Theorem sspn 9734

Description: The norm on a subspace is a restriction of the norm on the parent space.

Hypotheses

Ref	Expression
sspn.y	|- Y = (BaseSet` W)
sspn.n	|- N = (norm` U)
sspn.m	|- M = (norm` W)
sspn.h	|- H = (SubSp` U)

Assertion

Ref	Expression
sspn	|- ((U e. NrmCVec /\ W e. H) -> M = (N |` Y))

Proof of Theorem sspn

Step	Hyp	Ref	Expression
1		eqid 1884	. . . . . . . 8 |- (BaseSet` U) = (BaseSet` U)
2		sspn.n	. . . . . . . 8 |- N = (norm` U)
3	1, 2	nvf 9618	. . . . . . 7 |- (U e. NrmCVec -> N:(BaseSet` U)-->RR)
4		ffun 4565	. . . . . . 7 |- (N:(BaseSet` U)-->RR -> Fun N)
5		funres 4459	. . . . . . 7 |- (Fun N -> Fun (N |` Y))
6	3, 4, 5	3syl 24	. . . . . 6 |- (U e. NrmCVec -> Fun (N |` Y))
7	6	ad2antrr 440	. . . . 5 |- (((U e. NrmCVec /\ W e. H) /\ x e. Y) -> Fun (N |` Y))
8		sspn.h	. . . . . . . . . 10 |- H = (SubSp` U)
9	8	sspnv 9724	. . . . . . . . 9 |- ((U e. NrmCVec /\ W e. H) -> W e. NrmCVec)
10		sspn.y	. . . . . . . . . 10 |- Y = (BaseSet` W)
11		sspn.m	. . . . . . . . . 10 |- M = (norm` W)
12	10, 11	nvf 9618	. . . . . . . . 9 |- (W e. NrmCVec -> M:Y-->RR)
13	9, 12	syl 12	. . . . . . . 8 |- ((U e. NrmCVec /\ W e. H) -> M:Y-->RR)
14		ffn 4562	. . . . . . . 8 |- (M:Y-->RR -> M Fn Y)
15		fnresdm 4522	. . . . . . . 8 |- (M Fn Y -> (M |` Y) = M)
16	13, 14, 15	3syl 24	. . . . . . 7 |- ((U e. NrmCVec /\ W e. H) -> (M |` Y) = M)
17		eqid 1884	. . . . . . . . . 10 |- (+v` U) = (+v` U)
18		eqid 1884	. . . . . . . . . 10 |- (+v` W) = (+v` W)
19		eqid 1884	. . . . . . . . . 10 |- (.s` U) = (.s` U)
20		eqid 1884	. . . . . . . . . 10 |- (.s` W) = (.s` W)
21	17, 18, 19, 20, 2, 11, 8	isssp 9722	. . . . . . . . 9 |- (U e. NrmCVec -> (W e. H <-> (W e. NrmCVec /\ ((+v` W) C_ (+v` U) /\ (.s` W) C_ (.s` U) /\ M C_ N))))
22	21	simplbda 465	. . . . . . . 8 |- ((U e. NrmCVec /\ W e. H) -> ((+v` W) C_ (+v` U) /\ (.s` W) C_ (.s` U) /\ M C_ N))
23		simp3 878	. . . . . . . 8 |- (((+v` W) C_ (+v` U) /\ (.s` W) C_ (.s` U) /\ M C_ N) -> M C_ N)
24		ssres 4239	. . . . . . . 8 |- (M C_ N -> (M |` Y) C_ (N |` Y))
25	22, 23, 24	3syl 24	. . . . . . 7 |- ((U e. NrmCVec /\ W e. H) -> (M |` Y) C_ (N |` Y))
26	16, 25	eqsstr3d 2652	. . . . . 6 |- ((U e. NrmCVec /\ W e. H) -> M C_ (N |` Y))
27	26	adantr 425	. . . . 5 |- (((U e. NrmCVec /\ W e. H) /\ x e. Y) -> M C_ (N |` Y))
28		fdm 4567	. . . . . . . . 9 |- (M:Y-->RR -> dom M = Y)
29	12, 28	syl 12	. . . . . . . 8 |- (W e. NrmCVec -> dom M = Y)
30	29	eleq2d 1964	. . . . . . 7 |- (W e. NrmCVec -> (x e. dom M <-> x e. Y))
31	30	biimpar 461	. . . . . 6 |- ((W e. NrmCVec /\ x e. Y) -> x e. dom M)
32	31, 9	sylan 497	. . . . 5 |- (((U e. NrmCVec /\ W e. H) /\ x e. Y) -> x e. dom M)
33		funssfv 4692	. . . . 5 |- ((Fun (N |` Y) /\ M C_ (N |` Y) /\ x e. dom M) -> ((N |` Y)` x) = (M` x))
34	7, 27, 32, 33	syl111anc 1100	. . . 4 |- (((U e. NrmCVec /\ W e. H) /\ x e. Y) -> ((N |` Y)` x) = (M` x))
35	34	eqcomd 1889	. . 3 |- (((U e. NrmCVec /\ W e. H) /\ x e. Y) -> (M` x) = ((N |` Y)` x))
36	35	r19.21aiva 2176	. 2 |- ((U e. NrmCVec /\ W e. H) -> A.x e. Y (M` x) = ((N |` Y)` x))
37	9, 12, 14	3syl 24	. . 3 |- ((U e. NrmCVec /\ W e. H) -> M Fn Y)
38		ffn 4562	. . . . . 6 |- (N:(BaseSet` U)-->RR -> N Fn (BaseSet` U))
39	3, 38	syl 12	. . . . 5 |- (U e. NrmCVec -> N Fn (BaseSet` U))
40	39	adantr 425	. . . 4 |- ((U e. NrmCVec /\ W e. H) -> N Fn (BaseSet` U))
41	1, 10, 8	sspba 9725	. . . 4 |- ((U e. NrmCVec /\ W e. H) -> Y C_ (BaseSet` U))
42		fnssres 4526	. . . 4 |- ((N Fn (BaseSet` U) /\ Y C_ (BaseSet` U)) -> (N |` Y) Fn Y)
43	40, 41, 42	syl11anc 524	. . 3 |- ((U e. NrmCVec /\ W e. H) -> (N |` Y) Fn Y)
44		eqfnfv2 4767	. . 3 |- ((M Fn Y /\ (N |` Y) Fn Y) -> (M = (N |` Y) <-> A.x e. Y (M` x) = ((N |` Y)` x)))
45	37, 43, 44	syl11anc 524	. 2 |- ((U e. NrmCVec /\ W e. H) -> (M = (N |` Y) <-> A.x e. Y (M` x) = ((N |` Y)` x)))
46	36, 45	mpbird 213	1 |- ((U e. NrmCVec /\ W e. H) -> M = (N |` Y))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105   C_ wss 2593  dom cdm 3986   |` cres 3988  Fun wfun 3992   Fn wfn 3993  -->wf 3994  ` cfv 3998  RRcr 6385  NrmCVeccnv 9535  +vcpv 9536  BaseSetcba 9537  .scns 9538  normcnm 9541  SubSpcss 9719

This theorem is referenced by:  sspnval 9735

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fo 4012  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-gid 9317  df-nv 9543  df-va 9546  df-ba 9547  df-sm 9548  df-0v 9549  df-nm 9551  df-ssp 9720

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