrp )
17	4	sralmod 19187	. . . 4 |- ( S e. (SubRing ` W ) -> A e. LMod )
18	17	adantl 482	. . 3 |- ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) -> A e. LMod )
19	5, 8	srasca 19181	. . . 4 |- ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) -> ( W ↾s S ) = (Scalar ` A ) )
20		eqid 2622	. . . . 5 |- ( W ↾s S ) = ( W ↾s S )
21	20	subrgnrg 22477	. . . 4 |- ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) -> ( W ↾s S ) e. NrmRing )
22	19, 21	eqeltrrd 2702	. . 3 |- ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) -> (Scalar ` A ) e. NrmRing )
23	16, 18, 22	3jca 1242	. 2 |- ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) -> ( A e. NrmGrp /\ A e. LMod /\ (Scalar ` A ) e. NrmRing ) )
24		eqid 2622	. . . . . . . 8 |- ( norm ` W ) = ( norm ` W )
25		eqid 2622	. . . . . . . 8 |- (AbsVal ` W ) = (AbsVal ` W )
26	24, 25	nrgabv 22465	. . . . . . 7 |- ( W e. NrmRing -> ( norm ` W ) e. (AbsVal ` W ) )
27	26	ad2antrr 762	. . . . . 6 |- ( ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) /\ ( x e. ( Base ` (Scalar ` A ) ) /\ y e. ( Base ` A ) ) ) -> ( norm ` W ) e. (AbsVal ` W ) )
28	8	adantr 481	. . . . . . 7 |- ( ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) /\ ( x e. ( Base ` (Scalar ` A ) ) /\ y e. ( Base ` A ) ) ) -> S C_ ( Base ` W ) )
29		simprl 794	. . . . . . . 8 |- ( ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) /\ ( x e. ( Base ` (Scalar ` A ) ) /\ y e. ( Base ` A ) ) ) -> x e. ( Base ` (Scalar ` A ) ) )
30	20	subrgbas 18789	. . . . . . . . . . 11 |- ( S e. (SubRing ` W ) -> S = ( Base ` ( W ↾s S ) ) )
31	30	adantl 482	. . . . . . . . . 10 |- ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) -> S = ( Base ` ( W ↾s S ) ) )
32	19	fveq2d 6195	. . . . . . . . . 10 |- ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) -> ( Base ` ( W ↾s S ) ) = ( Base ` (Scalar ` A ) ) )
33	31, 32	eqtrd 2656	. . . . . . . . 9 |- ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) -> S = ( Base ` (Scalar ` A ) ) )
34	33	adantr 481	. . . . . . . 8 |- ( ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) /\ ( x e. ( Base ` (Scalar ` A ) ) /\ y e. ( Base ` A ) ) ) -> S = ( Base ` (Scalar ` A ) ) )
35	29, 34	eleqtrrd 2704	. . . . . . 7 |- ( ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) /\ ( x e. ( Base ` (Scalar ` A ) ) /\ y e. ( Base ` A ) ) ) -> x e. S )
36	28, 35	sseldd 3604	. . . . . 6 |- ( ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) /\ ( x e. ( Base ` (Scalar ` A ) ) /\ y e. ( Base ` A ) ) ) -> x e. ( Base ` W ) )
37		simprr 796	. . . . . . 7 |- ( ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) /\ ( x e. ( Base ` (Scalar ` A ) ) /\ y e. ( Base ` A ) ) ) -> y e. ( Base ` A ) )
38	9	adantr 481	. . . . . . 7 |- ( ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) /\ ( x e. ( Base ` (Scalar ` A ) ) /\ y e. ( Base ` A ) ) ) -> ( Base ` W ) = ( Base ` A ) )
39	37, 38	eleqtrrd 2704	. . . . . 6 |- ( ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) /\ ( x e. ( Base ` (Scalar ` A ) ) /\ y e. ( Base ` A ) ) ) -> y e. ( Base ` W ) )
40		eqid 2622	. . . . . . 7 |- ( .r ` W ) = ( .r ` W )
41	25, 6, 40	abvmul 18829	. . . . . 6 |- ( ( ( norm ` W ) e. (AbsVal ` W ) /\ x e. ( Base ` W ) /\ y e. ( Base ` W ) ) -> ( ( norm ` W ) ` ( x ( .r ` W ) y ) ) = ( ( ( norm ` W ) ` x ) x. ( ( norm ` W ) ` y ) ) )
42	27, 36, 39, 41	syl3anc 1326	. . . . 5 |- ( ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) /\ ( x e. ( Base ` (Scalar ` A ) ) /\ y e. ( Base ` A ) ) ) -> ( ( norm ` W ) ` ( x ( .r ` W ) y ) ) = ( ( ( norm ` W ) ` x ) x. ( ( norm ` W ) ` y ) ) )
43	9, 10, 12	nmpropd 22398	. . . . . . 7 |- ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) -> ( norm ` W ) = ( norm ` A ) )
44	43	adantr 481	. . . . . 6 |- ( ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) /\ ( x e. ( Base ` (Scalar ` A ) ) /\ y e. ( Base ` A ) ) ) -> ( norm ` W ) = ( norm ` A ) )
45	5, 8	sravsca 19182	. . . . . . 7 |- ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) -> ( .r ` W ) = ( .s ` A ) )
46	45	oveqdr 6674	. . . . . 6 |- ( ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) /\ ( x e. ( Base ` (Scalar ` A ) ) /\ y e. ( Base ` A ) ) ) -> ( x ( .r ` W ) y ) = ( x ( .s ` A ) y ) )
47	44, 46	fveq12d 6197	. . . . 5 |- ( ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) /\ ( x e. ( Base ` (Scalar ` A ) ) /\ y e. ( Base ` A ) ) ) -> ( ( norm ` W ) ` ( x ( .r ` W ) y ) ) = ( ( norm ` A ) ` ( x ( .s ` A ) y ) ) )
48	42, 47	eqtr3d 2658	. . . 4 |- ( ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) /\ ( x e. ( Base ` (Scalar ` A ) ) /\ y e. ( Base ` A ) ) ) -> ( ( ( norm ` W ) ` x ) x. ( ( norm ` W ) ` y ) ) = ( ( norm ` A ) ` ( x ( .s ` A ) y ) ) )
49		subrgsubg 18786	. . . . . . . 8 |- ( S e. (SubRing ` W ) -> S e. (SubGrp ` W ) )
50	49	ad2antlr 763	. . . . . . 7 |- ( ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) /\ ( x e. ( Base ` (Scalar ` A ) ) /\ y e. ( Base ` A ) ) ) -> S e. (SubGrp ` W ) )
51		eqid 2622	. . . . . . . 8 |- ( norm ` ( W ↾s S ) ) = ( norm ` ( W ↾s S ) )
52	20, 24, 51	subgnm2 22438	. . . . . . 7 |- ( ( S e. (SubGrp ` W ) /\ x e. S ) -> ( ( norm ` ( W ↾s S ) ) ` x ) = ( ( norm ` W ) ` x ) )
53	50, 35, 52	syl2anc 693	. . . . . 6 |- ( ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) /\ ( x e. ( Base ` (Scalar ` A ) ) /\ y e. ( Base ` A ) ) ) -> ( ( norm ` ( W ↾s S ) ) ` x ) = ( ( norm ` W ) ` x ) )
54	19	adantr 481	. . . . . . . 8 |- ( ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) /\ ( x e. ( Base ` (Scalar ` A ) ) /\ y e. ( Base ` A ) ) ) -> ( W ↾s S ) = (Scalar ` A ) )
55	54	fveq2d 6195	. . . . . . 7 |- ( ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) /\ ( x e. ( Base ` (Scalar ` A ) ) /\ y e. ( Base ` A ) ) ) -> ( norm ` ( W ↾s S ) ) = ( norm ` (Scalar ` A ) ) )
56	55	fveq1d 6193	. . . . . 6 |- ( ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) /\ ( x e. ( Base ` (Scalar ` A ) ) /\ y e. ( Base ` A ) ) ) -> ( ( norm ` ( W ↾s S ) ) ` x ) = ( ( norm ` (Scalar ` A ) ) ` x ) )
57	53, 56	eqtr3d 2658	. . . . 5 |- ( ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) /\ ( x e. ( Base ` (Scalar ` A ) ) /\ y e. ( Base ` A ) ) ) -> ( ( norm ` W ) ` x ) = ( ( norm ` (Scalar ` A ) ) ` x ) )
58	44	fveq1d 6193	. . . . 5 |- ( ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) /\ ( x e. ( Base ` (Scalar ` A ) ) /\ y e. ( Base ` A ) ) ) -> ( ( norm ` W ) ` y ) = ( ( norm ` A ) ` y ) )
59	57, 58	oveq12d 6668	. . . 4 |- ( ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) /\ ( x e. ( Base ` (Scalar ` A ) ) /\ y e. ( Base ` A ) ) ) -> ( ( ( norm ` W ) ` x ) x. ( ( norm ` W ) ` y ) ) = ( ( ( norm ` (Scalar ` A ) ) ` x ) x. ( ( norm ` A ) ` y ) ) )
60	48, 59	eqtr3d 2658	. . 3 |- ( ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) /\ ( x e. ( Base ` (Scalar ` A ) ) /\ y e. ( Base ` A ) ) ) -> ( ( norm ` A ) ` ( x ( .s ` A ) y ) ) = ( ( ( norm ` (Scalar ` A ) ) ` x ) x. ( ( norm ` A ) ` y ) ) )
61	60	ralrimivva 2971	. 2 |- ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) -> A. x e. ( Base ` (Scalar ` A ) ) A. y e. ( Base ` A ) ( ( norm ` A ) ` ( x ( .s ` A ) y ) ) = ( ( ( norm ` (Scalar ` A ) ) ` x ) x. ( ( norm ` A ) ` y ) ) )
62		eqid 2622	. . 3 |- ( Base ` A ) = ( Base ` A )
63		eqid 2622	. . 3 |- ( norm ` A ) = ( norm ` A )
64		eqid 2622	. . 3 |- ( .s ` A ) = ( .s ` A )
65		eqid 2622	. . 3 |- (Scalar ` A ) = (Scalar ` A )
66		eqid 2622	. . 3 |- ( Base ` (Scalar ` A ) ) = ( Base ` (Scalar ` A ) )
67		eqid 2622	. . 3 |- ( norm ` (Scalar ` A ) ) = ( norm ` (Scalar ` A ) )
68	62, 63, 64, 65, 66, 67	isnlm 22479	. 2 |- ( A e. NrmMod <-> ( ( A e. NrmGrp /\ A e. LMod /\ (Scalar ` A ) e. NrmRing ) /\ A. x e. ( Base ` (Scalar ` A ) ) A. y e. ( Base ` A ) ( ( norm ` A ) ` ( x ( .s ` A ) y ) ) = ( ( ( norm ` (Scalar ` A ) ) ` x ) x. ( ( norm ` A ) ` y ) ) ) )
69	23, 61, 68	sylanbrc 698	1 |- ( ( W e. NrmRing /\ S e. (SubRing ` W ) ) -> A e. NrmMod )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   X. cxp 5112   ` cfv 5888  (class class class)co 6650   x. cmul 9941   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   distcds 15950  SubGrpcsubg 17588  SubRingcsubrg 18776  AbsValcabv 18816   LModclmod 18863  subringAlg csra 19168   normcnm 22381  NrmGrpcngp 22382  NrmRingcnrg 22384  NrmModcnlm 22385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ico 12181  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ds 15964  df-rest 16083  df-topn 16084  df-0g 16102  df-topgen 16104  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-mgp 18490  df-ur 18502  df-ring 18549  df-subrg 18778  df-abv 18817  df-lmod 18865  df-sra 19172  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-xms 22125  df-ms 22126  df-nm 22387  df-ngp 22388  df-nrg 22390  df-nlm 22391

This theorem is referenced by:  rlmnlm  22492  srabn  23156