Theorem sspba 29089

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 2738	. . . . . 6 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
2		eqid 2738	. . . . . 6 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊)
3		eqid 2738	. . . . . 6 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
4		eqid 2738	. . . . . 6 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊)
5		eqid 2738	. . . . . 6 ⊢ (normCV‘𝑈) = (normCV‘𝑈)
6		eqid 2738	. . . . . 6 ⊢ (normCV‘𝑊) = (normCV‘𝑊)
7		sspba.h	. . . . . 6 ⊢ 𝐻 = (SubSp‘𝑈)
8	1, 2, 3, 4, 5, 6, 7	isssp 29086	. . . . 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 5848	. . 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 28966	. 2 ⊢ 𝑌 = ran ( +𝑣 ‘𝑊)
15		sspba.x	. . 3 ⊢ 𝑋 = (BaseSet‘𝑈)
16	15, 1	bafval 28966	. 2 ⊢ 𝑋 = ran ( +𝑣 ‘𝑈)
17	12, 14, 16	3sstr4g 3966	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑌 ⊆ 𝑋)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ⊆ wss 3887  ran crn 5590  ‘cfv 6433  NrmCVeccnv 28946   +_𝑣 cpv 28947  BaseSetcba 28948   ·_𝑠OLD cns 28949  norm_CVcnmcv 28952  SubSpcss 29083

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441  df-oprab 7279  df-1st 7831  df-2nd 7832  df-vc 28921  df-nv 28954  df-va 28957  df-ba 28958  df-sm 28959  df-nmcv 28962  df-ssp 29084

This theorem is referenced by:  sspg  29090  ssps  29092  sspmlem  29094  sspmval  29095  sspz  29097  sspn  29098  sspimsval  29100  minvecolem1  29236  minvecolem2  29237  minvecolem3  29238  minvecolem4b  29240  minvecolem4  29242  minvecolem5  29243  minvecolem6  29244  minvecolem7  29245