Theorem sspba 30930

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 2762	. . . . . 6 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
2		eqid 2762	. . . . . 6 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊)
3		eqid 2762	. . . . . 6 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
4		eqid 2762	. . . . . 6 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊)
5		eqid 2762	. . . . . 6 ⊢ (normCV‘𝑈) = (normCV‘𝑈)
6		eqid 2762	. . . . . 6 ⊢ (normCV‘𝑊) = (normCV‘𝑊)
7		sspba.h	. . . . . 6 ⊢ 𝐻 = (SubSp‘𝑈)
8	1, 2, 3, 4, 5, 6, 7	isssp 30927	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑊) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑊) ⊆ (normCV‘𝑈)))))
9	8	simplbda 503	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑊) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑊) ⊆ (normCV‘𝑈)))
10	9	simp1d 1155	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → ( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈))
11		rnss 5915	. . 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 30807	. 2 ⊢ 𝑌 = ran ( +𝑣 ‘𝑊)
15		sspba.x	. . 3 ⊢ 𝑋 = (BaseSet‘𝑈)
16	15, 1	bafval 30807	. 2 ⊢ 𝑋 = ran ( +𝑣 ‘𝑈)
17	12, 14, 16	3sstr4g 3989	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑌 ⊆ 𝑋)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142   ⊆ wss 3904  ran crn 5648  ‘cfv 6521  NrmCVeccnv 30787   +_𝑣 cpv 30788  BaseSetcba 30789   ·_𝑠OLD cns 30790  norm_CVcnmcv 30793  SubSpcss 30924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fo 6527  df-fv 6529  df-oprab 7400  df-1st 7970  df-2nd 7971  df-vc 30762  df-nv 30795  df-va 30798  df-ba 30799  df-sm 30800  df-nmcv 30803  df-ssp 30925

This theorem is referenced by:  sspg  30931  ssps  30933  sspmlem  30935  sspmval  30936  sspz  30938  sspn  30939  sspimsval  30941  minvecolem1  31077  minvecolem2  31078  minvecolem3  31079  minvecolem4b  31081  minvecolem4  31083  minvecolem5  31084  minvecolem6  31085  minvecolem7  31086