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Theorem sspval 27424
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 6148 . . . . . . 7 (𝑢 = 𝑈 → ( +𝑣𝑢) = ( +𝑣𝑈))
3 sspval.g . . . . . . 7 𝐺 = ( +𝑣𝑈)
42, 3syl6eqr 2673 . . . . . 6 (𝑢 = 𝑈 → ( +𝑣𝑢) = 𝐺)
54sseq2d 3612 . . . . 5 (𝑢 = 𝑈 → (( +𝑣𝑤) ⊆ ( +𝑣𝑢) ↔ ( +𝑣𝑤) ⊆ 𝐺))
6 fveq2 6148 . . . . . . 7 (𝑢 = 𝑈 → ( ·𝑠OLD𝑢) = ( ·𝑠OLD𝑈))
7 sspval.s . . . . . . 7 𝑆 = ( ·𝑠OLD𝑈)
86, 7syl6eqr 2673 . . . . . 6 (𝑢 = 𝑈 → ( ·𝑠OLD𝑢) = 𝑆)
98sseq2d 3612 . . . . 5 (𝑢 = 𝑈 → (( ·𝑠OLD𝑤) ⊆ ( ·𝑠OLD𝑢) ↔ ( ·𝑠OLD𝑤) ⊆ 𝑆))
10 fveq2 6148 . . . . . . 7 (𝑢 = 𝑈 → (normCV𝑢) = (normCV𝑈))
11 sspval.n . . . . . . 7 𝑁 = (normCV𝑈)
1210, 11syl6eqr 2673 . . . . . 6 (𝑢 = 𝑈 → (normCV𝑢) = 𝑁)
1312sseq2d 3612 . . . . 5 (𝑢 = 𝑈 → ((normCV𝑤) ⊆ (normCV𝑢) ↔ (normCV𝑤) ⊆ 𝑁))
145, 9, 133anbi123d 1396 . . . 4 (𝑢 = 𝑈 → ((( +𝑣𝑤) ⊆ ( +𝑣𝑢) ∧ ( ·𝑠OLD𝑤) ⊆ ( ·𝑠OLD𝑢) ∧ (normCV𝑤) ⊆ (normCV𝑢)) ↔ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)))
1514rabbidv 3177 . . 3 (𝑢 = 𝑈 → {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ ( +𝑣𝑢) ∧ ( ·𝑠OLD𝑤) ⊆ ( ·𝑠OLD𝑢) ∧ (normCV𝑤) ⊆ (normCV𝑢))} = {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)})
16 df-ssp 27423 . . 3 SubSp = (𝑢 ∈ NrmCVec ↦ {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ ( +𝑣𝑢) ∧ ( ·𝑠OLD𝑤) ⊆ ( ·𝑠OLD𝑢) ∧ (normCV𝑤) ⊆ (normCV𝑢))})
17 fvex 6158 . . . . . . . 8 ( +𝑣𝑈) ∈ V
183, 17eqeltri 2694 . . . . . . 7 𝐺 ∈ V
1918pwex 4808 . . . . . 6 𝒫 𝐺 ∈ V
20 fvex 6158 . . . . . . . 8 ( ·𝑠OLD𝑈) ∈ V
217, 20eqeltri 2694 . . . . . . 7 𝑆 ∈ V
2221pwex 4808 . . . . . 6 𝒫 𝑆 ∈ V
2319, 22xpex 6915 . . . . 5 (𝒫 𝐺 × 𝒫 𝑆) ∈ V
24 fvex 6158 . . . . . . 7 (normCV𝑈) ∈ V
2511, 24eqeltri 2694 . . . . . 6 𝑁 ∈ V
2625pwex 4808 . . . . 5 𝒫 𝑁 ∈ V
2723, 26xpex 6915 . . . 4 ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁) ∈ V
28 rabss 3658 . . . . 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 6158 . . . . . . . . . 10 ( +𝑣𝑤) ∈ V
3029elpw 4136 . . . . . . . . 9 (( +𝑣𝑤) ∈ 𝒫 𝐺 ↔ ( +𝑣𝑤) ⊆ 𝐺)
31 fvex 6158 . . . . . . . . . 10 ( ·𝑠OLD𝑤) ∈ V
3231elpw 4136 . . . . . . . . 9 (( ·𝑠OLD𝑤) ∈ 𝒫 𝑆 ↔ ( ·𝑠OLD𝑤) ⊆ 𝑆)
33 opelxpi 5108 . . . . . . . . 9 ((( +𝑣𝑤) ∈ 𝒫 𝐺 ∧ ( ·𝑠OLD𝑤) ∈ 𝒫 𝑆) → ⟨( +𝑣𝑤), ( ·𝑠OLD𝑤)⟩ ∈ (𝒫 𝐺 × 𝒫 𝑆))
3430, 32, 33syl2anbr 497 . . . . . . . 8 ((( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆) → ⟨( +𝑣𝑤), ( ·𝑠OLD𝑤)⟩ ∈ (𝒫 𝐺 × 𝒫 𝑆))
35 fvex 6158 . . . . . . . . . 10 (normCV𝑤) ∈ V
3635elpw 4136 . . . . . . . . 9 ((normCV𝑤) ∈ 𝒫 𝑁 ↔ (normCV𝑤) ⊆ 𝑁)
3736biimpri 218 . . . . . . . 8 ((normCV𝑤) ⊆ 𝑁 → (normCV𝑤) ∈ 𝒫 𝑁)
38 opelxpi 5108 . . . . . . . 8 ((⟨( +𝑣𝑤), ( ·𝑠OLD𝑤)⟩ ∈ (𝒫 𝐺 × 𝒫 𝑆) ∧ (normCV𝑤) ∈ 𝒫 𝑁) → ⟨⟨( +𝑣𝑤), ( ·𝑠OLD𝑤)⟩, (normCV𝑤)⟩ ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁))
3934, 37, 38syl2an 494 . . . . . . 7 (((( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆) ∧ (normCV𝑤) ⊆ 𝑁) → ⟨⟨( +𝑣𝑤), ( ·𝑠OLD𝑤)⟩, (normCV𝑤)⟩ ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁))
40393impa 1256 . . . . . 6 ((( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁) → ⟨⟨( +𝑣𝑤), ( ·𝑠OLD𝑤)⟩, (normCV𝑤)⟩ ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁))
41 eqid 2621 . . . . . . . 8 ( +𝑣𝑤) = ( +𝑣𝑤)
42 eqid 2621 . . . . . . . 8 ( ·𝑠OLD𝑤) = ( ·𝑠OLD𝑤)
43 eqid 2621 . . . . . . . 8 (normCV𝑤) = (normCV𝑤)
4441, 42, 43nvop 27377 . . . . . . 7 (𝑤 ∈ NrmCVec → 𝑤 = ⟨⟨( +𝑣𝑤), ( ·𝑠OLD𝑤)⟩, (normCV𝑤)⟩)
4544eleq1d 2683 . . . . . 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 2922 . . . 4 {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)} ⊆ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁)
4827, 47ssexi 4763 . . 3 {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)} ∈ V
4915, 16, 48fvmpt 6239 . 2 (𝑈 ∈ NrmCVec → (SubSp‘𝑈) = {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)})
501, 49syl5eq 2667 1 (𝑈 ∈ NrmCVec → 𝐻 = {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3186  wss 3555  𝒫 cpw 4130  cop 4154   × cxp 5072  cfv 5847  NrmCVeccnv 27285   +𝑣 cpv 27286   ·𝑠OLD cns 27288  normCVcnmcv 27291  SubSpcss 27422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fo 5853  df-fv 5855  df-oprab 6608  df-1st 7113  df-2nd 7114  df-vc 27260  df-nv 27293  df-va 27296  df-sm 27298  df-nmcv 27301  df-ssp 27423
This theorem is referenced by:  isssp  27425
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