Theorem sspval 26962

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 6103	. . . . . . 7 ⊢ (𝑢 = 𝑈 → ( +𝑣 ‘𝑢) = ( +𝑣 ‘𝑈))
3		sspval.g	. . . . . . 7 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
4	2, 3	syl6eqr 2662	. . . . . 6 ⊢ (𝑢 = 𝑈 → ( +𝑣 ‘𝑢) = 𝐺)
5	4	sseq2d 3596	. . . . 5 ⊢ (𝑢 = 𝑈 → (( +𝑣 ‘𝑤) ⊆ ( +𝑣 ‘𝑢) ↔ ( +𝑣 ‘𝑤) ⊆ 𝐺))
6		fveq2 6103	. . . . . . 7 ⊢ (𝑢 = 𝑈 → ( ·𝑠OLD ‘𝑢) = ( ·𝑠OLD ‘𝑈))
7		sspval.s	. . . . . . 7 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
8	6, 7	syl6eqr 2662	. . . . . 6 ⊢ (𝑢 = 𝑈 → ( ·𝑠OLD ‘𝑢) = 𝑆)
9	8	sseq2d 3596	. . . . 5 ⊢ (𝑢 = 𝑈 → (( ·𝑠OLD ‘𝑤) ⊆ ( ·𝑠OLD ‘𝑢) ↔ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆))
10		fveq2 6103	. . . . . . 7 ⊢ (𝑢 = 𝑈 → (normCV‘𝑢) = (normCV‘𝑈))
11		sspval.n	. . . . . . 7 ⊢ 𝑁 = (normCV‘𝑈)
12	10, 11	syl6eqr 2662	. . . . . 6 ⊢ (𝑢 = 𝑈 → (normCV‘𝑢) = 𝑁)
13	12	sseq2d 3596	. . . . 5 ⊢ (𝑢 = 𝑈 → ((normCV‘𝑤) ⊆ (normCV‘𝑢) ↔ (normCV‘𝑤) ⊆ 𝑁))
14	5, 9, 13	3anbi123d 1391	. . . 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 3164	. . 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 26961	. . 3 ⊢ SubSp = (𝑢 ∈ NrmCVec ↦ {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ ( +𝑣 ‘𝑢) ∧ ( ·𝑠OLD ‘𝑤) ⊆ ( ·𝑠OLD ‘𝑢) ∧ (normCV‘𝑤) ⊆ (normCV‘𝑢))})
17		fvex 6113	. . . . . . . 8 ⊢ ( +𝑣 ‘𝑈) ∈ V
18	3, 17	eqeltri 2684	. . . . . . 7 ⊢ 𝐺 ∈ V
19	18	pwex 4774	. . . . . 6 ⊢ 𝒫 𝐺 ∈ V
20		fvex 6113	. . . . . . . 8 ⊢ ( ·𝑠OLD ‘𝑈) ∈ V
21	7, 20	eqeltri 2684	. . . . . . 7 ⊢ 𝑆 ∈ V
22	21	pwex 4774	. . . . . 6 ⊢ 𝒫 𝑆 ∈ V
23	19, 22	xpex 6860	. . . . 5 ⊢ (𝒫 𝐺 × 𝒫 𝑆) ∈ V
24		fvex 6113	. . . . . . 7 ⊢ (normCV‘𝑈) ∈ V
25	11, 24	eqeltri 2684	. . . . . 6 ⊢ 𝑁 ∈ V
26	25	pwex 4774	. . . . 5 ⊢ 𝒫 𝑁 ∈ V
27	23, 26	xpex 6860	. . . 4 ⊢ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁) ∈ V
28		rabss 3642	. . . . 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)))
29		fvex 6113	. . . . . . . . . 10 ⊢ ( +𝑣 ‘𝑤) ∈ V
30	29	elpw 4114	. . . . . . . . 9 ⊢ (( +𝑣 ‘𝑤) ∈ 𝒫 𝐺 ↔ ( +𝑣 ‘𝑤) ⊆ 𝐺)
31		fvex 6113	. . . . . . . . . 10 ⊢ ( ·𝑠OLD ‘𝑤) ∈ V
32	31	elpw 4114	. . . . . . . . 9 ⊢ (( ·𝑠OLD ‘𝑤) ∈ 𝒫 𝑆 ↔ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆)
33		opelxpi 5072	. . . . . . . . 9 ⊢ ((( +𝑣 ‘𝑤) ∈ 𝒫 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ∈ 𝒫 𝑆) → ⟨( +𝑣 ‘𝑤), ( ·𝑠OLD ‘𝑤)⟩ ∈ (𝒫 𝐺 × 𝒫 𝑆))
34	30, 32, 33	syl2anbr 496	. . . . . . . 8 ⊢ ((( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆) → ⟨( +𝑣 ‘𝑤), ( ·𝑠OLD ‘𝑤)⟩ ∈ (𝒫 𝐺 × 𝒫 𝑆))
35		fvex 6113	. . . . . . . . . 10 ⊢ (normCV‘𝑤) ∈ V
36	35	elpw 4114	. . . . . . . . 9 ⊢ ((normCV‘𝑤) ∈ 𝒫 𝑁 ↔ (normCV‘𝑤) ⊆ 𝑁)
37	36	biimpri 217	. . . . . . . 8 ⊢ ((normCV‘𝑤) ⊆ 𝑁 → (normCV‘𝑤) ∈ 𝒫 𝑁)
38		opelxpi 5072	. . . . . . . 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))
39	34, 37, 38	syl2an 493	. . . . . . 7 ⊢ (((( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆) ∧ (normCV‘𝑤) ⊆ 𝑁) → ⟨⟨( +𝑣 ‘𝑤), ( ·𝑠OLD ‘𝑤)⟩, (normCV‘𝑤)⟩ ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁))
40	39	3impa 1251	. . . . . 6 ⊢ ((( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁) → ⟨⟨( +𝑣 ‘𝑤), ( ·𝑠OLD ‘𝑤)⟩, (normCV‘𝑤)⟩ ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁))
41		eqid 2610	. . . . . . . 8 ⊢ ( +𝑣 ‘𝑤) = ( +𝑣 ‘𝑤)
42		eqid 2610	. . . . . . . 8 ⊢ ( ·𝑠OLD ‘𝑤) = ( ·𝑠OLD ‘𝑤)
43		eqid 2610	. . . . . . . 8 ⊢ (normCV‘𝑤) = (normCV‘𝑤)
44	41, 42, 43	nvop 26915	. . . . . . 7 ⊢ (𝑤 ∈ NrmCVec → 𝑤 = ⟨⟨( +𝑣 ‘𝑤), ( ·𝑠OLD ‘𝑤)⟩, (normCV‘𝑤)⟩)
45	44	eleq1d 2672	. . . . . 6 ⊢ (𝑤 ∈ NrmCVec → (𝑤 ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁) ↔ ⟨⟨( +𝑣 ‘𝑤), ( ·𝑠OLD ‘𝑤)⟩, (normCV‘𝑤)⟩ ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁)))
46	40, 45	syl5ibr 235	. . . . 5 ⊢ (𝑤 ∈ NrmCVec → ((( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁) → 𝑤 ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁)))
47	28, 46	mprgbir 2911	. . . 4 ⊢ {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)} ⊆ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁)
48	27, 47	ssexi 4731	. . 3 ⊢ {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)} ∈ V
49	15, 16, 48	fvmpt 6191	. 2 ⊢ (𝑈 ∈ NrmCVec → (SubSp‘𝑈) = {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)})
50	1, 49	syl5eq 2656	1 ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)})

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  ⟨cop 4131   × cxp 5036  ‘cfv 5804  NrmCVeccnv 26823   +_𝑣 cpv 26824   ·_𝑠OLD cns 26826  norm_CVcnmcv 26829  SubSpcss 26960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-oprab 6553  df-1st 7059  df-2nd 7060  df-vc 26798  df-nv 26831  df-va 26834  df-sm 26836  df-nmcv 26839  df-ssp 26961

This theorem is referenced by:  isssp  26963