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Theorem sspba 26966
 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 2610 . . . . . 6 ( +𝑣𝑈) = ( +𝑣𝑈)
2 eqid 2610 . . . . . 6 ( +𝑣𝑊) = ( +𝑣𝑊)
3 eqid 2610 . . . . . 6 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
4 eqid 2610 . . . . . 6 ( ·𝑠OLD𝑊) = ( ·𝑠OLD𝑊)
5 eqid 2610 . . . . . 6 (normCV𝑈) = (normCV𝑈)
6 eqid 2610 . . . . . 6 (normCV𝑊) = (normCV𝑊)
7 sspba.h . . . . . 6 𝐻 = (SubSp‘𝑈)
81, 2, 3, 4, 5, 6, 7isssp 26963 . . . . 5 (𝑈 ∈ NrmCVec → (𝑊𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑊) ⊆ ( ·𝑠OLD𝑈) ∧ (normCV𝑊) ⊆ (normCV𝑈)))))
98simplbda 652 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑊) ⊆ ( ·𝑠OLD𝑈) ∧ (normCV𝑊) ⊆ (normCV𝑈)))
109simp1d 1066 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → ( +𝑣𝑊) ⊆ ( +𝑣𝑈))
11 rnss 5275 . . 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 26843 . 2 𝑌 = ran ( +𝑣𝑊)
15 sspba.x . . 3 𝑋 = (BaseSet‘𝑈)
1615, 1bafval 26843 . 2 𝑋 = ran ( +𝑣𝑈)
1712, 14, 163sstr4g 3609 1 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑌𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  ran crn 5039  ‘cfv 5804  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-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-rab 2905  df-v 3175  df-sbc 3403  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-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-fo 5810  df-fv 5812  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-nmcv 26839  df-ssp 26961 This theorem is referenced by:  sspg  26967  ssps  26969  sspmlem  26971  sspmval  26972  sspz  26974  sspn  26975  sspimsval  26977  sspph  27094  minvecolem1  27114  minvecolem2  27115  minvecolem3  27116  minvecolem4b  27118  minvecolem4  27120  minvecolem5  27121  minvecolem6  27122  minvecolem7  27123
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