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Theorem sspba 29711
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 2737 . . . . . 6 ( +𝑣 β€˜π‘ˆ) = ( +𝑣 β€˜π‘ˆ)
2 eqid 2737 . . . . . 6 ( +𝑣 β€˜π‘Š) = ( +𝑣 β€˜π‘Š)
3 eqid 2737 . . . . . 6 ( ·𝑠OLD β€˜π‘ˆ) = ( ·𝑠OLD β€˜π‘ˆ)
4 eqid 2737 . . . . . 6 ( ·𝑠OLD β€˜π‘Š) = ( ·𝑠OLD β€˜π‘Š)
5 eqid 2737 . . . . . 6 (normCVβ€˜π‘ˆ) = (normCVβ€˜π‘ˆ)
6 eqid 2737 . . . . . 6 (normCVβ€˜π‘Š) = (normCVβ€˜π‘Š)
7 sspba.h . . . . . 6 𝐻 = (SubSpβ€˜π‘ˆ)
81, 2, 3, 4, 5, 6, 7isssp 29708 . . . . 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 5899 . . 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 29588 . 2 π‘Œ = ran ( +𝑣 β€˜π‘Š)
15 sspba.x . . 3 𝑋 = (BaseSetβ€˜π‘ˆ)
1615, 1bafval 29588 . 2 𝑋 = ran ( +𝑣 β€˜π‘ˆ)
1712, 14, 163sstr4g 3994 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 3915  ran crn 5639  β€˜cfv 6501  NrmCVeccnv 29568   +𝑣 cpv 29569  BaseSetcba 29570   ·𝑠OLD cns 29571  normCVcnmcv 29574  SubSpcss 29705
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 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fo 6507  df-fv 6509  df-oprab 7366  df-1st 7926  df-2nd 7927  df-vc 29543  df-nv 29576  df-va 29579  df-ba 29580  df-sm 29581  df-nmcv 29584  df-ssp 29706
This theorem is referenced by:  sspg  29712  ssps  29714  sspmlem  29716  sspmval  29717  sspz  29719  sspn  29720  sspimsval  29722  minvecolem1  29858  minvecolem2  29859  minvecolem3  29860  minvecolem4b  29862  minvecolem4  29864  minvecolem5  29865  minvecolem6  29866  minvecolem7  29867
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