Theorem lssnlm 22505

Description: A subspace of a normed module is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)

Hypotheses

Ref	Expression
lssnlm.x	|- X = ( W ↾s U )
lssnlm.s	|- S = ( LSubSp ` W )

Assertion

Ref	Expression
lssnlm	|- ( ( W e. NrmMod /\ U e. S ) -> X e. NrmMod )

Proof of Theorem lssnlm

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

Step	Hyp	Ref	Expression
1		nlmngp 22481	. . . . 5 |- ( W e. NrmMod -> W e. NrmGrp )
2	1	adantr 481	. . . 4 |- ( ( W e. NrmMod /\ U e. S ) -> W e. NrmGrp )
3		nlmlmod 22482	. . . . 5 |- ( W e. NrmMod -> W e. LMod )
4		lssnlm.s	. . . . . 6 |- S = ( LSubSp ` W )
5	4	lsssubg 18957	. . . . 5 |- ( ( W e. LMod /\ U e. S ) -> U e. (SubGrp ` W ) )
6	3, 5	sylan 488	. . . 4 |- ( ( W e. NrmMod /\ U e. S ) -> U e. (SubGrp ` W ) )
7		lssnlm.x	. . . . 5 |- X = ( W ↾s U )
8	7	subgngp 22439	. . . 4 |- ( ( W e. NrmGrp /\ U e. (SubGrp ` W ) ) -> X e. NrmGrp )
9	2, 6, 8	syl2anc 693	. . 3 |- ( ( W e. NrmMod /\ U e. S ) -> X e. NrmGrp )
10	7, 4	lsslmod 18960	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> X e. LMod )
11	3, 10	sylan 488	. . 3 |- ( ( W e. NrmMod /\ U e. S ) -> X e. LMod )
12		eqid 2622	. . . . . 6 |- (Scalar ` W ) = (Scalar ` W )
13	7, 12	resssca 16031	. . . . 5 |- ( U e. S -> (Scalar ` W ) = (Scalar ` X ) )
14	13	adantl 482	. . . 4 |- ( ( W e. NrmMod /\ U e. S ) -> (Scalar ` W ) = (Scalar ` X ) )
15	12	nlmnrg 22483	. . . . 5 |- ( W e. NrmMod -> (Scalar ` W ) e. NrmRing )
16	15	adantr 481	. . . 4 |- ( ( W e. NrmMod /\ U e. S ) -> (Scalar ` W ) e. NrmRing )
17	14, 16	eqeltrrd 2702	. . 3 |- ( ( W e. NrmMod /\ U e. S ) -> (Scalar ` X ) e. NrmRing )
18	9, 11, 17	3jca 1242	. 2 |- ( ( W e. NrmMod /\ U e. S ) -> ( X e. NrmGrp /\ X e. LMod /\ (Scalar ` X ) e. NrmRing ) )
19		simpll 790	. . . . 5 |- ( ( ( W e. NrmMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ y e. ( Base ` X ) ) ) -> W e. NrmMod )
20		simprl 794	. . . . . 6 |- ( ( ( W e. NrmMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ y e. ( Base ` X ) ) ) -> x e. ( Base ` (Scalar ` X ) ) )
21	14	adantr 481	. . . . . . 7 |- ( ( ( W e. NrmMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ y e. ( Base ` X ) ) ) -> (Scalar ` W ) = (Scalar ` X ) )
22	21	fveq2d 6195	. . . . . 6 |- ( ( ( W e. NrmMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ y e. ( Base ` X ) ) ) -> ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` X ) ) )
23	20, 22	eleqtrrd 2704	. . . . 5 |- ( ( ( W e. NrmMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ y e. ( Base ` X ) ) ) -> x e. ( Base ` (Scalar ` W ) ) )
24	6	adantr 481	. . . . . . 7 |- ( ( ( W e. NrmMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ y e. ( Base ` X ) ) ) -> U e. (SubGrp ` W ) )
25		eqid 2622	. . . . . . . 8 |- ( Base ` W ) = ( Base ` W )
26	25	subgss 17595	. . . . . . 7 |- ( U e. (SubGrp ` W ) -> U C_ ( Base ` W ) )
27	24, 26	syl 17	. . . . . 6 |- ( ( ( W e. NrmMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ y e. ( Base ` X ) ) ) -> U C_ ( Base ` W ) )
28		simprr 796	. . . . . . 7 |- ( ( ( W e. NrmMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ y e. ( Base ` X ) ) ) -> y e. ( Base ` X ) )
29	7	subgbas 17598	. . . . . . . 8 |- ( U e. (SubGrp ` W ) -> U = ( Base ` X ) )
30	24, 29	syl 17	. . . . . . 7 |- ( ( ( W e. NrmMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ y e. ( Base ` X ) ) ) -> U = ( Base ` X ) )
31	28, 30	eleqtrrd 2704	. . . . . 6 |- ( ( ( W e. NrmMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ y e. ( Base ` X ) ) ) -> y e. U )
32	27, 31	sseldd 3604	. . . . 5 |- ( ( ( W e. NrmMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ y e. ( Base ` X ) ) ) -> y e. ( Base ` W ) )
33		eqid 2622	. . . . . 6 |- ( norm ` W ) = ( norm ` W )
34		eqid 2622	. . . . . 6 |- ( .s ` W ) = ( .s ` W )
35		eqid 2622	. . . . . 6 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
36		eqid 2622	. . . . . 6 |- ( norm ` (Scalar ` W ) ) = ( norm ` (Scalar ` W ) )
37	25, 33, 34, 12, 35, 36	nmvs 22480	. . . . 5 |- ( ( W e. NrmMod /\ x e. ( Base ` (Scalar ` W ) ) /\ y e. ( Base ` W ) ) -> ( ( norm ` W ) ` ( x ( .s ` W ) y ) ) = ( ( ( norm ` (Scalar ` W ) ) ` x ) x. ( ( norm ` W ) ` y ) ) )
38	19, 23, 32, 37	syl3anc 1326	. . . 4 |- ( ( ( W e. NrmMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ y e. ( Base ` X ) ) ) -> ( ( norm ` W ) ` ( x ( .s ` W ) y ) ) = ( ( ( norm ` (Scalar ` W ) ) ` x ) x. ( ( norm ` W ) ` y ) ) )
39		simplr 792	. . . . . . . 8 |- ( ( ( W e. NrmMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ y e. ( Base ` X ) ) ) -> U e. S )
40	7, 34	ressvsca 16032	. . . . . . . 8 |- ( U e. S -> ( .s ` W ) = ( .s ` X ) )
41	39, 40	syl 17	. . . . . . 7 |- ( ( ( W e. NrmMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ y e. ( Base ` X ) ) ) -> ( .s ` W ) = ( .s ` X ) )
42	41	oveqd 6667	. . . . . 6 |- ( ( ( W e. NrmMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ y e. ( Base ` X ) ) ) -> ( x ( .s ` W ) y ) = ( x ( .s ` X ) y ) )
43	42	fveq2d 6195	. . . . 5 |- ( ( ( W e. NrmMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ y e. ( Base ` X ) ) ) -> ( ( norm ` X ) ` ( x ( .s ` W ) y ) ) = ( ( norm ` X ) ` ( x ( .s ` X ) y ) ) )
44	3	ad2antrr 762	. . . . . . 7 |- ( ( ( W e. NrmMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ y e. ( Base ` X ) ) ) -> W e. LMod )
45	12, 34, 35, 4	lssvscl 18955	. . . . . . 7 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. U ) ) -> ( x ( .s ` W ) y ) e. U )
46	44, 39, 23, 31, 45	syl22anc 1327	. . . . . 6 |- ( ( ( W e. NrmMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ y e. ( Base ` X ) ) ) -> ( x ( .s ` W ) y ) e. U )
47		eqid 2622	. . . . . . 7 |- ( norm ` X ) = ( norm ` X )
48	7, 33, 47	subgnm2 22438	. . . . . 6 |- ( ( U e. (SubGrp ` W ) /\ ( x ( .s ` W ) y ) e. U ) -> ( ( norm ` X ) ` ( x ( .s ` W ) y ) ) = ( ( norm ` W ) ` ( x ( .s ` W ) y ) ) )
49	24, 46, 48	syl2anc 693	. . . . 5 |- ( ( ( W e. NrmMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ y e. ( Base ` X ) ) ) -> ( ( norm ` X ) ` ( x ( .s ` W ) y ) ) = ( ( norm ` W ) ` ( x ( .s ` W ) y ) ) )
50	43, 49	eqtr3d 2658	. . . 4 |- ( ( ( W e. NrmMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ y e. ( Base ` X ) ) ) -> ( ( norm ` X ) ` ( x ( .s ` X ) y ) ) = ( ( norm ` W ) ` ( x ( .s ` W ) y ) ) )
51	21	eqcomd 2628	. . . . . . 7 |- ( ( ( W e. NrmMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ y e. ( Base ` X ) ) ) -> (Scalar ` X ) = (Scalar ` W ) )
52	51	fveq2d 6195	. . . . . 6 |- ( ( ( W e. NrmMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ y e. ( Base ` X ) ) ) -> ( norm ` (Scalar ` X ) ) = ( norm ` (Scalar ` W ) ) )
53	52	fveq1d 6193	. . . . 5 |- ( ( ( W e. NrmMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ y e. ( Base ` X ) ) ) -> ( ( norm ` (Scalar ` X ) ) ` x ) = ( ( norm ` (Scalar ` W ) ) ` x ) )
54	7, 33, 47	subgnm2 22438	. . . . . 6 |- ( ( U e. (SubGrp ` W ) /\ y e. U ) -> ( ( norm ` X ) ` y ) = ( ( norm ` W ) ` y ) )
55	24, 31, 54	syl2anc 693	. . . . 5 |- ( ( ( W e. NrmMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ y e. ( Base ` X ) ) ) -> ( ( norm ` X ) ` y ) = ( ( norm ` W ) ` y ) )
56	53, 55	oveq12d 6668	. . . 4 |- ( ( ( W e. NrmMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ y e. ( Base ` X ) ) ) -> ( ( ( norm ` (Scalar ` X ) ) ` x ) x. ( ( norm ` X ) ` y ) ) = ( ( ( norm ` (Scalar ` W ) ) ` x ) x. ( ( norm ` W ) ` y ) ) )
57	38, 50, 56	3eqtr4d 2666	. . 3 |- ( ( ( W e. NrmMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ y e. ( Base ` X ) ) ) -> ( ( norm ` X ) ` ( x ( .s ` X ) y ) ) = ( ( ( norm ` (Scalar ` X ) ) ` x ) x. ( ( norm ` X ) ` y ) ) )
58	57	ralrimivva 2971	. 2 |- ( ( W e. NrmMod /\ U e. S ) -> A. x e. ( Base ` (Scalar ` X ) ) A. y e. ( Base ` X ) ( ( norm ` X ) ` ( x ( .s ` X ) y ) ) = ( ( ( norm ` (Scalar ` X ) ) ` x ) x. ( ( norm ` X ) ` y ) ) )
59		eqid 2622	. . 3 |- ( Base ` X ) = ( Base ` X )
60		eqid 2622	. . 3 |- ( .s ` X ) = ( .s ` X )
61		eqid 2622	. . 3 |- (Scalar ` X ) = (Scalar ` X )
62		eqid 2622	. . 3 |- ( Base ` (Scalar ` X ) ) = ( Base ` (Scalar ` X ) )
63		eqid 2622	. . 3 |- ( norm ` (Scalar ` X ) ) = ( norm ` (Scalar ` X ) )
64	59, 47, 60, 61, 62, 63	isnlm 22479	. 2 |- ( X e. NrmMod <-> ( ( X e. NrmGrp /\ X e. LMod /\ (Scalar ` X ) e. NrmRing ) /\ A. x e. ( Base ` (Scalar ` X ) ) A. y e. ( Base ` X ) ( ( norm ` X ) ` ( x ( .s ` X ) y ) ) = ( ( ( norm ` (Scalar ` X ) ) ` x ) x. ( ( norm ` X ) ` y ) ) ) )
65	18, 58, 64	sylanbrc 698	1 |- ( ( W e. NrmMod /\ U e. S ) -> X e. NrmMod )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   ` cfv 5888  (class class class)co 6650   x. cmul 9941   Basecbs 15857   ↾_s cress 15858  Scalarcsca 15944   .scvsca 15945  SubGrpcsubg 17588   LModclmod 18863   LSubSpclss 18932   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-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-sca 15957  df-vsca 15958  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-lmod 18865  df-lss 18933  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-nlm 22391

This theorem is referenced by:  lssnvc  22506