Theorem sspba 31338

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 2760	. . . . . 6 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
2		eqid 2760	. . . . . 6 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊)
3		eqid 2760	. . . . . 6 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
4		eqid 2760	. . . . . 6 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊)
5		eqid 2760	. . . . . 6 ⊢ (normCV‘𝑈) = (normCV‘𝑈)
6		eqid 2760	. . . . . 6 ⊢ (normCV‘𝑊) = (normCV‘𝑊)
7		sspba.h	. . . . . 6 ⊢ 𝐻 = (SubSp‘𝑈)
8	1, 2, 3, 4, 5, 6, 7	isssp 31335	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑊) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑊) ⊆ (normCV‘𝑈)))))
9	8	simplbda 505	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑊) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑊) ⊆ (normCV‘𝑈)))
10	9	simp1d 1160	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → ( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈))
11		rnss 5922	. . 3 ⊢ (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) → ran ( +𝑣 ‘𝑊) ⊆ ran ( +𝑣 ‘𝑈))
12	10, 11	syl 18	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → ran ( +𝑣 ‘𝑊) ⊆ ran ( +𝑣 ‘𝑈))
13		sspba.y	. . 3 ⊢ 𝑌 = (BaseSet‘𝑊)
14	13, 2	bafval 31215	. 2 ⊢ 𝑌 = ran ( +𝑣 ‘𝑊)
15		sspba.x	. . 3 ⊢ 𝑋 = (BaseSet‘𝑈)
16	15, 1	bafval 31215	. 2 ⊢ 𝑋 = ran ( +𝑣 ‘𝑈)
17	12, 14, 16	3sstr4g 3962	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑌 ⊆ 𝑋)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ⊆ wss 3896  ran crn 5652  ‘cfv 6543  NrmCVeccnv 31195   +_𝑣 cpv 31196  BaseSetcba 31197   ·_𝑠OLD cns 31198  norm_CVcnmcv 31201  SubSpcss 31332

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-oprab 7428  df-1st 8005  df-2nd 8006  df-vc 31170  df-nv 31203  df-va 31206  df-ba 31207  df-sm 31208  df-nmcv 31211  df-ssp 31333

This theorem is used by:  sspg  31339  ssps  31341  sspmlem  31343  sspmval  31344  sspz  31346  sspn  31347  sspimsval  31349  minvecolem1  31485  minvecolem2  31486  minvecolem3  31487  minvecolem4b  31489  minvecolem4  31491  minvecolem5  31492  minvecolem6  31493  minvecolem7  31494