Theorem sspba 29290

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 ∧ 𝑊 ∈ 𝐻) → 𝑌 ⊆ 𝑋)

Proof of Theorem sspba

Step	Hyp	Ref	Expression
1		eqid 2736	. . . . . 6 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
2		eqid 2736	. . . . . 6 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊)
3		eqid 2736	. . . . . 6 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
4		eqid 2736	. . . . . 6 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊)
5		eqid 2736	. . . . . 6 ⊢ (normCV‘𝑈) = (normCV‘𝑈)
6		eqid 2736	. . . . . 6 ⊢ (normCV‘𝑊) = (normCV‘𝑊)
7		sspba.h	. . . . . 6 ⊢ 𝐻 = (SubSp‘𝑈)
8	1, 2, 3, 4, 5, 6, 7	isssp 29287	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑊) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑊) ⊆ (normCV‘𝑈)))))
9	8	simplbda 500	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑊) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑊) ⊆ (normCV‘𝑈)))
10	9	simp1d 1141	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → ( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈))
11		rnss 5874	. . 3 ⊢ (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) → ran ( +𝑣 ‘𝑊) ⊆ ran ( +𝑣 ‘𝑈))
12	10, 11	syl 17	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → ran ( +𝑣 ‘𝑊) ⊆ ran ( +𝑣 ‘𝑈))
13		sspba.y	. . 3 ⊢ 𝑌 = (BaseSet‘𝑊)
14	13, 2	bafval 29167	. 2 ⊢ 𝑌 = ran ( +𝑣 ‘𝑊)
15		sspba.x	. . 3 ⊢ 𝑋 = (BaseSet‘𝑈)
16	15, 1	bafval 29167	. 2 ⊢ 𝑋 = ran ( +𝑣 ‘𝑈)
17	12, 14, 16	3sstr4g 3976	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑌 ⊆ 𝑋)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   ⊆ wss 3897  ran crn 5615  ‘cfv 6473  NrmCVeccnv 29147   +_𝑣 cpv 29148  BaseSetcba 29149   ·_𝑠OLD cns 29150  norm_CVcnmcv 29153  SubSpcss 29284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-br 5090  df-opab 5152  df-mpt 5173  df-id 5512  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-fo 6479  df-fv 6481  df-oprab 7333  df-1st 7891  df-2nd 7892  df-vc 29122  df-nv 29155  df-va 29158  df-ba 29159  df-sm 29160  df-nmcv 29163  df-ssp 29285

This theorem is referenced by:  sspg  29291  ssps  29293  sspmlem  29295  sspmval  29296  sspz  29298  sspn  29299  sspimsval  29301  minvecolem1  29437  minvecolem2  29438  minvecolem3  29439  minvecolem4b  29441  minvecolem4  29443  minvecolem5  29444  minvecolem6  29445  minvecolem7  29446