Theorem sspval 28986

Description: The set of all subspaces of a normed complex vector space. (Contributed by NM, 26-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
sspval.g	⊢ 𝐺 = ( +𝑣 ‘𝑈)
sspval.s	⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
sspval.n	⊢ 𝑁 = (normCV‘𝑈)
sspval.h	⊢ 𝐻 = (SubSp‘𝑈)

Assertion

Ref	Expression
sspval	⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)})

Distinct variable groups:   𝑤,𝐺   𝑤,𝑁   𝑤,𝑆   𝑤,𝑈

Allowed substitution hint:   𝐻(𝑤)

Proof of Theorem sspval

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sspval.h	. 2 ⊢ 𝐻 = (SubSp‘𝑈)
2		fveq2 6756	. . . . . . 7 ⊢ (𝑢 = 𝑈 → ( +𝑣 ‘𝑢) = ( +𝑣 ‘𝑈))
3		sspval.g	. . . . . . 7 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
4	2, 3	eqtr4di 2797	. . . . . 6 ⊢ (𝑢 = 𝑈 → ( +𝑣 ‘𝑢) = 𝐺)
5	4	sseq2d 3949	. . . . 5 ⊢ (𝑢 = 𝑈 → (( +𝑣 ‘𝑤) ⊆ ( +𝑣 ‘𝑢) ↔ ( +𝑣 ‘𝑤) ⊆ 𝐺))
6		fveq2 6756	. . . . . . 7 ⊢ (𝑢 = 𝑈 → ( ·𝑠OLD ‘𝑢) = ( ·𝑠OLD ‘𝑈))
7		sspval.s	. . . . . . 7 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
8	6, 7	eqtr4di 2797	. . . . . 6 ⊢ (𝑢 = 𝑈 → ( ·𝑠OLD ‘𝑢) = 𝑆)
9	8	sseq2d 3949	. . . . 5 ⊢ (𝑢 = 𝑈 → (( ·𝑠OLD ‘𝑤) ⊆ ( ·𝑠OLD ‘𝑢) ↔ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆))
10		fveq2 6756	. . . . . . 7 ⊢ (𝑢 = 𝑈 → (normCV‘𝑢) = (normCV‘𝑈))
11		sspval.n	. . . . . . 7 ⊢ 𝑁 = (normCV‘𝑈)
12	10, 11	eqtr4di 2797	. . . . . 6 ⊢ (𝑢 = 𝑈 → (normCV‘𝑢) = 𝑁)
13	12	sseq2d 3949	. . . . 5 ⊢ (𝑢 = 𝑈 → ((normCV‘𝑤) ⊆ (normCV‘𝑢) ↔ (normCV‘𝑤) ⊆ 𝑁))
14	5, 9, 13	3anbi123d 1434	. . . 4 ⊢ (𝑢 = 𝑈 → ((( +𝑣 ‘𝑤) ⊆ ( +𝑣 ‘𝑢) ∧ ( ·𝑠OLD ‘𝑤) ⊆ ( ·𝑠OLD ‘𝑢) ∧ (normCV‘𝑤) ⊆ (normCV‘𝑢)) ↔ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)))
15	14	rabbidv 3404	. . 3 ⊢ (𝑢 = 𝑈 → {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ ( +𝑣 ‘𝑢) ∧ ( ·𝑠OLD ‘𝑤) ⊆ ( ·𝑠OLD ‘𝑢) ∧ (normCV‘𝑤) ⊆ (normCV‘𝑢))} = {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)})
16		df-ssp 28985	. . 3 ⊢ SubSp = (𝑢 ∈ NrmCVec ↦ {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ ( +𝑣 ‘𝑢) ∧ ( ·𝑠OLD ‘𝑤) ⊆ ( ·𝑠OLD ‘𝑢) ∧ (normCV‘𝑤) ⊆ (normCV‘𝑢))})
17	3	fvexi 6770	. . . . . . 7 ⊢ 𝐺 ∈ V
18	17	pwex 5298	. . . . . 6 ⊢ 𝒫 𝐺 ∈ V
19	7	fvexi 6770	. . . . . . 7 ⊢ 𝑆 ∈ V
20	19	pwex 5298	. . . . . 6 ⊢ 𝒫 𝑆 ∈ V
21	18, 20	xpex 7581	. . . . 5 ⊢ (𝒫 𝐺 × 𝒫 𝑆) ∈ V
22	11	fvexi 6770	. . . . . 6 ⊢ 𝑁 ∈ V
23	22	pwex 5298	. . . . 5 ⊢ 𝒫 𝑁 ∈ V
24	21, 23	xpex 7581	. . . 4 ⊢ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁) ∈ V
25		rabss 4001	. . . . 5 ⊢ ({𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)} ⊆ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁) ↔ ∀𝑤 ∈ NrmCVec ((( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁) → 𝑤 ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁)))
26		fvex 6769	. . . . . . . . . 10 ⊢ ( +𝑣 ‘𝑤) ∈ V
27	26	elpw 4534	. . . . . . . . 9 ⊢ (( +𝑣 ‘𝑤) ∈ 𝒫 𝐺 ↔ ( +𝑣 ‘𝑤) ⊆ 𝐺)
28		fvex 6769	. . . . . . . . . 10 ⊢ ( ·𝑠OLD ‘𝑤) ∈ V
29	28	elpw 4534	. . . . . . . . 9 ⊢ (( ·𝑠OLD ‘𝑤) ∈ 𝒫 𝑆 ↔ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆)
30		opelxpi 5617	. . . . . . . . 9 ⊢ ((( +𝑣 ‘𝑤) ∈ 𝒫 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ∈ 𝒫 𝑆) → ⟨( +𝑣 ‘𝑤), ( ·𝑠OLD ‘𝑤)⟩ ∈ (𝒫 𝐺 × 𝒫 𝑆))
31	27, 29, 30	syl2anbr 598	. . . . . . . 8 ⊢ ((( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆) → ⟨( +𝑣 ‘𝑤), ( ·𝑠OLD ‘𝑤)⟩ ∈ (𝒫 𝐺 × 𝒫 𝑆))
32		fvex 6769	. . . . . . . . . 10 ⊢ (normCV‘𝑤) ∈ V
33	32	elpw 4534	. . . . . . . . 9 ⊢ ((normCV‘𝑤) ∈ 𝒫 𝑁 ↔ (normCV‘𝑤) ⊆ 𝑁)
34	33	biimpri 227	. . . . . . . 8 ⊢ ((normCV‘𝑤) ⊆ 𝑁 → (normCV‘𝑤) ∈ 𝒫 𝑁)
35		opelxpi 5617	. . . . . . . 8 ⊢ ((⟨( +𝑣 ‘𝑤), ( ·𝑠OLD ‘𝑤)⟩ ∈ (𝒫 𝐺 × 𝒫 𝑆) ∧ (normCV‘𝑤) ∈ 𝒫 𝑁) → ⟨⟨( +𝑣 ‘𝑤), ( ·𝑠OLD ‘𝑤)⟩, (normCV‘𝑤)⟩ ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁))
36	31, 34, 35	syl2an 595	. . . . . . 7 ⊢ (((( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆) ∧ (normCV‘𝑤) ⊆ 𝑁) → ⟨⟨( +𝑣 ‘𝑤), ( ·𝑠OLD ‘𝑤)⟩, (normCV‘𝑤)⟩ ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁))
37	36	3impa 1108	. . . . . 6 ⊢ ((( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁) → ⟨⟨( +𝑣 ‘𝑤), ( ·𝑠OLD ‘𝑤)⟩, (normCV‘𝑤)⟩ ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁))
38		eqid 2738	. . . . . . . 8 ⊢ ( +𝑣 ‘𝑤) = ( +𝑣 ‘𝑤)
39		eqid 2738	. . . . . . . 8 ⊢ ( ·𝑠OLD ‘𝑤) = ( ·𝑠OLD ‘𝑤)
40		eqid 2738	. . . . . . . 8 ⊢ (normCV‘𝑤) = (normCV‘𝑤)
41	38, 39, 40	nvop 28939	. . . . . . 7 ⊢ (𝑤 ∈ NrmCVec → 𝑤 = ⟨⟨( +𝑣 ‘𝑤), ( ·𝑠OLD ‘𝑤)⟩, (normCV‘𝑤)⟩)
42	41	eleq1d 2823	. . . . . 6 ⊢ (𝑤 ∈ NrmCVec → (𝑤 ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁) ↔ ⟨⟨( +𝑣 ‘𝑤), ( ·𝑠OLD ‘𝑤)⟩, (normCV‘𝑤)⟩ ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁)))
43	37, 42	syl5ibr 245	. . . . 5 ⊢ (𝑤 ∈ NrmCVec → ((( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁) → 𝑤 ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁)))
44	25, 43	mprgbir 3078	. . . 4 ⊢ {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)} ⊆ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁)
45	24, 44	ssexi 5241	. . 3 ⊢ {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)} ∈ V
46	15, 16, 45	fvmpt 6857	. 2 ⊢ (𝑈 ∈ NrmCVec → (SubSp‘𝑈) = {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)})
47	1, 46	syl5eq 2791	1 ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)})

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  {crab 3067   ⊆ wss 3883  𝒫 cpw 4530  ⟨cop 4564   × cxp 5578  ‘cfv 6418  NrmCVeccnv 28847   +_𝑣 cpv 28848   ·_𝑠OLD cns 28850  norm_CVcnmcv 28853  SubSpcss 28984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fo 6424  df-fv 6426  df-oprab 7259  df-1st 7804  df-2nd 7805  df-vc 28822  df-nv 28855  df-va 28858  df-sm 28860  df-nmcv 28863  df-ssp 28985

This theorem is referenced by:  isssp  28987