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Theorem sspn 27575
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 27565 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑊 ∈ NrmCVec)
3 sspn.y . . . . 5 𝑌 = (BaseSet‘𝑊)
4 sspn.m . . . . 5 𝑀 = (normCV𝑊)
53, 4nvf 27499 . . . 4 (𝑊 ∈ NrmCVec → 𝑀:𝑌⟶ℝ)
62, 5syl 17 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑀:𝑌⟶ℝ)
7 ffn 6043 . . 3 (𝑀:𝑌⟶ℝ → 𝑀 Fn 𝑌)
86, 7syl 17 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑀 Fn 𝑌)
9 eqid 2621 . . . . . 6 (BaseSet‘𝑈) = (BaseSet‘𝑈)
10 sspn.n . . . . . 6 𝑁 = (normCV𝑈)
119, 10nvf 27499 . . . . 5 (𝑈 ∈ NrmCVec → 𝑁:(BaseSet‘𝑈)⟶ℝ)
12 ffn 6043 . . . . 5 (𝑁:(BaseSet‘𝑈)⟶ℝ → 𝑁 Fn (BaseSet‘𝑈))
1311, 12syl 17 . . . 4 (𝑈 ∈ NrmCVec → 𝑁 Fn (BaseSet‘𝑈))
1413adantr 481 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑁 Fn (BaseSet‘𝑈))
159, 3, 1sspba 27566 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑌 ⊆ (BaseSet‘𝑈))
16 fnssres 6002 . . 3 ((𝑁 Fn (BaseSet‘𝑈) ∧ 𝑌 ⊆ (BaseSet‘𝑈)) → (𝑁𝑌) Fn 𝑌)
1714, 15, 16syl2anc 693 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑁𝑌) Fn 𝑌)
18 ffun 6046 . . . . . . 7 (𝑁:(BaseSet‘𝑈)⟶ℝ → Fun 𝑁)
1911, 18syl 17 . . . . . 6 (𝑈 ∈ NrmCVec → Fun 𝑁)
20 funres 5927 . . . . . 6 (Fun 𝑁 → Fun (𝑁𝑌))
2119, 20syl 17 . . . . 5 (𝑈 ∈ NrmCVec → Fun (𝑁𝑌))
2221ad2antrr 762 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ 𝑥𝑌) → Fun (𝑁𝑌))
23 fnresdm 5998 . . . . . . 7 (𝑀 Fn 𝑌 → (𝑀𝑌) = 𝑀)
248, 23syl 17 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑀𝑌) = 𝑀)
25 eqid 2621 . . . . . . . . . 10 ( +𝑣𝑈) = ( +𝑣𝑈)
26 eqid 2621 . . . . . . . . . 10 ( +𝑣𝑊) = ( +𝑣𝑊)
27 eqid 2621 . . . . . . . . . 10 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
28 eqid 2621 . . . . . . . . . 10 ( ·𝑠OLD𝑊) = ( ·𝑠OLD𝑊)
2925, 26, 27, 28, 10, 4, 1isssp 27563 . . . . . . . . 9 (𝑈 ∈ NrmCVec → (𝑊𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑊) ⊆ ( ·𝑠OLD𝑈) ∧ 𝑀𝑁))))
3029simplbda 654 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑊) ⊆ ( ·𝑠OLD𝑈) ∧ 𝑀𝑁))
3130simp3d 1074 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑀𝑁)
32 ssres 5422 . . . . . . 7 (𝑀𝑁 → (𝑀𝑌) ⊆ (𝑁𝑌))
3331, 32syl 17 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝑀𝑌) ⊆ (𝑁𝑌))
3424, 33eqsstr3d 3638 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑀 ⊆ (𝑁𝑌))
3534adantr 481 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ 𝑥𝑌) → 𝑀 ⊆ (𝑁𝑌))
36 fdm 6049 . . . . . . . 8 (𝑀:𝑌⟶ℝ → dom 𝑀 = 𝑌)
375, 36syl 17 . . . . . . 7 (𝑊 ∈ NrmCVec → dom 𝑀 = 𝑌)
3837eleq2d 2686 . . . . . 6 (𝑊 ∈ NrmCVec → (𝑥 ∈ dom 𝑀𝑥𝑌))
3938biimpar 502 . . . . 5 ((𝑊 ∈ NrmCVec ∧ 𝑥𝑌) → 𝑥 ∈ dom 𝑀)
402, 39sylan 488 . . . 4 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ 𝑥𝑌) → 𝑥 ∈ dom 𝑀)
41 funssfv 6207 . . . 4 ((Fun (𝑁𝑌) ∧ 𝑀 ⊆ (𝑁𝑌) ∧ 𝑥 ∈ dom 𝑀) → ((𝑁𝑌)‘𝑥) = (𝑀𝑥))
4222, 35, 40, 41syl3anc 1325 . . 3 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ 𝑥𝑌) → ((𝑁𝑌)‘𝑥) = (𝑀𝑥))
4342eqcomd 2627 . 2 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ 𝑥𝑌) → (𝑀𝑥) = ((𝑁𝑌)‘𝑥))
448, 17, 43eqfnfvd 6312 1 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑀 = (𝑁𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1037   = wceq 1482  wcel 1989  wss 3572  dom cdm 5112  cres 5114  Fun wfun 5880   Fn wfn 5881  wf 5882  cfv 5886  cr 9932  NrmCVeccnv 27423   +𝑣 cpv 27424  BaseSetcba 27425   ·𝑠OLD cns 27426  normCVcnmcv 27429  SubSpcss 27560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-1st 7165  df-2nd 7166  df-vc 27398  df-nv 27431  df-va 27434  df-ba 27435  df-sm 27436  df-0v 27437  df-nmcv 27439  df-ssp 27561
This theorem is referenced by:  sspnval  27576
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