MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sspba Structured version   Visualization version   GIF version
Theorem sspba 29980
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 2733 . . . . . 6 ( +𝑣 β€˜π‘ˆ) = ( +𝑣 β€˜π‘ˆ)
2 eqid 2733 . . . . . 6 ( +𝑣 β€˜π‘Š) = ( +𝑣 β€˜π‘Š)
3 eqid 2733 . . . . . 6 ( ·𝑠OLD β€˜π‘ˆ) = ( ·𝑠OLD β€˜π‘ˆ)
4 eqid 2733 . . . . . 6 ( ·𝑠OLD β€˜π‘Š) = ( ·𝑠OLD β€˜π‘Š)
5 eqid 2733 . . . . . 6 (normCVβ€˜π‘ˆ) = (normCVβ€˜π‘ˆ)
6 eqid 2733 . . . . . 6 (normCVβ€˜π‘Š) = (normCVβ€˜π‘Š)
7 sspba.h . . . . . 6 𝐻 = (SubSpβ€˜π‘ˆ)
81, 2, 3, 4, 5, 6, 7isssp 29977 . . . . 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 501 . . . 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 1143 . . 3 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ 𝐻) β†’ ( +𝑣 β€˜π‘Š) βŠ† ( +𝑣 β€˜π‘ˆ))
11 rnss 5939 . . 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 29857 . 2 π‘Œ = ran ( +𝑣 β€˜π‘Š)
15 sspba.x . . 3 𝑋 = (BaseSetβ€˜π‘ˆ)
1615, 1bafval 29857 . 2 𝑋 = ran ( +𝑣 β€˜π‘ˆ)
1712, 14, 163sstr4g 4028 1 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ 𝐻) β†’ π‘Œ βŠ† 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   βŠ† wss 3949  ran crn 5678  β€˜cfv 6544  NrmCVeccnv 29837   +𝑣 cpv 29838  BaseSetcba 29839   ·𝑠OLD cns 29840  normCVcnmcv 29843  SubSpcss 29974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-oprab 7413  df-1st 7975  df-2nd 7976  df-vc 29812  df-nv 29845  df-va 29848  df-ba 29849  df-sm 29850  df-nmcv 29853  df-ssp 29975
This theorem is referenced by:  sspg  29981  ssps  29983  sspmlem  29985  sspmval  29986  sspz  29988  sspn  29989  sspimsval  29991  minvecolem1  30127  minvecolem2  30128  minvecolem3  30129  minvecolem4b  30131  minvecolem4  30133  minvecolem5  30134  minvecolem6  30135  minvecolem7  30136
  Copyright terms: Public domain W3C validator