MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sspn Structured version   Visualization version   GIF version

Theorem sspn 30755
Description: The norm on a subspace is a restriction of the norm on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
sspn.y 𝑌 = (BaseSet‘𝑊)
sspn.n 𝑁 = (normCV𝑈)
sspn.m 𝑀 = (normCV𝑊)
sspn.h 𝐻 = (SubSp‘𝑈)
Assertion
Ref Expression
sspn ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑀 = (𝑁𝑌))

Proof of Theorem sspn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sspn.h . . . . 5 𝐻 = (SubSp‘𝑈)
21sspnv 30745 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑊 ∈ NrmCVec)
3 sspn.y . . . . 5 𝑌 = (BaseSet‘𝑊)
4 sspn.m . . . . 5 𝑀 = (normCV𝑊)
53, 4nvf 30679 . . . 4 (𝑊 ∈ NrmCVec → 𝑀:𝑌⟶ℝ)
62, 5syl 17 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑀:𝑌⟶ℝ)
76ffnd 6737 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑀 Fn 𝑌)
8 eqid 2737 . . . . . 6 (BaseSet‘𝑈) = (BaseSet‘𝑈)
9 sspn.n . . . . . 6 𝑁 = (normCV𝑈)
108, 9nvf 30679 . . . . 5 (𝑈 ∈ NrmCVec → 𝑁:(BaseSet‘𝑈)⟶ℝ)
1110ffnd 6737 . . . 4 (𝑈 ∈ NrmCVec → 𝑁 Fn (BaseSet‘𝑈))
1211adantr 480 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑁 Fn (BaseSet‘𝑈))
138, 3, 1sspba 30746 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑌 ⊆ (BaseSet‘𝑈))
14 fnssres 6691 . . 3 ((𝑁 Fn (BaseSet‘𝑈) ∧ 𝑌 ⊆ (BaseSet‘𝑈)) → (𝑁𝑌) Fn 𝑌)
1512, 13, 14syl2anc 584 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑁𝑌) Fn 𝑌)
1610ffund 6740 . . . . . 6 (𝑈 ∈ NrmCVec → Fun 𝑁)
1716funresd 6609 . . . . 5 (𝑈 ∈ NrmCVec → Fun (𝑁𝑌))
1817ad2antrr 726 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ 𝑥𝑌) → Fun (𝑁𝑌))
19 fnresdm 6687 . . . . . . 7 (𝑀 Fn 𝑌 → (𝑀𝑌) = 𝑀)
207, 19syl 17 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑀𝑌) = 𝑀)
21 eqid 2737 . . . . . . . . . 10 ( +𝑣𝑈) = ( +𝑣𝑈)
22 eqid 2737 . . . . . . . . . 10 ( +𝑣𝑊) = ( +𝑣𝑊)
23 eqid 2737 . . . . . . . . . 10 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
24 eqid 2737 . . . . . . . . . 10 ( ·𝑠OLD𝑊) = ( ·𝑠OLD𝑊)
2521, 22, 23, 24, 9, 4, 1isssp 30743 . . . . . . . . 9 (𝑈 ∈ NrmCVec → (𝑊𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑊) ⊆ ( ·𝑠OLD𝑈) ∧ 𝑀𝑁))))
2625simplbda 499 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑊) ⊆ ( ·𝑠OLD𝑈) ∧ 𝑀𝑁))
2726simp3d 1145 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑀𝑁)
28 ssres 6021 . . . . . . 7 (𝑀𝑁 → (𝑀𝑌) ⊆ (𝑁𝑌))
2927, 28syl 17 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑀𝑌) ⊆ (𝑁𝑌))
3020, 29eqsstrrd 4019 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑀 ⊆ (𝑁𝑌))
3130adantr 480 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ 𝑥𝑌) → 𝑀 ⊆ (𝑁𝑌))
325fdmd 6746 . . . . . . 7 (𝑊 ∈ NrmCVec → dom 𝑀 = 𝑌)
3332eleq2d 2827 . . . . . 6 (𝑊 ∈ NrmCVec → (𝑥 ∈ dom 𝑀𝑥𝑌))
3433biimpar 477 . . . . 5 ((𝑊 ∈ NrmCVec ∧ 𝑥𝑌) → 𝑥 ∈ dom 𝑀)
352, 34sylan 580 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ 𝑥𝑌) → 𝑥 ∈ dom 𝑀)
36 funssfv 6927 . . . 4 ((Fun (𝑁𝑌) ∧ 𝑀 ⊆ (𝑁𝑌) ∧ 𝑥 ∈ dom 𝑀) → ((𝑁𝑌)‘𝑥) = (𝑀𝑥))
3718, 31, 35, 36syl3anc 1373 . . 3 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ 𝑥𝑌) → ((𝑁𝑌)‘𝑥) = (𝑀𝑥))
3837eqcomd 2743 . 2 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ 𝑥𝑌) → (𝑀𝑥) = ((𝑁𝑌)‘𝑥))
397, 15, 38eqfnfvd 7054 1 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑀 = (𝑁𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wss 3951  dom cdm 5685  cres 5687  Fun wfun 6555   Fn wfn 6556  wf 6557  cfv 6561  cr 11154  NrmCVeccnv 30603   +𝑣 cpv 30604  BaseSetcba 30605   ·𝑠OLD cns 30606  normCVcnmcv 30609  SubSpcss 30740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-1st 8014  df-2nd 8015  df-vc 30578  df-nv 30611  df-va 30614  df-ba 30615  df-sm 30616  df-0v 30617  df-nmcv 30619  df-ssp 30741
This theorem is referenced by:  sspnval  30756
  Copyright terms: Public domain W3C validator