Theorem isssp 31333

Description: The predicate "is a subspace." (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)

Hypotheses

Ref	Expression
isssp.g	⊢ 𝐺 = ( +𝑣 ‘𝑈)
isssp.f	⊢ 𝐹 = ( +𝑣 ‘𝑊)
isssp.s	⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
isssp.r	⊢ 𝑅 = ( ·𝑠OLD ‘𝑊)
isssp.n	⊢ 𝑁 = (normCV‘𝑈)
isssp.m	⊢ 𝑀 = (normCV‘𝑊)
isssp.h	⊢ 𝐻 = (SubSp‘𝑈)

Assertion

Ref	Expression
isssp	⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (𝐹 ⊆ 𝐺 ∧ 𝑅 ⊆ 𝑆 ∧ 𝑀 ⊆ 𝑁))))

Proof of Theorem isssp

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isssp.g	. . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
2		isssp.s	. . . 4 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
3		isssp.n	. . . 4 ⊢ 𝑁 = (normCV‘𝑈)
4		isssp.h	. . . 4 ⊢ 𝐻 = (SubSp‘𝑈)
5	1, 2, 3, 4	sspval 31332	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)})
6	5	eleq2d 2846	. 2 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ 𝑊 ∈ {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)}))
7		fveq2 6888	. . . . . 6 ⊢ (𝑤 = 𝑊 → ( +𝑣 ‘𝑤) = ( +𝑣 ‘𝑊))
8		isssp.f	. . . . . 6 ⊢ 𝐹 = ( +𝑣 ‘𝑊)
9	7, 8	eqtr4di 2813	. . . . 5 ⊢ (𝑤 = 𝑊 → ( +𝑣 ‘𝑤) = 𝐹)
10	9	sseq1d 3940	. . . 4 ⊢ (𝑤 = 𝑊 → (( +𝑣 ‘𝑤) ⊆ 𝐺 ↔ 𝐹 ⊆ 𝐺))
11		fveq2 6888	. . . . . 6 ⊢ (𝑤 = 𝑊 → ( ·𝑠OLD ‘𝑤) = ( ·𝑠OLD ‘𝑊))
12		isssp.r	. . . . . 6 ⊢ 𝑅 = ( ·𝑠OLD ‘𝑊)
13	11, 12	eqtr4di 2813	. . . . 5 ⊢ (𝑤 = 𝑊 → ( ·𝑠OLD ‘𝑤) = 𝑅)
14	13	sseq1d 3940	. . . 4 ⊢ (𝑤 = 𝑊 → (( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ↔ 𝑅 ⊆ 𝑆))
15		fveq2 6888	. . . . . 6 ⊢ (𝑤 = 𝑊 → (normCV‘𝑤) = (normCV‘𝑊))
16		isssp.m	. . . . . 6 ⊢ 𝑀 = (normCV‘𝑊)
17	15, 16	eqtr4di 2813	. . . . 5 ⊢ (𝑤 = 𝑊 → (normCV‘𝑤) = 𝑀)
18	17	sseq1d 3940	. . . 4 ⊢ (𝑤 = 𝑊 → ((normCV‘𝑤) ⊆ 𝑁 ↔ 𝑀 ⊆ 𝑁))
19	10, 14, 18	3anbi123d 1464	. . 3 ⊢ (𝑤 = 𝑊 → ((( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁) ↔ (𝐹 ⊆ 𝐺 ∧ 𝑅 ⊆ 𝑆 ∧ 𝑀 ⊆ 𝑁)))
20	19	elrab 3644	. 2 ⊢ (𝑊 ∈ {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)} ↔ (𝑊 ∈ NrmCVec ∧ (𝐹 ⊆ 𝐺 ∧ 𝑅 ⊆ 𝑆 ∧ 𝑀 ⊆ 𝑁)))
21	6, 20	bitrdi 290	1 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (𝐹 ⊆ 𝐺 ∧ 𝑅 ⊆ 𝑆 ∧ 𝑀 ⊆ 𝑁))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  {crab 3412   ⊆ wss 3896  ‘cfv 6541  NrmCVeccnv 31193   +_𝑣 cpv 31194   ·_𝑠OLD cns 31196  norm_CVcnmcv 31199  SubSpcss 31330

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 5326  ax-pr 5390  ax-un 7753

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 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-fo 6547  df-fv 6549  df-oprab 7426  df-1st 8003  df-2nd 8004  df-vc 31168  df-nv 31201  df-va 31204  df-sm 31206  df-nmcv 31209  df-ssp 31331

This theorem is used by:  sspid  31334  sspnv  31335  sspba  31336  sspg  31337  ssps  31339  sspn  31345  hhsst  31875  hhsssh2  31879