Theorem sspba 28133

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 2825	. . . . . 6 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
2		eqid 2825	. . . . . 6 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊)
3		eqid 2825	. . . . . 6 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
4		eqid 2825	. . . . . 6 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊)
5		eqid 2825	. . . . . 6 ⊢ (normCV‘𝑈) = (normCV‘𝑈)
6		eqid 2825	. . . . . 6 ⊢ (normCV‘𝑊) = (normCV‘𝑊)
7		sspba.h	. . . . . 6 ⊢ 𝐻 = (SubSp‘𝑈)
8	1, 2, 3, 4, 5, 6, 7	isssp 28130	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑊) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑊) ⊆ (normCV‘𝑈)))))
9	8	simplbda 495	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑊) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑊) ⊆ (normCV‘𝑈)))
10	9	simp1d 1176	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → ( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈))
11		rnss 5590	. . 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 28010	. 2 ⊢ 𝑌 = ran ( +𝑣 ‘𝑊)
15		sspba.x	. . 3 ⊢ 𝑋 = (BaseSet‘𝑈)
16	15, 1	bafval 28010	. 2 ⊢ 𝑋 = ran ( +𝑣 ‘𝑈)
17	12, 14, 16	3sstr4g 3871	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑌 ⊆ 𝑋)

Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164   ⊆ wss 3798  ran crn 5347  ‘cfv 6127  NrmCVeccnv 27990   +_𝑣 cpv 27991  BaseSetcba 27992   ·_𝑠OLD cns 27993  norm_CVcnmcv 27996  SubSpcss 28127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fo 6133  df-fv 6135  df-oprab 6914  df-1st 7433  df-2nd 7434  df-vc 27965  df-nv 27998  df-va 28001  df-ba 28002  df-sm 28003  df-nmcv 28006  df-ssp 28128

This theorem is referenced by:  sspg  28134  ssps  28136  sspmlem  28138  sspmval  28139  sspz  28141  sspn  28142  sspimsval  28144  sspphOLD  28261  minvecolem1  28281  minvecolem2  28282  minvecolem3  28283  minvecolem4b  28285  minvecolem4  28287  minvecolem5  28288  minvecolem6  28289  minvecolem7  28290