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Theorem pjpm 20488
 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 2759 . . . . 5 (ocv‘𝑊) = (ocv‘𝑊)
4 eqid 2759 . . . . 5 (proj1𝑊) = (proj1𝑊)
5 pjpm.k . . . . 5 𝐾 = (proj‘𝑊)
61, 2, 3, 4, 5pjfval 20486 . . . 4 𝐾 = ((𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) ∩ (V × (𝑉m 𝑉)))
7 inss1 4136 . . . 4 ((𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) ∩ (V × (𝑉m 𝑉))) ⊆ (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥)))
86, 7eqsstri 3929 . . 3 𝐾 ⊆ (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥)))
9 funmpt 6379 . . 3 Fun (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥)))
10 funss 6360 . . 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 2759 . . . . . 6 (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) = (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥)))
13 ovexd 7192 . . . . . 6 (𝑥𝐿 → (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥)) ∈ V)
1412, 13fmpti 6874 . . . . 5 (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))):𝐿⟶V
15 fssxp 6525 . . . . 5 ((𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))):𝐿⟶V → (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) ⊆ (𝐿 × V))
16 ssrin 4141 . . . . 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 3929 . . 3 𝐾 ⊆ ((𝐿 × V) ∩ (V × (𝑉m 𝑉)))
19 inxp 5679 . . . 4 ((𝐿 × V) ∩ (V × (𝑉m 𝑉))) = ((𝐿 ∩ V) × (V ∩ (𝑉m 𝑉)))
20 inv1 4294 . . . . 5 (𝐿 ∩ V) = 𝐿
21 incom 4109 . . . . . 6 (V ∩ (𝑉m 𝑉)) = ((𝑉m 𝑉) ∩ V)
22 inv1 4294 . . . . . 6 ((𝑉m 𝑉) ∩ V) = (𝑉m 𝑉)
2321, 22eqtri 2782 . . . . 5 (V ∩ (𝑉m 𝑉)) = (𝑉m 𝑉)
2420, 23xpeq12i 5557 . . . 4 ((𝐿 ∩ V) × (V ∩ (𝑉m 𝑉))) = (𝐿 × (𝑉m 𝑉))
2519, 24eqtri 2782 . . 3 ((𝐿 × V) ∩ (V × (𝑉m 𝑉))) = (𝐿 × (𝑉m 𝑉))
2618, 25sseqtri 3931 . 2 𝐾 ⊆ (𝐿 × (𝑉m 𝑉))
27 ovex 7190 . . 3 (𝑉m 𝑉) ∈ V
282fvexi 6678 . . 3 𝐿 ∈ V
2927, 28elpm 8469 . 2 (𝐾 ∈ ((𝑉m 𝑉) ↑pm 𝐿) ↔ (Fun 𝐾𝐾 ⊆ (𝐿 × (𝑉m 𝑉))))
3011, 26, 29mpbir2an 710 1 𝐾 ∈ ((𝑉m 𝑉) ↑pm 𝐿)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1539   ∈ wcel 2112  Vcvv 3410   ∩ cin 3860   ⊆ wss 3861   ↦ cmpt 5117   × cxp 5527  Fun wfun 6335  ⟶wf 6337  ‘cfv 6341  (class class class)co 7157   ↑m cmap 8423   ↑pm cpm 8424  Basecbs 16556  proj1cpj1 18842  LSubSpclss 19786  ocvcocv 20440  projcpj 20480 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5174  ax-nul 5181  ax-pow 5239  ax-pr 5303  ax-un 7466 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3700  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-nul 4229  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4803  df-br 5038  df-opab 5100  df-mpt 5118  df-id 5435  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-fv 6349  df-ov 7160  df-oprab 7161  df-mpo 7162  df-pm 8426  df-pj 20483 This theorem is referenced by: (None)
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