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

Theorem sspba 28808
Description: The base set of a subspace is included in the parent base set. (Contributed by NM, 27-Jan-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
sspba.x 𝑋 = (BaseSet‘𝑈)
sspba.y 𝑌 = (BaseSet‘𝑊)
sspba.h 𝐻 = (SubSp‘𝑈)
Assertion
Ref Expression
sspba ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑌𝑋)

Proof of Theorem sspba
StepHypRef Expression
1 eqid 2737 . . . . . 6 ( +𝑣𝑈) = ( +𝑣𝑈)
2 eqid 2737 . . . . . 6 ( +𝑣𝑊) = ( +𝑣𝑊)
3 eqid 2737 . . . . . 6 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
4 eqid 2737 . . . . . 6 ( ·𝑠OLD𝑊) = ( ·𝑠OLD𝑊)
5 eqid 2737 . . . . . 6 (normCV𝑈) = (normCV𝑈)
6 eqid 2737 . . . . . 6 (normCV𝑊) = (normCV𝑊)
7 sspba.h . . . . . 6 𝐻 = (SubSp‘𝑈)
81, 2, 3, 4, 5, 6, 7isssp 28805 . . . . 5 (𝑈 ∈ NrmCVec → (𝑊𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑊) ⊆ ( ·𝑠OLD𝑈) ∧ (normCV𝑊) ⊆ (normCV𝑈)))))
98simplbda 503 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑊) ⊆ ( ·𝑠OLD𝑈) ∧ (normCV𝑊) ⊆ (normCV𝑈)))
109simp1d 1144 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → ( +𝑣𝑊) ⊆ ( +𝑣𝑈))
11 rnss 5808 . . 3 (( +𝑣𝑊) ⊆ ( +𝑣𝑈) → ran ( +𝑣𝑊) ⊆ ran ( +𝑣𝑈))
1210, 11syl 17 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → ran ( +𝑣𝑊) ⊆ ran ( +𝑣𝑈))
13 sspba.y . . 3 𝑌 = (BaseSet‘𝑊)
1413, 2bafval 28685 . 2 𝑌 = ran ( +𝑣𝑊)
15 sspba.x . . 3 𝑋 = (BaseSet‘𝑈)
1615, 1bafval 28685 . 2 𝑋 = ran ( +𝑣𝑈)
1712, 14, 163sstr4g 3946 1 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑌𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2110  wss 3866  ran crn 5552  cfv 6380  NrmCVeccnv 28665   +𝑣 cpv 28666  BaseSetcba 28667   ·𝑠OLD cns 28668  normCVcnmcv 28671  SubSpcss 28802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fo 6386  df-fv 6388  df-oprab 7217  df-1st 7761  df-2nd 7762  df-vc 28640  df-nv 28673  df-va 28676  df-ba 28677  df-sm 28678  df-nmcv 28681  df-ssp 28803
This theorem is referenced by:  sspg  28809  ssps  28811  sspmlem  28813  sspmval  28814  sspz  28816  sspn  28817  sspimsval  28819  minvecolem1  28955  minvecolem2  28956  minvecolem3  28957  minvecolem4b  28959  minvecolem4  28961  minvecolem5  28962  minvecolem6  28963  minvecolem7  28964
  Copyright terms: Public domain W3C validator