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Theorem sspval 30551
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 = ( +𝑣𝑈)
sspval.s 𝑆 = ( ·𝑠OLD𝑈)
sspval.n 𝑁 = (normCV𝑈)
sspval.h 𝐻 = (SubSp‘𝑈)
Assertion
Ref Expression
sspval (𝑈 ∈ NrmCVec → 𝐻 = {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 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.
StepHypRef Expression
1 sspval.h . 2 𝐻 = (SubSp‘𝑈)
2 fveq2 6900 . . . . . . 7 (𝑢 = 𝑈 → ( +𝑣𝑢) = ( +𝑣𝑈))
3 sspval.g . . . . . . 7 𝐺 = ( +𝑣𝑈)
42, 3eqtr4di 2785 . . . . . 6 (𝑢 = 𝑈 → ( +𝑣𝑢) = 𝐺)
54sseq2d 4012 . . . . 5 (𝑢 = 𝑈 → (( +𝑣𝑤) ⊆ ( +𝑣𝑢) ↔ ( +𝑣𝑤) ⊆ 𝐺))
6 fveq2 6900 . . . . . . 7 (𝑢 = 𝑈 → ( ·𝑠OLD𝑢) = ( ·𝑠OLD𝑈))
7 sspval.s . . . . . . 7 𝑆 = ( ·𝑠OLD𝑈)
86, 7eqtr4di 2785 . . . . . 6 (𝑢 = 𝑈 → ( ·𝑠OLD𝑢) = 𝑆)
98sseq2d 4012 . . . . 5 (𝑢 = 𝑈 → (( ·𝑠OLD𝑤) ⊆ ( ·𝑠OLD𝑢) ↔ ( ·𝑠OLD𝑤) ⊆ 𝑆))
10 fveq2 6900 . . . . . . 7 (𝑢 = 𝑈 → (normCV𝑢) = (normCV𝑈))
11 sspval.n . . . . . . 7 𝑁 = (normCV𝑈)
1210, 11eqtr4di 2785 . . . . . 6 (𝑢 = 𝑈 → (normCV𝑢) = 𝑁)
1312sseq2d 4012 . . . . 5 (𝑢 = 𝑈 → ((normCV𝑤) ⊆ (normCV𝑢) ↔ (normCV𝑤) ⊆ 𝑁))
145, 9, 133anbi123d 1432 . . . 4 (𝑢 = 𝑈 → ((( +𝑣𝑤) ⊆ ( +𝑣𝑢) ∧ ( ·𝑠OLD𝑤) ⊆ ( ·𝑠OLD𝑢) ∧ (normCV𝑤) ⊆ (normCV𝑢)) ↔ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)))
1514rabbidv 3436 . . 3 (𝑢 = 𝑈 → {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ ( +𝑣𝑢) ∧ ( ·𝑠OLD𝑤) ⊆ ( ·𝑠OLD𝑢) ∧ (normCV𝑤) ⊆ (normCV𝑢))} = {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)})
16 df-ssp 30550 . . 3 SubSp = (𝑢 ∈ NrmCVec ↦ {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ ( +𝑣𝑢) ∧ ( ·𝑠OLD𝑤) ⊆ ( ·𝑠OLD𝑢) ∧ (normCV𝑤) ⊆ (normCV𝑢))})
173fvexi 6914 . . . . . . 7 𝐺 ∈ V
1817pwex 5382 . . . . . 6 𝒫 𝐺 ∈ V
197fvexi 6914 . . . . . . 7 𝑆 ∈ V
2019pwex 5382 . . . . . 6 𝒫 𝑆 ∈ V
2118, 20xpex 7759 . . . . 5 (𝒫 𝐺 × 𝒫 𝑆) ∈ V
2211fvexi 6914 . . . . . 6 𝑁 ∈ V
2322pwex 5382 . . . . 5 𝒫 𝑁 ∈ V
2421, 23xpex 7759 . . . 4 ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁) ∈ V
25 rabss 4067 . . . . 5 ({𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)} ⊆ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁) ↔ ∀𝑤 ∈ NrmCVec ((( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁) → 𝑤 ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁)))
26 fvex 6913 . . . . . . . . . 10 ( +𝑣𝑤) ∈ V
2726elpw 4608 . . . . . . . . 9 (( +𝑣𝑤) ∈ 𝒫 𝐺 ↔ ( +𝑣𝑤) ⊆ 𝐺)
28 fvex 6913 . . . . . . . . . 10 ( ·𝑠OLD𝑤) ∈ V
2928elpw 4608 . . . . . . . . 9 (( ·𝑠OLD𝑤) ∈ 𝒫 𝑆 ↔ ( ·𝑠OLD𝑤) ⊆ 𝑆)
30 opelxpi 5717 . . . . . . . . 9 ((( +𝑣𝑤) ∈ 𝒫 𝐺 ∧ ( ·𝑠OLD𝑤) ∈ 𝒫 𝑆) → ⟨( +𝑣𝑤), ( ·𝑠OLD𝑤)⟩ ∈ (𝒫 𝐺 × 𝒫 𝑆))
3127, 29, 30syl2anbr 597 . . . . . . . 8 ((( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆) → ⟨( +𝑣𝑤), ( ·𝑠OLD𝑤)⟩ ∈ (𝒫 𝐺 × 𝒫 𝑆))
32 fvex 6913 . . . . . . . . . 10 (normCV𝑤) ∈ V
3332elpw 4608 . . . . . . . . 9 ((normCV𝑤) ∈ 𝒫 𝑁 ↔ (normCV𝑤) ⊆ 𝑁)
3433biimpri 227 . . . . . . . 8 ((normCV𝑤) ⊆ 𝑁 → (normCV𝑤) ∈ 𝒫 𝑁)
35 opelxpi 5717 . . . . . . . 8 ((⟨( +𝑣𝑤), ( ·𝑠OLD𝑤)⟩ ∈ (𝒫 𝐺 × 𝒫 𝑆) ∧ (normCV𝑤) ∈ 𝒫 𝑁) → ⟨⟨( +𝑣𝑤), ( ·𝑠OLD𝑤)⟩, (normCV𝑤)⟩ ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁))
3631, 34, 35syl2an 594 . . . . . . 7 (((( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆) ∧ (normCV𝑤) ⊆ 𝑁) → ⟨⟨( +𝑣𝑤), ( ·𝑠OLD𝑤)⟩, (normCV𝑤)⟩ ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁))
37363impa 1107 . . . . . 6 ((( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁) → ⟨⟨( +𝑣𝑤), ( ·𝑠OLD𝑤)⟩, (normCV𝑤)⟩ ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁))
38 eqid 2727 . . . . . . . 8 ( +𝑣𝑤) = ( +𝑣𝑤)
39 eqid 2727 . . . . . . . 8 ( ·𝑠OLD𝑤) = ( ·𝑠OLD𝑤)
40 eqid 2727 . . . . . . . 8 (normCV𝑤) = (normCV𝑤)
4138, 39, 40nvop 30504 . . . . . . 7 (𝑤 ∈ NrmCVec → 𝑤 = ⟨⟨( +𝑣𝑤), ( ·𝑠OLD𝑤)⟩, (normCV𝑤)⟩)
4241eleq1d 2813 . . . . . 6 (𝑤 ∈ NrmCVec → (𝑤 ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁) ↔ ⟨⟨( +𝑣𝑤), ( ·𝑠OLD𝑤)⟩, (normCV𝑤)⟩ ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁)))
4337, 42imbitrrid 245 . . . . 5 (𝑤 ∈ NrmCVec → ((( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁) → 𝑤 ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁)))
4425, 43mprgbir 3064 . . . 4 {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)} ⊆ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁)
4524, 44ssexi 5324 . . 3 {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)} ∈ V
4615, 16, 45fvmpt 7008 . 2 (𝑈 ∈ NrmCVec → (SubSp‘𝑈) = {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)})
471, 46eqtrid 2779 1 (𝑈 ∈ NrmCVec → 𝐻 = {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  {crab 3428  wss 3947  𝒫 cpw 4604  cop 4636   × cxp 5678  cfv 6551  NrmCVeccnv 30412   +𝑣 cpv 30413   ·𝑠OLD cns 30415  normCVcnmcv 30418  SubSpcss 30549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-fo 6557  df-fv 6559  df-oprab 7428  df-1st 7997  df-2nd 7998  df-vc 30387  df-nv 30420  df-va 30423  df-sm 30425  df-nmcv 30428  df-ssp 30550
This theorem is referenced by:  isssp  30552
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