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Theorem sspval 28509
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 6661 . . . . . . 7 (𝑢 = 𝑈 → ( +𝑣𝑢) = ( +𝑣𝑈))
3 sspval.g . . . . . . 7 𝐺 = ( +𝑣𝑈)
42, 3syl6eqr 2877 . . . . . 6 (𝑢 = 𝑈 → ( +𝑣𝑢) = 𝐺)
54sseq2d 3985 . . . . 5 (𝑢 = 𝑈 → (( +𝑣𝑤) ⊆ ( +𝑣𝑢) ↔ ( +𝑣𝑤) ⊆ 𝐺))
6 fveq2 6661 . . . . . . 7 (𝑢 = 𝑈 → ( ·𝑠OLD𝑢) = ( ·𝑠OLD𝑈))
7 sspval.s . . . . . . 7 𝑆 = ( ·𝑠OLD𝑈)
86, 7syl6eqr 2877 . . . . . 6 (𝑢 = 𝑈 → ( ·𝑠OLD𝑢) = 𝑆)
98sseq2d 3985 . . . . 5 (𝑢 = 𝑈 → (( ·𝑠OLD𝑤) ⊆ ( ·𝑠OLD𝑢) ↔ ( ·𝑠OLD𝑤) ⊆ 𝑆))
10 fveq2 6661 . . . . . . 7 (𝑢 = 𝑈 → (normCV𝑢) = (normCV𝑈))
11 sspval.n . . . . . . 7 𝑁 = (normCV𝑈)
1210, 11syl6eqr 2877 . . . . . 6 (𝑢 = 𝑈 → (normCV𝑢) = 𝑁)
1312sseq2d 3985 . . . . 5 (𝑢 = 𝑈 → ((normCV𝑤) ⊆ (normCV𝑢) ↔ (normCV𝑤) ⊆ 𝑁))
145, 9, 133anbi123d 1433 . . . 4 (𝑢 = 𝑈 → ((( +𝑣𝑤) ⊆ ( +𝑣𝑢) ∧ ( ·𝑠OLD𝑤) ⊆ ( ·𝑠OLD𝑢) ∧ (normCV𝑤) ⊆ (normCV𝑢)) ↔ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)))
1514rabbidv 3465 . . 3 (𝑢 = 𝑈 → {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ ( +𝑣𝑢) ∧ ( ·𝑠OLD𝑤) ⊆ ( ·𝑠OLD𝑢) ∧ (normCV𝑤) ⊆ (normCV𝑢))} = {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)})
16 df-ssp 28508 . . 3 SubSp = (𝑢 ∈ NrmCVec ↦ {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ ( +𝑣𝑢) ∧ ( ·𝑠OLD𝑤) ⊆ ( ·𝑠OLD𝑢) ∧ (normCV𝑤) ⊆ (normCV𝑢))})
173fvexi 6675 . . . . . . 7 𝐺 ∈ V
1817pwex 5268 . . . . . 6 𝒫 𝐺 ∈ V
197fvexi 6675 . . . . . . 7 𝑆 ∈ V
2019pwex 5268 . . . . . 6 𝒫 𝑆 ∈ V
2118, 20xpex 7470 . . . . 5 (𝒫 𝐺 × 𝒫 𝑆) ∈ V
2211fvexi 6675 . . . . . 6 𝑁 ∈ V
2322pwex 5268 . . . . 5 𝒫 𝑁 ∈ V
2421, 23xpex 7470 . . . 4 ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁) ∈ V
25 rabss 4034 . . . . 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 6674 . . . . . . . . . 10 ( +𝑣𝑤) ∈ V
2726elpw 4526 . . . . . . . . 9 (( +𝑣𝑤) ∈ 𝒫 𝐺 ↔ ( +𝑣𝑤) ⊆ 𝐺)
28 fvex 6674 . . . . . . . . . 10 ( ·𝑠OLD𝑤) ∈ V
2928elpw 4526 . . . . . . . . 9 (( ·𝑠OLD𝑤) ∈ 𝒫 𝑆 ↔ ( ·𝑠OLD𝑤) ⊆ 𝑆)
30 opelxpi 5579 . . . . . . . . 9 ((( +𝑣𝑤) ∈ 𝒫 𝐺 ∧ ( ·𝑠OLD𝑤) ∈ 𝒫 𝑆) → ⟨( +𝑣𝑤), ( ·𝑠OLD𝑤)⟩ ∈ (𝒫 𝐺 × 𝒫 𝑆))
3127, 29, 30syl2anbr 601 . . . . . . . 8 ((( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆) → ⟨( +𝑣𝑤), ( ·𝑠OLD𝑤)⟩ ∈ (𝒫 𝐺 × 𝒫 𝑆))
32 fvex 6674 . . . . . . . . . 10 (normCV𝑤) ∈ V
3332elpw 4526 . . . . . . . . 9 ((normCV𝑤) ∈ 𝒫 𝑁 ↔ (normCV𝑤) ⊆ 𝑁)
3433biimpri 231 . . . . . . . 8 ((normCV𝑤) ⊆ 𝑁 → (normCV𝑤) ∈ 𝒫 𝑁)
35 opelxpi 5579 . . . . . . . 8 ((⟨( +𝑣𝑤), ( ·𝑠OLD𝑤)⟩ ∈ (𝒫 𝐺 × 𝒫 𝑆) ∧ (normCV𝑤) ∈ 𝒫 𝑁) → ⟨⟨( +𝑣𝑤), ( ·𝑠OLD𝑤)⟩, (normCV𝑤)⟩ ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁))
3631, 34, 35syl2an 598 . . . . . . 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 2824 . . . . . . . 8 ( +𝑣𝑤) = ( +𝑣𝑤)
39 eqid 2824 . . . . . . . 8 ( ·𝑠OLD𝑤) = ( ·𝑠OLD𝑤)
40 eqid 2824 . . . . . . . 8 (normCV𝑤) = (normCV𝑤)
4138, 39, 40nvop 28462 . . . . . . 7 (𝑤 ∈ NrmCVec → 𝑤 = ⟨⟨( +𝑣𝑤), ( ·𝑠OLD𝑤)⟩, (normCV𝑤)⟩)
4241eleq1d 2900 . . . . . 6 (𝑤 ∈ NrmCVec → (𝑤 ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁) ↔ ⟨⟨( +𝑣𝑤), ( ·𝑠OLD𝑤)⟩, (normCV𝑤)⟩ ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁)))
4337, 42syl5ibr 249 . . . . 5 (𝑤 ∈ NrmCVec → ((( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁) → 𝑤 ∈ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁)))
4425, 43mprgbir 3148 . . . 4 {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)} ⊆ ((𝒫 𝐺 × 𝒫 𝑆) × 𝒫 𝑁)
4524, 44ssexi 5212 . . 3 {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)} ∈ V
4615, 16, 45fvmpt 6759 . 2 (𝑈 ∈ NrmCVec → (SubSp‘𝑈) = {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)})
471, 46syl5eq 2871 1 (𝑈 ∈ NrmCVec → 𝐻 = {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  {crab 3137  wss 3919  𝒫 cpw 4522  cop 4556   × cxp 5540  cfv 6343  NrmCVeccnv 28370   +𝑣 cpv 28371   ·𝑠OLD cns 28373  normCVcnmcv 28376  SubSpcss 28507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fo 6349  df-fv 6351  df-oprab 7153  df-1st 7684  df-2nd 7685  df-vc 28345  df-nv 28378  df-va 28381  df-sm 28383  df-nmcv 28386  df-ssp 28508
This theorem is referenced by:  isssp  28510
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