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Theorem sspval 27706
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 6229 . . . . . . 7 (𝑢 = 𝑈 → ( +𝑣𝑢) = ( +𝑣𝑈))
3 sspval.g . . . . . . 7 𝐺 = ( +𝑣𝑈)
42, 3syl6eqr 2703 . . . . . 6 (𝑢 = 𝑈 → ( +𝑣𝑢) = 𝐺)
54sseq2d 3666 . . . . 5 (𝑢 = 𝑈 → (( +𝑣𝑤) ⊆ ( +𝑣𝑢) ↔ ( +𝑣𝑤) ⊆ 𝐺))
6 fveq2 6229 . . . . . . 7 (𝑢 = 𝑈 → ( ·𝑠OLD𝑢) = ( ·𝑠OLD𝑈))
7 sspval.s . . . . . . 7 𝑆 = ( ·𝑠OLD𝑈)
86, 7syl6eqr 2703 . . . . . 6 (𝑢 = 𝑈 → ( ·𝑠OLD𝑢) = 𝑆)
98sseq2d 3666 . . . . 5 (𝑢 = 𝑈 → (( ·𝑠OLD𝑤) ⊆ ( ·𝑠OLD𝑢) ↔ ( ·𝑠OLD𝑤) ⊆ 𝑆))
10 fveq2 6229 . . . . . . 7 (𝑢 = 𝑈 → (normCV𝑢) = (normCV𝑈))
11 sspval.n . . . . . . 7 𝑁 = (normCV𝑈)
1210, 11syl6eqr 2703 . . . . . 6 (𝑢 = 𝑈 → (normCV𝑢) = 𝑁)
1312sseq2d 3666 . . . . 5 (𝑢 = 𝑈 → ((normCV𝑤) ⊆ (normCV𝑢) ↔ (normCV𝑤) ⊆ 𝑁))
145, 9, 133anbi123d 1439 . . . 4 (𝑢 = 𝑈 → ((( +𝑣𝑤) ⊆ ( +𝑣𝑢) ∧ ( ·𝑠OLD𝑤) ⊆ ( ·𝑠OLD𝑢) ∧ (normCV𝑤) ⊆ (normCV𝑢)) ↔ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)))
1514rabbidv 3220 . . 3 (𝑢 = 𝑈 → {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ ( +𝑣𝑢) ∧ ( ·𝑠OLD𝑤) ⊆ ( ·𝑠OLD𝑢) ∧ (normCV𝑤) ⊆ (normCV𝑢))} = {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)})
16 df-ssp 27705 . . 3 SubSp = (𝑢 ∈ NrmCVec ↦ {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ ( +𝑣𝑢) ∧ ( ·𝑠OLD𝑤) ⊆ ( ·𝑠OLD𝑢) ∧ (normCV𝑤) ⊆ (normCV𝑢))})
17 fvex 6239 . . . . . . . 8 ( +𝑣𝑈) ∈ V
183, 17eqeltri 2726 . . . . . . 7 𝐺 ∈ V
1918pwex 4878 . . . . . 6 𝒫 𝐺 ∈ V
20 fvex 6239 . . . . . . . 8 ( ·𝑠OLD𝑈) ∈ V
217, 20eqeltri 2726 . . . . . . 7 𝑆 ∈ V
2221pwex 4878 . . . . . 6 𝒫 𝑆 ∈ V
2319, 22xpex 7004 . . . . 5 (𝒫 𝐺 × 𝒫 𝑆) ∈ V
24 fvex 6239 . . . . . . 7 (normCV𝑈) ∈ V
2511, 24eqeltri 2726 . . . . . 6 𝑁 ∈ V
2625pwex 4878 . . . . 5 𝒫 𝑁 ∈ V
2723, 26xpex 7004 . . . 4 ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁) ∈ V
28 rabss 3712 . . . . 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)))
29 fvex 6239 . . . . . . . . . 10 ( +𝑣𝑤) ∈ V
3029elpw 4197 . . . . . . . . 9 (( +𝑣𝑤) ∈ 𝒫 𝐺 ↔ ( +𝑣𝑤) ⊆ 𝐺)
31 fvex 6239 . . . . . . . . . 10 ( ·𝑠OLD𝑤) ∈ V
3231elpw 4197 . . . . . . . . 9 (( ·𝑠OLD𝑤) ∈ 𝒫 𝑆 ↔ ( ·𝑠OLD𝑤) ⊆ 𝑆)
33 opelxpi 5182 . . . . . . . . 9 ((( +𝑣𝑤) ∈ 𝒫 𝐺 ∧ ( ·𝑠OLD𝑤) ∈ 𝒫 𝑆) → ⟨( +𝑣𝑤), ( ·𝑠OLD𝑤)⟩ ∈ (𝒫 𝐺 × 𝒫 𝑆))
3430, 32, 33syl2anbr 496 . . . . . . . 8 ((( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆) → ⟨( +𝑣𝑤), ( ·𝑠OLD𝑤)⟩ ∈ (𝒫 𝐺 × 𝒫 𝑆))
35 fvex 6239 . . . . . . . . . 10 (normCV𝑤) ∈ V
3635elpw 4197 . . . . . . . . 9 ((normCV𝑤) ∈ 𝒫 𝑁 ↔ (normCV𝑤) ⊆ 𝑁)
3736biimpri 218 . . . . . . . 8 ((normCV𝑤) ⊆ 𝑁 → (normCV𝑤) ∈ 𝒫 𝑁)
38 opelxpi 5182 . . . . . . . 8 ((⟨( +𝑣𝑤), ( ·𝑠OLD𝑤)⟩ ∈ (𝒫 𝐺 × 𝒫 𝑆) ∧ (normCV𝑤) ∈ 𝒫 𝑁) → ⟨⟨( +𝑣𝑤), ( ·𝑠OLD𝑤)⟩, (normCV𝑤)⟩ ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁))
3934, 37, 38syl2an 493 . . . . . . 7 (((( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆) ∧ (normCV𝑤) ⊆ 𝑁) → ⟨⟨( +𝑣𝑤), ( ·𝑠OLD𝑤)⟩, (normCV𝑤)⟩ ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁))
40393impa 1278 . . . . . 6 ((( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁) → ⟨⟨( +𝑣𝑤), ( ·𝑠OLD𝑤)⟩, (normCV𝑤)⟩ ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁))
41 eqid 2651 . . . . . . . 8 ( +𝑣𝑤) = ( +𝑣𝑤)
42 eqid 2651 . . . . . . . 8 ( ·𝑠OLD𝑤) = ( ·𝑠OLD𝑤)
43 eqid 2651 . . . . . . . 8 (normCV𝑤) = (normCV𝑤)
4441, 42, 43nvop 27659 . . . . . . 7 (𝑤 ∈ NrmCVec → 𝑤 = ⟨⟨( +𝑣𝑤), ( ·𝑠OLD𝑤)⟩, (normCV𝑤)⟩)
4544eleq1d 2715 . . . . . 6 (𝑤 ∈ NrmCVec → (𝑤 ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁) ↔ ⟨⟨( +𝑣𝑤), ( ·𝑠OLD𝑤)⟩, (normCV𝑤)⟩ ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁)))
4640, 45syl5ibr 236 . . . . 5 (𝑤 ∈ NrmCVec → ((( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁) → 𝑤 ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁)))
4728, 46mprgbir 2956 . . . 4 {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)} ⊆ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁)
4827, 47ssexi 4836 . . 3 {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)} ∈ V
4915, 16, 48fvmpt 6321 . 2 (𝑈 ∈ NrmCVec → (SubSp‘𝑈) = {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)})
501, 49syl5eq 2697 1 (𝑈 ∈ NrmCVec → 𝐻 = {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  {crab 2945  Vcvv 3231  wss 3607  𝒫 cpw 4191  cop 4216   × cxp 5141  cfv 5926  NrmCVeccnv 27567   +𝑣 cpv 27568   ·𝑠OLD cns 27570  normCVcnmcv 27573  SubSpcss 27704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934  df-oprab 6694  df-1st 7210  df-2nd 7211  df-vc 27542  df-nv 27575  df-va 27578  df-sm 27580  df-nmcv 27583  df-ssp 27705
This theorem is referenced by:  isssp  27707
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