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Theorem pjpm 19974
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 2621 . . . . 5 (ocv‘𝑊) = (ocv‘𝑊)
4 eqid 2621 . . . . 5 (proj1𝑊) = (proj1𝑊)
5 pjpm.k . . . . 5 𝐾 = (proj‘𝑊)
61, 2, 3, 4, 5pjfval 19972 . . . 4 𝐾 = ((𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) ∩ (V × (𝑉𝑚 𝑉)))
7 inss1 3813 . . . 4 ((𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) ∩ (V × (𝑉𝑚 𝑉))) ⊆ (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥)))
86, 7eqsstri 3616 . . 3 𝐾 ⊆ (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥)))
9 funmpt 5886 . . 3 Fun (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥)))
10 funss 5868 . . 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 2621 . . . . . 6 (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) = (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥)))
13 ovex 6635 . . . . . . 7 (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥)) ∈ V
1413a1i 11 . . . . . 6 (𝑥𝐿 → (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥)) ∈ V)
1512, 14fmpti 6341 . . . . 5 (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))):𝐿⟶V
16 fssxp 6019 . . . . 5 ((𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))):𝐿⟶V → (𝑥𝐿 ↦ (𝑥(proj1𝑊)((ocv‘𝑊)‘𝑥))) ⊆ (𝐿 × V))
17 ssrin 3818 . . . . 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 3616 . . 3 𝐾 ⊆ ((𝐿 × V) ∩ (V × (𝑉𝑚 𝑉)))
20 inxp 5216 . . . 4 ((𝐿 × V) ∩ (V × (𝑉𝑚 𝑉))) = ((𝐿 ∩ V) × (V ∩ (𝑉𝑚 𝑉)))
21 inv1 3944 . . . . 5 (𝐿 ∩ V) = 𝐿
22 incom 3785 . . . . . 6 (V ∩ (𝑉𝑚 𝑉)) = ((𝑉𝑚 𝑉) ∩ V)
23 inv1 3944 . . . . . 6 ((𝑉𝑚 𝑉) ∩ V) = (𝑉𝑚 𝑉)
2422, 23eqtri 2643 . . . . 5 (V ∩ (𝑉𝑚 𝑉)) = (𝑉𝑚 𝑉)
2521, 24xpeq12i 5099 . . . 4 ((𝐿 ∩ V) × (V ∩ (𝑉𝑚 𝑉))) = (𝐿 × (𝑉𝑚 𝑉))
2620, 25eqtri 2643 . . 3 ((𝐿 × V) ∩ (V × (𝑉𝑚 𝑉))) = (𝐿 × (𝑉𝑚 𝑉))
2719, 26sseqtri 3618 . 2 𝐾 ⊆ (𝐿 × (𝑉𝑚 𝑉))
28 ovex 6635 . . 3 (𝑉𝑚 𝑉) ∈ V
29 fvex 6160 . . . 4 (LSubSp‘𝑊) ∈ V
302, 29eqeltri 2694 . . 3 𝐿 ∈ V
3128, 30elpm 7835 . 2 (𝐾 ∈ ((𝑉𝑚 𝑉) ↑pm 𝐿) ↔ (Fun 𝐾𝐾 ⊆ (𝐿 × (𝑉𝑚 𝑉))))
3211, 27, 31mpbir2an 954 1 𝐾 ∈ ((𝑉𝑚 𝑉) ↑pm 𝐿)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  Vcvv 3186  cin 3555  wss 3556  cmpt 4675   × cxp 5074  Fun wfun 5843  wf 5845  cfv 5849  (class class class)co 6607  𝑚 cmap 7805  pm cpm 7806  Basecbs 15784  proj1cpj1 17974  LSubSpclss 18854  ocvcocv 19926  projcpj 19966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-pm 7808  df-pj 19969
This theorem is referenced by: (None)
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