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Theorem sspn 26975
 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 26965 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑊 ∈ NrmCVec)
3 sspn.y . . . . 5 𝑌 = (BaseSet‘𝑊)
4 sspn.m . . . . 5 𝑀 = (normCV𝑊)
53, 4nvf 26899 . . . 4 (𝑊 ∈ NrmCVec → 𝑀:𝑌⟶ℝ)
62, 5syl 17 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑀:𝑌⟶ℝ)
7 ffn 5958 . . 3 (𝑀:𝑌⟶ℝ → 𝑀 Fn 𝑌)
86, 7syl 17 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑀 Fn 𝑌)
9 eqid 2610 . . . . . 6 (BaseSet‘𝑈) = (BaseSet‘𝑈)
10 sspn.n . . . . . 6 𝑁 = (normCV𝑈)
119, 10nvf 26899 . . . . 5 (𝑈 ∈ NrmCVec → 𝑁:(BaseSet‘𝑈)⟶ℝ)
12 ffn 5958 . . . . 5 (𝑁:(BaseSet‘𝑈)⟶ℝ → 𝑁 Fn (BaseSet‘𝑈))
1311, 12syl 17 . . . 4 (𝑈 ∈ NrmCVec → 𝑁 Fn (BaseSet‘𝑈))
1413adantr 480 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑁 Fn (BaseSet‘𝑈))
159, 3, 1sspba 26966 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑌 ⊆ (BaseSet‘𝑈))
16 fnssres 5918 . . 3 ((𝑁 Fn (BaseSet‘𝑈) ∧ 𝑌 ⊆ (BaseSet‘𝑈)) → (𝑁𝑌) Fn 𝑌)
1714, 15, 16syl2anc 691 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑁𝑌) Fn 𝑌)
18 ffun 5961 . . . . . . 7 (𝑁:(BaseSet‘𝑈)⟶ℝ → Fun 𝑁)
1911, 18syl 17 . . . . . 6 (𝑈 ∈ NrmCVec → Fun 𝑁)
20 funres 5843 . . . . . 6 (Fun 𝑁 → Fun (𝑁𝑌))
2119, 20syl 17 . . . . 5 (𝑈 ∈ NrmCVec → Fun (𝑁𝑌))
2221ad2antrr 758 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ 𝑥𝑌) → Fun (𝑁𝑌))
23 fnresdm 5914 . . . . . . 7 (𝑀 Fn 𝑌 → (𝑀𝑌) = 𝑀)
248, 23syl 17 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑀𝑌) = 𝑀)
25 eqid 2610 . . . . . . . . . 10 ( +𝑣𝑈) = ( +𝑣𝑈)
26 eqid 2610 . . . . . . . . . 10 ( +𝑣𝑊) = ( +𝑣𝑊)
27 eqid 2610 . . . . . . . . . 10 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
28 eqid 2610 . . . . . . . . . 10 ( ·𝑠OLD𝑊) = ( ·𝑠OLD𝑊)
2925, 26, 27, 28, 10, 4, 1isssp 26963 . . . . . . . . 9 (𝑈 ∈ NrmCVec → (𝑊𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑊) ⊆ ( ·𝑠OLD𝑈) ∧ 𝑀𝑁))))
3029simplbda 652 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑊) ⊆ ( ·𝑠OLD𝑈) ∧ 𝑀𝑁))
3130simp3d 1068 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑀𝑁)
32 ssres 5344 . . . . . . 7 (𝑀𝑁 → (𝑀𝑌) ⊆ (𝑁𝑌))
3331, 32syl 17 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑀𝑌) ⊆ (𝑁𝑌))
3424, 33eqsstr3d 3603 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑀 ⊆ (𝑁𝑌))
3534adantr 480 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ 𝑥𝑌) → 𝑀 ⊆ (𝑁𝑌))
36 fdm 5964 . . . . . . . 8 (𝑀:𝑌⟶ℝ → dom 𝑀 = 𝑌)
375, 36syl 17 . . . . . . 7 (𝑊 ∈ NrmCVec → dom 𝑀 = 𝑌)
3837eleq2d 2673 . . . . . 6 (𝑊 ∈ NrmCVec → (𝑥 ∈ dom 𝑀𝑥𝑌))
3938biimpar 501 . . . . 5 ((𝑊 ∈ NrmCVec ∧ 𝑥𝑌) → 𝑥 ∈ dom 𝑀)
402, 39sylan 487 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ 𝑥𝑌) → 𝑥 ∈ dom 𝑀)
41 funssfv 6119 . . . 4 ((Fun (𝑁𝑌) ∧ 𝑀 ⊆ (𝑁𝑌) ∧ 𝑥 ∈ dom 𝑀) → ((𝑁𝑌)‘𝑥) = (𝑀𝑥))
4222, 35, 40, 41syl3anc 1318 . . 3 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ 𝑥𝑌) → ((𝑁𝑌)‘𝑥) = (𝑀𝑥))
4342eqcomd 2616 . 2 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ 𝑥𝑌) → (𝑀𝑥) = ((𝑁𝑌)‘𝑥))
448, 17, 43eqfnfvd 6222 1 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑀 = (𝑁𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  dom cdm 5038   ↾ cres 5040  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  ℝcr 9814  NrmCVeccnv 26823   +𝑣 cpv 26824  BaseSetcba 26825   ·𝑠OLD cns 26826  normCVcnmcv 26829  SubSpcss 26960 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-1st 7059  df-2nd 7060  df-vc 26798  df-nv 26831  df-va 26834  df-ba 26835  df-sm 26836  df-0v 26837  df-nmcv 26839  df-ssp 26961 This theorem is referenced by:  sspnval  26976
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