MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pjpm Structured version   Visualization version   GIF version

Theorem pjpm 21746
Description: The projection map is a partial function from subspaces of the pre-Hilbert space to total operators. (Contributed by Mario Carneiro, 16-Oct-2015.)
Hypotheses
Ref Expression
pjpm.v 𝑉 = (Base‘𝑊)
pjpm.l 𝐿 = (LSubSp‘𝑊)
pjpm.k 𝐾 = (proj‘𝑊)
Assertion
Ref Expression
pjpm 𝐾 ∈ ((𝑉m 𝑉) ↑pm 𝐿)

Proof of Theorem pjpm
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pjpm.v . . . . 5 𝑉 = (Base‘𝑊)
2 pjpm.l . . . . 5 𝐿 = (LSubSp‘𝑊)
3 eqid 2735 . . . . 5 (ocv‘𝑊) = (ocv‘𝑊)
4 eqid 2735 . . . . 5 (proj1𝑊) = (proj1𝑊)
5 pjpm.k . . . . 5 𝐾 = (proj‘𝑊)
61, 2, 3, 4, 5pjfval 21744 . . . 4 𝐾 = ((𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) ∩ (V × (𝑉m 𝑉)))
7 inss1 4245 . . . 4 ((𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) ∩ (V × (𝑉m 𝑉))) ⊆ (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥)))
86, 7eqsstri 4030 . . 3 𝐾 ⊆ (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥)))
9 funmpt 6606 . . 3 Fun (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥)))
10 funss 6587 . . 3 (𝐾 ⊆ (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) → (Fun (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) → Fun 𝐾))
118, 9, 10mp2 9 . 2 Fun 𝐾
12 eqid 2735 . . . . . 6 (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) = (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥)))
13 ovexd 7466 . . . . . 6 (𝑥𝐿 → (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥)) ∈ V)
1412, 13fmpti 7132 . . . . 5 (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))):𝐿⟶V
15 fssxp 6764 . . . . 5 ((𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))):𝐿⟶V → (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) ⊆ (𝐿 × V))
16 ssrin 4250 . . . . 5 ((𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) ⊆ (𝐿 × V) → ((𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) ∩ (V × (𝑉m 𝑉))) ⊆ ((𝐿 × V) ∩ (V × (𝑉m 𝑉))))
1714, 15, 16mp2b 10 . . . 4 ((𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) ∩ (V × (𝑉m 𝑉))) ⊆ ((𝐿 × V) ∩ (V × (𝑉m 𝑉)))
186, 17eqsstri 4030 . . 3 𝐾 ⊆ ((𝐿 × V) ∩ (V × (𝑉m 𝑉)))
19 inxp 5845 . . . 4 ((𝐿 × V) ∩ (V × (𝑉m 𝑉))) = ((𝐿 ∩ V) × (V ∩ (𝑉m 𝑉)))
20 inv1 4404 . . . . 5 (𝐿 ∩ V) = 𝐿
21 incom 4217 . . . . . 6 (V ∩ (𝑉m 𝑉)) = ((𝑉m 𝑉) ∩ V)
22 inv1 4404 . . . . . 6 ((𝑉m 𝑉) ∩ V) = (𝑉m 𝑉)
2321, 22eqtri 2763 . . . . 5 (V ∩ (𝑉m 𝑉)) = (𝑉m 𝑉)
2420, 23xpeq12i 5717 . . . 4 ((𝐿 ∩ V) × (V ∩ (𝑉m 𝑉))) = (𝐿 × (𝑉m 𝑉))
2519, 24eqtri 2763 . . 3 ((𝐿 × V) ∩ (V × (𝑉m 𝑉))) = (𝐿 × (𝑉m 𝑉))
2618, 25sseqtri 4032 . 2 𝐾 ⊆ (𝐿 × (𝑉m 𝑉))
27 ovex 7464 . . 3 (𝑉m 𝑉) ∈ V
282fvexi 6921 . . 3 𝐿 ∈ V
2927, 28elpm 8912 . 2 (𝐾 ∈ ((𝑉m 𝑉) ↑pm 𝐿) ↔ (Fun 𝐾𝐾 ⊆ (𝐿 × (𝑉m 𝑉))))
3011, 26, 29mpbir2an 711 1 𝐾 ∈ ((𝑉m 𝑉) ↑pm 𝐿)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  Vcvv 3478  cin 3962  wss 3963  cmpt 5231   × cxp 5687  Fun wfun 6557  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865  pm cpm 8866  Basecbs 17245  proj1cpj1 19668  LSubSpclss 20947  ocvcocv 21696  projcpj 21738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-pm 8868  df-pj 21741
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator