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Theorem pjpm 21728
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 2737 . . . . 5 (ocv‘𝑊) = (ocv‘𝑊)
4 eqid 2737 . . . . 5 (proj1𝑊) = (proj1𝑊)
5 pjpm.k . . . . 5 𝐾 = (proj‘𝑊)
61, 2, 3, 4, 5pjfval 21726 . . . 4 𝐾 = ((𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) ∩ (V × (𝑉m 𝑉)))
7 inss1 4237 . . . 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 6604 . . 3 Fun (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥)))
10 funss 6585 . . 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 2737 . . . . . 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 6763 . . . . 5 ((𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))):𝐿⟶V → (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) ⊆ (𝐿 × V))
16 ssrin 4242 . . . . 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 5842 . . . 4 ((𝐿 × V) ∩ (V × (𝑉m 𝑉))) = ((𝐿 ∩ V) × (V ∩ (𝑉m 𝑉)))
20 inv1 4398 . . . . 5 (𝐿 ∩ V) = 𝐿
21 incom 4209 . . . . . 6 (V ∩ (𝑉m 𝑉)) = ((𝑉m 𝑉) ∩ V)
22 inv1 4398 . . . . . 6 ((𝑉m 𝑉) ∩ V) = (𝑉m 𝑉)
2321, 22eqtri 2765 . . . . 5 (V ∩ (𝑉m 𝑉)) = (𝑉m 𝑉)
2420, 23xpeq12i 5713 . . . 4 ((𝐿 ∩ V) × (V ∩ (𝑉m 𝑉))) = (𝐿 × (𝑉m 𝑉))
2519, 24eqtri 2765 . . 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 6920 . . 3 𝐿 ∈ V
2927, 28elpm 8913 . 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 1540  wcel 2108  Vcvv 3480  cin 3950  wss 3951  cmpt 5225   × cxp 5683  Fun wfun 6555  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  pm cpm 8867  Basecbs 17247  proj1cpj1 19653  LSubSpclss 20929  ocvcocv 21678  projcpj 21720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-pm 8869  df-pj 21723
This theorem is referenced by: (None)
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