Theorem sspn 8395

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

Hypotheses

Ref	Expression
sspn.y	|- Y = (Base` 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		funssfv 3735	. . . . . 6 |- ((Fun (N |` Y) /\ M (_ (N |` Y) /\ x e. dom M) -> ((N |` Y)` x) = (M` x))
2		eqid 1475	. . . . . . . . 9 |- (Base` U) = (Base` U)
3		sspn.n	. . . . . . . . 9 |- N = (norm` U)
4	2, 3	nvf 8286	. . . . . . . 8 |- (U e. NrmCVec -> N:(Base` U)-->RR)
5		ffun 3629	. . . . . . . 8 |- (N:(Base` U)-->RR -> Fun N)
6		funres 3551	. . . . . . . 8 |- (Fun N -> Fun (N |` Y))
7	4, 5, 6	3syl 20	. . . . . . 7 |- (U e. NrmCVec -> Fun (N |` Y))
8	7	ad2antrr 404	. . . . . 6 |- (((U e. NrmCVec /\ W e. H) /\ x e. Y) -> Fun (N |` Y))
9		sspn.h	. . . . . . . . . . 11 |- H = (SubSp` U)
10	9	sspnv 8385	. . . . . . . . . 10 |- ((U e. NrmCVec /\ W e. H) -> W e. NrmCVec)
11		sspn.y	. . . . . . . . . . 11 |- Y = (Base` W)
12		sspn.m	. . . . . . . . . . 11 |- M = (norm` W)
13	11, 12	nvf 8286	. . . . . . . . . 10 |- (W e. NrmCVec -> M:Y-->RR)
14	10, 13	syl 10	. . . . . . . . 9 |- ((U e. NrmCVec /\ W e. H) -> M:Y-->RR)
15		ffn 3627	. . . . . . . . 9 |- (M:Y-->RR -> M Fn Y)
16		fnresdm 3596	. . . . . . . . 9 |- (M Fn Y -> (M |` Y) = M)
17	14, 15, 16	3syl 20	. . . . . . . 8 |- ((U e. NrmCVec /\ W e. H) -> (M |` Y) = M)
18		eqid 1475	. . . . . . . . . . 11 |- (+v` U) = (+v` U)
19		eqid 1475	. . . . . . . . . . 11 |- (+v` W) = (+v` W)
20		eqid 1475	. . . . . . . . . . 11 |- (.s` U) = (.s` U)
21		eqid 1475	. . . . . . . . . . 11 |- (.s` W) = (.s` W)
22	18, 19, 20, 21, 3, 12, 9	isssp 8383	. . . . . . . . . 10 |- (U e. NrmCVec -> (W e. H <-> (W e. NrmCVec /\ ((+v` W) (_ (+v` U) /\ (.s` W) (_ (.s` U) /\ M (_ N))))
23	22	pm3.27bda 421	. . . . . . . . 9 |- ((U e. NrmCVec /\ W e. H) -> ((+v` W) (_ (+v` U) /\ (.s` W) (_ (.s` U) /\ M (_ N))
24		3simp3 790	. . . . . . . . 9 |- (((+v` W) (_ (+v` U) /\ (.s` W) (_ (.s` U) /\ M (_ N) -> M (_ N)
25		ssres 3385	. . . . . . . . 9 |- (M (_ N -> (M |` Y) (_ (N |` Y))
26	23, 24, 25	3syl 20	. . . . . . . 8 |- ((U e. NrmCVec /\ W e. H) -> (M |` Y) (_ (N |` Y))
27	17, 26	eqsstr3d 2096	. . . . . . 7 |- ((U e. NrmCVec /\ W e. H) -> M (_ (N |` Y))
28	27	adantr 389	. . . . . 6 |- (((U e. NrmCVec /\ W e. H) /\ x e. Y) -> M (_ (N |` Y))
29		fdm 3631	. . . . . . . . . 10 |- (M:Y-->RR -> dom M = Y)
30	13, 29	syl 10	. . . . . . . . 9 |- (W e. NrmCVec -> dom M = Y)
31	30	eleq2d 1541	. . . . . . . 8 |- (W e. NrmCVec -> (x e. dom M <-> x e. Y))
32	31	biimpar 417	. . . . . . 7 |- ((W e. NrmCVec /\ x e. Y) -> x e. dom M)
33	32, 10	sylan 448	. . . . . 6 |- (((U e. NrmCVec /\ W e. H) /\ x e. Y) -> x e. dom M)
34	1, 8, 28, 33	syl3anc 858	. . . . 5 |- (((U e. NrmCVec /\ W e. H) /\ x e. Y) -> ((N |` Y)` x) = (M` x))
35	34	eqcomd 1480	. . . 4 |- (((U e. NrmCVec /\ W e. H) /\ x e. Y) -> (M` x) = ((N |` Y)` x))
36	35	r19.21aiva 1714	. . 3 |- ((U e. NrmCVec /\ W e. H) -> A.x e. Y (M` x) = ((N |` Y)` x))
37		eqid 1475	. . 3 |- Y = Y
38	36, 37	jctil 292	. 2 |- ((U e. NrmCVec /\ W e. H) -> (Y = Y /\ A.x e. Y (M` x) = ((N |` Y)` x)))
39		eqfnfv 3797	. . 3 |- ((M Fn Y /\ (N |` Y) Fn Y) -> (M = (N |` Y) <-> (Y = Y /\ A.x e. Y (M` x) = ((N |` Y)` x))))
40	10, 13, 15	3syl 20	. . 3 |- ((U e. NrmCVec /\ W e. H) -> M Fn Y)
41		fnssres 3600	. . . 4 |- ((N Fn (Base` U) /\ Y (_ (Base` U)) -> (N |` Y) Fn Y)
42		ffn 3627	. . . . . 6 |- (N:(Base` U)-->RR -> N Fn (Base` U))
43	4, 42	syl 10	. . . . 5 |- (U e. NrmCVec -> N Fn (Base` U))
44	43	adantr 389	. . . 4 |- ((U e. NrmCVec /\ W e. H) -> N Fn (Base` U))
45	2, 11, 9	sspba 8386	. . . 4 |- ((U e. NrmCVec /\ W e. H) -> Y (_ (Base` U))
46	41, 44, 45	sylanc 471	. . 3 |- ((U e. NrmCVec /\ W e. H) -> (N |` Y) Fn Y)
47	39, 40, 46	sylanc 471	. 2 |- ((U e. NrmCVec /\ W e. H) -> (M = (N |` Y) <-> (Y = Y /\ A.x e. Y (M` x) = ((N |` Y)` x))))
48	38, 47	mpbird 196	1 |- ((U e. NrmCVec /\ W e. H) -> M = (N |` Y))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  A.wral 1645   (_ wss 2047  dom cdm 3170   |` cres 3172  Fun wfun 3176   Fn wfn 3177  -->wf 3178  ` cfv 3182  RRcr 5233  NrmCVeccnv 8203  +vcpv 8204  Basecba 8205  .scns 8206  normcnm 8209  SubSpcss 8380

This theorem is referenced by:  sspnval 8396

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fo 3196  df-fv 3198  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-grp 8037  df-gid 8038  df-nv 8211  df-va 8214  df-ba 8215  df-sm 8216  df-0v 8217  df-nm 8219  df-ssp 8381

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