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Theorem pjpm 19871
 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 2610 . . . . 5 (ocv‘𝑊) = (ocv‘𝑊)
4 eqid 2610 . . . . 5 (proj1𝑊) = (proj1𝑊)
5 pjpm.k . . . . 5 𝐾 = (proj‘𝑊)
61, 2, 3, 4, 5pjfval 19869 . . . 4 𝐾 = ((𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) ∩ (V × (𝑉𝑚 𝑉)))
7 inss1 3795 . . . 4 ((𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) ∩ (V × (𝑉𝑚 𝑉))) ⊆ (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥)))
86, 7eqsstri 3598 . . 3 𝐾 ⊆ (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥)))
9 funmpt 5840 . . 3 Fun (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥)))
10 funss 5822 . . 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 2610 . . . . . 6 (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) = (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥)))
13 ovex 6577 . . . . . . 7 (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥)) ∈ V
1413a1i 11 . . . . . 6 (𝑥𝐿 → (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥)) ∈ V)
1512, 14fmpti 6291 . . . . 5 (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))):𝐿⟶V
16 fssxp 5973 . . . . 5 ((𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))):𝐿⟶V → (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) ⊆ (𝐿 × V))
17 ssrin 3800 . . . . 5 ((𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) ⊆ (𝐿 × V) → ((𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) ∩ (V × (𝑉𝑚 𝑉))) ⊆ ((𝐿 × V) ∩ (V × (𝑉𝑚 𝑉))))
1815, 16, 17mp2b 10 . . . 4 ((𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) ∩ (V × (𝑉𝑚 𝑉))) ⊆ ((𝐿 × V) ∩ (V × (𝑉𝑚 𝑉)))
196, 18eqsstri 3598 . . 3 𝐾 ⊆ ((𝐿 × V) ∩ (V × (𝑉𝑚 𝑉)))
20 inxp 5176 . . . 4 ((𝐿 × V) ∩ (V × (𝑉𝑚 𝑉))) = ((𝐿 ∩ V) × (V ∩ (𝑉𝑚 𝑉)))
21 inv1 3922 . . . . 5 (𝐿 ∩ V) = 𝐿
22 incom 3767 . . . . . 6 (V ∩ (𝑉𝑚 𝑉)) = ((𝑉𝑚 𝑉) ∩ V)
23 inv1 3922 . . . . . 6 ((𝑉𝑚 𝑉) ∩ V) = (𝑉𝑚 𝑉)
2422, 23eqtri 2632 . . . . 5 (V ∩ (𝑉𝑚 𝑉)) = (𝑉𝑚 𝑉)
2521, 24xpeq12i 5061 . . . 4 ((𝐿 ∩ V) × (V ∩ (𝑉𝑚 𝑉))) = (𝐿 × (𝑉𝑚 𝑉))
2620, 25eqtri 2632 . . 3 ((𝐿 × V) ∩ (V × (𝑉𝑚 𝑉))) = (𝐿 × (𝑉𝑚 𝑉))
2719, 26sseqtri 3600 . 2 𝐾 ⊆ (𝐿 × (𝑉𝑚 𝑉))
28 ovex 6577 . . 3 (𝑉𝑚 𝑉) ∈ V
29 fvex 6113 . . . 4 (LSubSp‘𝑊) ∈ V
302, 29eqeltri 2684 . . 3 𝐿 ∈ V
3128, 30elpm 7774 . 2 (𝐾 ∈ ((𝑉𝑚 𝑉) ↑pm 𝐿) ↔ (Fun 𝐾𝐾 ⊆ (𝐿 × (𝑉𝑚 𝑉))))
3211, 27, 31mpbir2an 957 1 𝐾 ∈ ((𝑉𝑚 𝑉) ↑pm 𝐿)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540   ↦ cmpt 4643   × cxp 5036  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744   ↑pm cpm 7745  Basecbs 15695  proj1cpj1 17873  LSubSpclss 18753  ocvcocv 19823  projcpj 19863 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-pm 7747  df-pj 19866 This theorem is referenced by: (None)
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