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Theorem sspba 30756
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 2735 . . . . . 6 ( +𝑣𝑈) = ( +𝑣𝑈)
2 eqid 2735 . . . . . 6 ( +𝑣𝑊) = ( +𝑣𝑊)
3 eqid 2735 . . . . . 6 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
4 eqid 2735 . . . . . 6 ( ·𝑠OLD𝑊) = ( ·𝑠OLD𝑊)
5 eqid 2735 . . . . . 6 (normCV𝑈) = (normCV𝑈)
6 eqid 2735 . . . . . 6 (normCV𝑊) = (normCV𝑊)
7 sspba.h . . . . . 6 𝐻 = (SubSp‘𝑈)
81, 2, 3, 4, 5, 6, 7isssp 30753 . . . . 5 (𝑈 ∈ NrmCVec → (𝑊𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑊) ⊆ ( ·𝑠OLD𝑈) ∧ (normCV𝑊) ⊆ (normCV𝑈)))))
98simplbda 499 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑊) ⊆ ( ·𝑠OLD𝑈) ∧ (normCV𝑊) ⊆ (normCV𝑈)))
109simp1d 1141 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → ( +𝑣𝑊) ⊆ ( +𝑣𝑈))
11 rnss 5953 . . 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 30633 . 2 𝑌 = ran ( +𝑣𝑊)
15 sspba.x . . 3 𝑋 = (BaseSet‘𝑈)
1615, 1bafval 30633 . 2 𝑋 = ran ( +𝑣𝑈)
1712, 14, 163sstr4g 4041 1 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑌𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wss 3963  ran crn 5690  cfv 6563  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-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-rab 3434  df-v 3480  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-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-fo 6569  df-fv 6571  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-nmcv 30629  df-ssp 30751
This theorem is referenced by:  sspg  30757  ssps  30759  sspmlem  30761  sspmval  30762  sspz  30764  sspn  30765  sspimsval  30767  minvecolem1  30903  minvecolem2  30904  minvecolem3  30905  minvecolem4b  30907  minvecolem4  30909  minvecolem5  30910  minvecolem6  30911  minvecolem7  30912
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