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Theorem sspn 30765
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 30755 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑊 ∈ NrmCVec)
3 sspn.y . . . . 5 𝑌 = (BaseSet‘𝑊)
4 sspn.m . . . . 5 𝑀 = (normCV𝑊)
53, 4nvf 30689 . . . 4 (𝑊 ∈ NrmCVec → 𝑀:𝑌⟶ℝ)
62, 5syl 17 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑀:𝑌⟶ℝ)
76ffnd 6738 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑀 Fn 𝑌)
8 eqid 2735 . . . . . 6 (BaseSet‘𝑈) = (BaseSet‘𝑈)
9 sspn.n . . . . . 6 𝑁 = (normCV𝑈)
108, 9nvf 30689 . . . . 5 (𝑈 ∈ NrmCVec → 𝑁:(BaseSet‘𝑈)⟶ℝ)
1110ffnd 6738 . . . 4 (𝑈 ∈ NrmCVec → 𝑁 Fn (BaseSet‘𝑈))
1211adantr 480 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑁 Fn (BaseSet‘𝑈))
138, 3, 1sspba 30756 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑌 ⊆ (BaseSet‘𝑈))
14 fnssres 6692 . . 3 ((𝑁 Fn (BaseSet‘𝑈) ∧ 𝑌 ⊆ (BaseSet‘𝑈)) → (𝑁𝑌) Fn 𝑌)
1512, 13, 14syl2anc 584 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑁𝑌) Fn 𝑌)
1610ffund 6741 . . . . . 6 (𝑈 ∈ NrmCVec → Fun 𝑁)
1716funresd 6611 . . . . 5 (𝑈 ∈ NrmCVec → Fun (𝑁𝑌))
1817ad2antrr 726 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ 𝑥𝑌) → Fun (𝑁𝑌))
19 fnresdm 6688 . . . . . . 7 (𝑀 Fn 𝑌 → (𝑀𝑌) = 𝑀)
207, 19syl 17 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑀𝑌) = 𝑀)
21 eqid 2735 . . . . . . . . . 10 ( +𝑣𝑈) = ( +𝑣𝑈)
22 eqid 2735 . . . . . . . . . 10 ( +𝑣𝑊) = ( +𝑣𝑊)
23 eqid 2735 . . . . . . . . . 10 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
24 eqid 2735 . . . . . . . . . 10 ( ·𝑠OLD𝑊) = ( ·𝑠OLD𝑊)
2521, 22, 23, 24, 9, 4, 1isssp 30753 . . . . . . . . 9 (𝑈 ∈ NrmCVec → (𝑊𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑊) ⊆ ( ·𝑠OLD𝑈) ∧ 𝑀𝑁))))
2625simplbda 499 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑊) ⊆ ( ·𝑠OLD𝑈) ∧ 𝑀𝑁))
2726simp3d 1143 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑀𝑁)
28 ssres 6024 . . . . . . 7 (𝑀𝑁 → (𝑀𝑌) ⊆ (𝑁𝑌))
2927, 28syl 17 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑀𝑌) ⊆ (𝑁𝑌))
3020, 29eqsstrrd 4035 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑀 ⊆ (𝑁𝑌))
3130adantr 480 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ 𝑥𝑌) → 𝑀 ⊆ (𝑁𝑌))
325fdmd 6747 . . . . . . 7 (𝑊 ∈ NrmCVec → dom 𝑀 = 𝑌)
3332eleq2d 2825 . . . . . 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 6928 . . . 4 ((Fun (𝑁𝑌) ∧ 𝑀 ⊆ (𝑁𝑌) ∧ 𝑥 ∈ dom 𝑀) → ((𝑁𝑌)‘𝑥) = (𝑀𝑥))
3718, 31, 35, 36syl3anc 1370 . . 3 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ 𝑥𝑌) → ((𝑁𝑌)‘𝑥) = (𝑀𝑥))
3837eqcomd 2741 . 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 1086   = wceq 1537  wcel 2106  wss 3963  dom cdm 5689  cres 5691  Fun wfun 6557   Fn wfn 6558  wf 6559  cfv 6563  cr 11152  NrmCVeccnv 30613   +𝑣 cpv 30614  BaseSetcba 30615   ·𝑠OLD cns 30616  normCVcnmcv 30619  SubSpcss 30750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-1st 8013  df-2nd 8014  df-vc 30588  df-nv 30621  df-va 30624  df-ba 30625  df-sm 30626  df-0v 30627  df-nmcv 30629  df-ssp 30751
This theorem is referenced by:  sspnval  30766
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