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

Theorem sspba 27470
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 2621 . . . . . 6 ( +𝑣𝑈) = ( +𝑣𝑈)
2 eqid 2621 . . . . . 6 ( +𝑣𝑊) = ( +𝑣𝑊)
3 eqid 2621 . . . . . 6 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
4 eqid 2621 . . . . . 6 ( ·𝑠OLD𝑊) = ( ·𝑠OLD𝑊)
5 eqid 2621 . . . . . 6 (normCV𝑈) = (normCV𝑈)
6 eqid 2621 . . . . . 6 (normCV𝑊) = (normCV𝑊)
7 sspba.h . . . . . 6 𝐻 = (SubSp‘𝑈)
81, 2, 3, 4, 5, 6, 7isssp 27467 . . . . 5 (𝑈 ∈ NrmCVec → (𝑊𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑊) ⊆ ( ·𝑠OLD𝑈) ∧ (normCV𝑊) ⊆ (normCV𝑈)))))
98simplbda 653 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (( +𝑣𝑊) ⊆ ( +𝑣𝑈) ∧ ( ·𝑠OLD𝑊) ⊆ ( ·𝑠OLD𝑈) ∧ (normCV𝑊) ⊆ (normCV𝑈)))
109simp1d 1071 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → ( +𝑣𝑊) ⊆ ( +𝑣𝑈))
11 rnss 5324 . . 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 27347 . 2 𝑌 = ran ( +𝑣𝑊)
15 sspba.x . . 3 𝑋 = (BaseSet‘𝑈)
1615, 1bafval 27347 . 2 𝑋 = ran ( +𝑣𝑈)
1712, 14, 163sstr4g 3631 1 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑌𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wss 3560  ran crn 5085  cfv 5857  NrmCVeccnv 27327   +𝑣 cpv 27328  BaseSetcba 27329   ·𝑠OLD cns 27330  normCVcnmcv 27333  SubSpcss 27464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fo 5863  df-fv 5865  df-oprab 6619  df-1st 7128  df-2nd 7129  df-vc 27302  df-nv 27335  df-va 27338  df-ba 27339  df-sm 27340  df-nmcv 27343  df-ssp 27465
This theorem is referenced by:  sspg  27471  ssps  27473  sspmlem  27475  sspmval  27476  sspz  27478  sspn  27479  sspimsval  27481  sspph  27598  minvecolem1  27618  minvecolem2  27619  minvecolem3  27620  minvecolem4b  27622  minvecolem4  27624  minvecolem5  27625  minvecolem6  27626  minvecolem7  27627
  Copyright terms: Public domain W3C validator