Theorem sspval 30670

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 6886	. . . . . . 7 ⊢ (𝑢 = 𝑈 → ( +𝑣 ‘𝑢) = ( +𝑣 ‘𝑈))
3		sspval.g	. . . . . . 7 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
4	2, 3	eqtr4di 2787	. . . . . 6 ⊢ (𝑢 = 𝑈 → ( +𝑣 ‘𝑢) = 𝐺)
5	4	sseq2d 3996	. . . . 5 ⊢ (𝑢 = 𝑈 → (( +𝑣 ‘𝑤) ⊆ ( +𝑣 ‘𝑢) ↔ ( +𝑣 ‘𝑤) ⊆ 𝐺))
6		fveq2 6886	. . . . . . 7 ⊢ (𝑢 = 𝑈 → ( ·𝑠OLD ‘𝑢) = ( ·𝑠OLD ‘𝑈))
7		sspval.s	. . . . . . 7 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
8	6, 7	eqtr4di 2787	. . . . . 6 ⊢ (𝑢 = 𝑈 → ( ·𝑠OLD ‘𝑢) = 𝑆)
9	8	sseq2d 3996	. . . . 5 ⊢ (𝑢 = 𝑈 → (( ·𝑠OLD ‘𝑤) ⊆ ( ·𝑠OLD ‘𝑢) ↔ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆))
10		fveq2 6886	. . . . . . 7 ⊢ (𝑢 = 𝑈 → (normCV‘𝑢) = (normCV‘𝑈))
11		sspval.n	. . . . . . 7 ⊢ 𝑁 = (normCV‘𝑈)
12	10, 11	eqtr4di 2787	. . . . . 6 ⊢ (𝑢 = 𝑈 → (normCV‘𝑢) = 𝑁)
13	12	sseq2d 3996	. . . . 5 ⊢ (𝑢 = 𝑈 → ((normCV‘𝑤) ⊆ (normCV‘𝑢) ↔ (normCV‘𝑤) ⊆ 𝑁))
14	5, 9, 13	3anbi123d 1437	. . . 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 3427	. . 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 30669	. . 3 ⊢ SubSp = (𝑢 ∈ NrmCVec ↦ {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ ( +𝑣 ‘𝑢) ∧ ( ·𝑠OLD ‘𝑤) ⊆ ( ·𝑠OLD ‘𝑢) ∧ (normCV‘𝑤) ⊆ (normCV‘𝑢))})
17	3	fvexi 6900	. . . . . . 7 ⊢ 𝐺 ∈ V
18	17	pwex 5360	. . . . . 6 ⊢ 𝒫 𝐺 ∈ V
19	7	fvexi 6900	. . . . . . 7 ⊢ 𝑆 ∈ V
20	19	pwex 5360	. . . . . 6 ⊢ 𝒫 𝑆 ∈ V
21	18, 20	xpex 7755	. . . . 5 ⊢ (𝒫 𝐺 × 𝒫 𝑆) ∈ V
22	11	fvexi 6900	. . . . . 6 ⊢ 𝑁 ∈ V
23	22	pwex 5360	. . . . 5 ⊢ 𝒫 𝑁 ∈ V
24	21, 23	xpex 7755	. . . 4 ⊢ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁) ∈ V
25		rabss 4052	. . . . 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 6899	. . . . . . . . . 10 ⊢ ( +𝑣 ‘𝑤) ∈ V
27	26	elpw 4584	. . . . . . . . 9 ⊢ (( +𝑣 ‘𝑤) ∈ 𝒫 𝐺 ↔ ( +𝑣 ‘𝑤) ⊆ 𝐺)
28		fvex 6899	. . . . . . . . . 10 ⊢ ( ·𝑠OLD ‘𝑤) ∈ V
29	28	elpw 4584	. . . . . . . . 9 ⊢ (( ·𝑠OLD ‘𝑤) ∈ 𝒫 𝑆 ↔ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆)
30		opelxpi 5702	. . . . . . . . 9 ⊢ ((( +𝑣 ‘𝑤) ∈ 𝒫 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ∈ 𝒫 𝑆) → ⟨( +𝑣 ‘𝑤), ( ·𝑠OLD ‘𝑤)⟩ ∈ (𝒫 𝐺 × 𝒫 𝑆))
31	27, 29, 30	syl2anbr 599	. . . . . . . 8 ⊢ ((( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆) → ⟨( +𝑣 ‘𝑤), ( ·𝑠OLD ‘𝑤)⟩ ∈ (𝒫 𝐺 × 𝒫 𝑆))
32		fvex 6899	. . . . . . . . . 10 ⊢ (normCV‘𝑤) ∈ V
33	32	elpw 4584	. . . . . . . . 9 ⊢ ((normCV‘𝑤) ∈ 𝒫 𝑁 ↔ (normCV‘𝑤) ⊆ 𝑁)
34	33	biimpri 228	. . . . . . . 8 ⊢ ((normCV‘𝑤) ⊆ 𝑁 → (normCV‘𝑤) ∈ 𝒫 𝑁)
35		opelxpi 5702	. . . . . . . 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 596	. . . . . . 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 1109	. . . . . 6 ⊢ ((( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁) → ⟨⟨( +𝑣 ‘𝑤), ( ·𝑠OLD ‘𝑤)⟩, (normCV‘𝑤)⟩ ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁))
38		eqid 2734	. . . . . . . 8 ⊢ ( +𝑣 ‘𝑤) = ( +𝑣 ‘𝑤)
39		eqid 2734	. . . . . . . 8 ⊢ ( ·𝑠OLD ‘𝑤) = ( ·𝑠OLD ‘𝑤)
40		eqid 2734	. . . . . . . 8 ⊢ (normCV‘𝑤) = (normCV‘𝑤)
41	38, 39, 40	nvop 30623	. . . . . . 7 ⊢ (𝑤 ∈ NrmCVec → 𝑤 = ⟨⟨( +𝑣 ‘𝑤), ( ·𝑠OLD ‘𝑤)⟩, (normCV‘𝑤)⟩)
42	41	eleq1d 2818	. . . . . 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	imbitrrid 246	. . . . 5 ⊢ (𝑤 ∈ NrmCVec → ((( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁) → 𝑤 ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁)))
44	25, 43	mprgbir 3057	. . . 4 ⊢ {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)} ⊆ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁)
45	24, 44	ssexi 5302	. . 3 ⊢ {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)} ∈ V
46	15, 16, 45	fvmpt 6996	. 2 ⊢ (𝑈 ∈ NrmCVec → (SubSp‘𝑈) = {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)})
47	1, 46	eqtrid 2781	1 ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)})

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  {crab 3419   ⊆ wss 3931  𝒫 cpw 4580  ⟨cop 4612   × cxp 5663  ‘cfv 6541  NrmCVeccnv 30531   +_𝑣 cpv 30532   ·_𝑠OLD cns 30534  norm_CVcnmcv 30537  SubSpcss 30668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-fo 6547  df-fv 6549  df-oprab 7417  df-1st 7996  df-2nd 7997  df-vc 30506  df-nv 30539  df-va 30542  df-sm 30544  df-nmcv 30547  df-ssp 30669

This theorem is referenced by:  isssp  30671