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Theorem pjpm 20246
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 2752 . . . . 5 (ocv‘𝑊) = (ocv‘𝑊)
4 eqid 2752 . . . . 5 (proj1𝑊) = (proj1𝑊)
5 pjpm.k . . . . 5 𝐾 = (proj‘𝑊)
61, 2, 3, 4, 5pjfval 20244 . . . 4 𝐾 = ((𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) ∩ (V × (𝑉𝑚 𝑉)))
7 inss1 3968 . . . 4 ((𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) ∩ (V × (𝑉𝑚 𝑉))) ⊆ (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥)))
86, 7eqsstri 3768 . . 3 𝐾 ⊆ (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥)))
9 funmpt 6079 . . 3 Fun (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥)))
10 funss 6060 . . 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 2752 . . . . . 6 (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) = (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥)))
13 ovexd 6835 . . . . . 6 (𝑥𝐿 → (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥)) ∈ V)
1412, 13fmpti 6538 . . . . 5 (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))):𝐿⟶V
15 fssxp 6213 . . . . 5 ((𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))):𝐿⟶V → (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) ⊆ (𝐿 × V))
16 ssrin 3973 . . . . 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 3768 . . 3 𝐾 ⊆ ((𝐿 × V) ∩ (V × (𝑉𝑚 𝑉)))
19 inxp 5402 . . . 4 ((𝐿 × V) ∩ (V × (𝑉𝑚 𝑉))) = ((𝐿 ∩ V) × (V ∩ (𝑉𝑚 𝑉)))
20 inv1 4105 . . . . 5 (𝐿 ∩ V) = 𝐿
21 incom 3940 . . . . . 6 (V ∩ (𝑉𝑚 𝑉)) = ((𝑉𝑚 𝑉) ∩ V)
22 inv1 4105 . . . . . 6 ((𝑉𝑚 𝑉) ∩ V) = (𝑉𝑚 𝑉)
2321, 22eqtri 2774 . . . . 5 (V ∩ (𝑉𝑚 𝑉)) = (𝑉𝑚 𝑉)
2420, 23xpeq12i 5286 . . . 4 ((𝐿 ∩ V) × (V ∩ (𝑉𝑚 𝑉))) = (𝐿 × (𝑉𝑚 𝑉))
2519, 24eqtri 2774 . . 3 ((𝐿 × V) ∩ (V × (𝑉𝑚 𝑉))) = (𝐿 × (𝑉𝑚 𝑉))
2618, 25sseqtri 3770 . 2 𝐾 ⊆ (𝐿 × (𝑉𝑚 𝑉))
27 ovex 6833 . . 3 (𝑉𝑚 𝑉) ∈ V
28 fvex 6354 . . . 4 (LSubSp‘𝑊) ∈ V
292, 28eqeltri 2827 . . 3 𝐿 ∈ V
3027, 29elpm 8046 . 2 (𝐾 ∈ ((𝑉𝑚 𝑉) ↑pm 𝐿) ↔ (Fun 𝐾𝐾 ⊆ (𝐿 × (𝑉𝑚 𝑉))))
3111, 26, 30mpbir2an 993 1 𝐾 ∈ ((𝑉𝑚 𝑉) ↑pm 𝐿)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1624  wcel 2131  Vcvv 3332  cin 3706  wss 3707  cmpt 4873   × cxp 5256  Fun wfun 6035  wf 6037  cfv 6041  (class class class)co 6805  𝑚 cmap 8015  pm cpm 8016  Basecbs 16051  proj1cpj1 18242  LSubSpclss 19126  ocvcocv 20198  projcpj 20238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-pm 8018  df-pj 20241
This theorem is referenced by: (None)
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