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Theorem sspn 28441
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 28431 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑊 ∈ NrmCVec)
3 sspn.y . . . . 5 𝑌 = (BaseSet‘𝑊)
4 sspn.m . . . . 5 𝑀 = (normCV𝑊)
53, 4nvf 28365 . . . 4 (𝑊 ∈ NrmCVec → 𝑀:𝑌⟶ℝ)
62, 5syl 17 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑀:𝑌⟶ℝ)
76ffnd 6509 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑀 Fn 𝑌)
8 eqid 2821 . . . . . 6 (BaseSet‘𝑈) = (BaseSet‘𝑈)
9 sspn.n . . . . . 6 𝑁 = (normCV𝑈)
108, 9nvf 28365 . . . . 5 (𝑈 ∈ NrmCVec → 𝑁:(BaseSet‘𝑈)⟶ℝ)
1110ffnd 6509 . . . 4 (𝑈 ∈ NrmCVec → 𝑁 Fn (BaseSet‘𝑈))
1211adantr 481 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑁 Fn (BaseSet‘𝑈))
138, 3, 1sspba 28432 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑌 ⊆ (BaseSet‘𝑈))
14 fnssres 6464 . . 3 ((𝑁 Fn (BaseSet‘𝑈) ∧ 𝑌 ⊆ (BaseSet‘𝑈)) → (𝑁𝑌) Fn 𝑌)
1512, 13, 14syl2anc 584 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑁𝑌) Fn 𝑌)
1610ffund 6512 . . . . . 6 (𝑈 ∈ NrmCVec → Fun 𝑁)
17 funres 6391 . . . . . 6 (Fun 𝑁 → Fun (𝑁𝑌))
1816, 17syl 17 . . . . 5 (𝑈 ∈ NrmCVec → Fun (𝑁𝑌))
1918ad2antrr 722 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ 𝑥𝑌) → Fun (𝑁𝑌))
20 fnresdm 6460 . . . . . . 7 (𝑀 Fn 𝑌 → (𝑀𝑌) = 𝑀)
217, 20syl 17 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑀𝑌) = 𝑀)
22 eqid 2821 . . . . . . . . . 10 ( +𝑣𝑈) = ( +𝑣𝑈)
23 eqid 2821 . . . . . . . . . 10 ( +𝑣𝑊) = ( +𝑣𝑊)
24 eqid 2821 . . . . . . . . . 10 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
25 eqid 2821 . . . . . . . . . 10 ( ·𝑠OLD𝑊) = ( ·𝑠OLD𝑊)
2622, 23, 24, 25, 9, 4, 1isssp 28429 . . . . . . . . 9 (𝑈 ∈ NrmCVec → (𝑊𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑊) ⊆ ( ·𝑠OLD𝑈) ∧ 𝑀𝑁))))
2726simplbda 500 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑊) ⊆ ( ·𝑠OLD𝑈) ∧ 𝑀𝑁))
2827simp3d 1136 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑀𝑁)
29 ssres 5874 . . . . . . 7 (𝑀𝑁 → (𝑀𝑌) ⊆ (𝑁𝑌))
3028, 29syl 17 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑀𝑌) ⊆ (𝑁𝑌))
3121, 30eqsstrrd 4005 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑀 ⊆ (𝑁𝑌))
3231adantr 481 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ 𝑥𝑌) → 𝑀 ⊆ (𝑁𝑌))
335fdmd 6517 . . . . . . 7 (𝑊 ∈ NrmCVec → dom 𝑀 = 𝑌)
3433eleq2d 2898 . . . . . 6 (𝑊 ∈ NrmCVec → (𝑥 ∈ dom 𝑀𝑥𝑌))
3534biimpar 478 . . . . 5 ((𝑊 ∈ NrmCVec ∧ 𝑥𝑌) → 𝑥 ∈ dom 𝑀)
362, 35sylan 580 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ 𝑥𝑌) → 𝑥 ∈ dom 𝑀)
37 funssfv 6685 . . . 4 ((Fun (𝑁𝑌) ∧ 𝑀 ⊆ (𝑁𝑌) ∧ 𝑥 ∈ dom 𝑀) → ((𝑁𝑌)‘𝑥) = (𝑀𝑥))
3819, 32, 36, 37syl3anc 1363 . . 3 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ 𝑥𝑌) → ((𝑁𝑌)‘𝑥) = (𝑀𝑥))
3938eqcomd 2827 . 2 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ 𝑥𝑌) → (𝑀𝑥) = ((𝑁𝑌)‘𝑥))
407, 15, 39eqfnfvd 6798 1 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑀 = (𝑁𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wss 3935  dom cdm 5549  cres 5551  Fun wfun 6343   Fn wfn 6344  wf 6345  cfv 6349  cr 10525  NrmCVeccnv 28289   +𝑣 cpv 28290  BaseSetcba 28291   ·𝑠OLD cns 28292  normCVcnmcv 28295  SubSpcss 28426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7148  df-oprab 7149  df-1st 7680  df-2nd 7681  df-vc 28264  df-nv 28297  df-va 28300  df-ba 28301  df-sm 28302  df-0v 28303  df-nmcv 28305  df-ssp 28427
This theorem is referenced by:  sspnval  28442
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