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Theorem sspba 29967
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 = (BaseSetβ€˜π‘ˆ)
sspba.y π‘Œ = (BaseSetβ€˜π‘Š)
sspba.h 𝐻 = (SubSpβ€˜π‘ˆ)
Assertion
Ref Expression
sspba ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ 𝐻) β†’ π‘Œ βŠ† 𝑋)
Proof of Theorem sspba
StepHypRef Expression
1 eqid 2732 . . . . . 6 ( +𝑣 β€˜π‘ˆ) = ( +𝑣 β€˜π‘ˆ)
2 eqid 2732 . . . . . 6 ( +𝑣 β€˜π‘Š) = ( +𝑣 β€˜π‘Š)
3 eqid 2732 . . . . . 6 ( ·𝑠OLD β€˜π‘ˆ) = ( ·𝑠OLD β€˜π‘ˆ)
4 eqid 2732 . . . . . 6 ( ·𝑠OLD β€˜π‘Š) = ( ·𝑠OLD β€˜π‘Š)
5 eqid 2732 . . . . . 6 (normCVβ€˜π‘ˆ) = (normCVβ€˜π‘ˆ)
6 eqid 2732 . . . . . 6 (normCVβ€˜π‘Š) = (normCVβ€˜π‘Š)
7 sspba.h . . . . . 6 𝐻 = (SubSpβ€˜π‘ˆ)
81, 2, 3, 4, 5, 6, 7isssp 29964 . . . . 5 (π‘ˆ ∈ NrmCVec β†’ (π‘Š ∈ 𝐻 ↔ (π‘Š ∈ NrmCVec ∧ (( +𝑣 β€˜π‘Š) βŠ† ( +𝑣 β€˜π‘ˆ) ∧ ( ·𝑠OLD β€˜π‘Š) βŠ† ( ·𝑠OLD β€˜π‘ˆ) ∧ (normCVβ€˜π‘Š) βŠ† (normCVβ€˜π‘ˆ)))))
98simplbda 500 . . . 4 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ 𝐻) β†’ (( +𝑣 β€˜π‘Š) βŠ† ( +𝑣 β€˜π‘ˆ) ∧ ( ·𝑠OLD β€˜π‘Š) βŠ† ( ·𝑠OLD β€˜π‘ˆ) ∧ (normCVβ€˜π‘Š) βŠ† (normCVβ€˜π‘ˆ)))
109simp1d 1142 . . 3 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ 𝐻) β†’ ( +𝑣 β€˜π‘Š) βŠ† ( +𝑣 β€˜π‘ˆ))
11 rnss 5936 . . 3 (( +𝑣 β€˜π‘Š) βŠ† ( +𝑣 β€˜π‘ˆ) β†’ ran ( +𝑣 β€˜π‘Š) βŠ† ran ( +𝑣 β€˜π‘ˆ))
1210, 11syl 17 . 2 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ 𝐻) β†’ ran ( +𝑣 β€˜π‘Š) βŠ† ran ( +𝑣 β€˜π‘ˆ))
13 sspba.y . . 3 π‘Œ = (BaseSetβ€˜π‘Š)
1413, 2bafval 29844 . 2 π‘Œ = ran ( +𝑣 β€˜π‘Š)
15 sspba.x . . 3 𝑋 = (BaseSetβ€˜π‘ˆ)
1615, 1bafval 29844 . 2 𝑋 = ran ( +𝑣 β€˜π‘ˆ)
1712, 14, 163sstr4g 4026 1 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ 𝐻) β†’ π‘Œ βŠ† 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   βŠ† wss 3947  ran crn 5676  β€˜cfv 6540  NrmCVeccnv 29824   +𝑣 cpv 29825  BaseSetcba 29826   ·𝑠OLD cns 29827  normCVcnmcv 29830  SubSpcss 29961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546  df-fv 6548  df-oprab 7409  df-1st 7971  df-2nd 7972  df-vc 29799  df-nv 29832  df-va 29835  df-ba 29836  df-sm 29837  df-nmcv 29840  df-ssp 29962
This theorem is referenced by:  sspg  29968  ssps  29970  sspmlem  29972  sspmval  29973  sspz  29975  sspn  29976  sspimsval  29978  minvecolem1  30114  minvecolem2  30115  minvecolem3  30116  minvecolem4b  30118  minvecolem4  30120  minvecolem5  30121  minvecolem6  30122  minvecolem7  30123
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