Theorem pjpm 20052

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 ∈ ((𝑉 ↑𝑚 𝑉) ↑pm 𝐿)

Proof of Theorem pjpm

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pjpm.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
2		pjpm.l	. . . . 5 ⊢ 𝐿 = (LSubSp‘𝑊)
3		eqid 2622	. . . . 5 ⊢ (ocv‘𝑊) = (ocv‘𝑊)
4		eqid 2622	. . . . 5 ⊢ (proj1‘𝑊) = (proj1‘𝑊)
5		pjpm.k	. . . . 5 ⊢ 𝐾 = (proj‘𝑊)
6	1, 2, 3, 4, 5	pjfval 20050	. . . 4 ⊢ 𝐾 = ((𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))) ∩ (V × (𝑉 ↑𝑚 𝑉)))
7		inss1 3833	. . . 4 ⊢ ((𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))) ∩ (V × (𝑉 ↑𝑚 𝑉))) ⊆ (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥)))
8	6, 7	eqsstri 3635	. . 3 ⊢ 𝐾 ⊆ (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥)))
9		funmpt 5926	. . 3 ⊢ Fun (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥)))
10		funss 5907	. . 3 ⊢ (𝐾 ⊆ (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))) → (Fun (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))) → Fun 𝐾))
11	8, 9, 10	mp2 9	. 2 ⊢ Fun 𝐾
12		eqid 2622	. . . . . 6 ⊢ (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))) = (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥)))
13		ovexd 6680	. . . . . 6 ⊢ (𝑥 ∈ 𝐿 → (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥)) ∈ V)
14	12, 13	fmpti 6383	. . . . 5 ⊢ (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))):𝐿⟶V
15		fssxp 6060	. . . . 5 ⊢ ((𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))):𝐿⟶V → (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))) ⊆ (𝐿 × V))
16		ssrin 3838	. . . . 5 ⊢ ((𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))) ⊆ (𝐿 × V) → ((𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))) ∩ (V × (𝑉 ↑𝑚 𝑉))) ⊆ ((𝐿 × V) ∩ (V × (𝑉 ↑𝑚 𝑉))))
17	14, 15, 16	mp2b 10	. . . 4 ⊢ ((𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))) ∩ (V × (𝑉 ↑𝑚 𝑉))) ⊆ ((𝐿 × V) ∩ (V × (𝑉 ↑𝑚 𝑉)))
18	6, 17	eqsstri 3635	. . 3 ⊢ 𝐾 ⊆ ((𝐿 × V) ∩ (V × (𝑉 ↑𝑚 𝑉)))
19		inxp 5254	. . . 4 ⊢ ((𝐿 × V) ∩ (V × (𝑉 ↑𝑚 𝑉))) = ((𝐿 ∩ V) × (V ∩ (𝑉 ↑𝑚 𝑉)))
20		inv1 3970	. . . . 5 ⊢ (𝐿 ∩ V) = 𝐿
21		incom 3805	. . . . . 6 ⊢ (V ∩ (𝑉 ↑𝑚 𝑉)) = ((𝑉 ↑𝑚 𝑉) ∩ V)
22		inv1 3970	. . . . . 6 ⊢ ((𝑉 ↑𝑚 𝑉) ∩ V) = (𝑉 ↑𝑚 𝑉)
23	21, 22	eqtri 2644	. . . . 5 ⊢ (V ∩ (𝑉 ↑𝑚 𝑉)) = (𝑉 ↑𝑚 𝑉)
24	20, 23	xpeq12i 5137	. . . 4 ⊢ ((𝐿 ∩ V) × (V ∩ (𝑉 ↑𝑚 𝑉))) = (𝐿 × (𝑉 ↑𝑚 𝑉))
25	19, 24	eqtri 2644	. . 3 ⊢ ((𝐿 × V) ∩ (V × (𝑉 ↑𝑚 𝑉))) = (𝐿 × (𝑉 ↑𝑚 𝑉))
26	18, 25	sseqtri 3637	. 2 ⊢ 𝐾 ⊆ (𝐿 × (𝑉 ↑𝑚 𝑉))
27		ovex 6678	. . 3 ⊢ (𝑉 ↑𝑚 𝑉) ∈ V
28		fvex 6201	. . . 4 ⊢ (LSubSp‘𝑊) ∈ V
29	2, 28	eqeltri 2697	. . 3 ⊢ 𝐿 ∈ V
30	27, 29	elpm 7888	. 2 ⊢ (𝐾 ∈ ((𝑉 ↑𝑚 𝑉) ↑pm 𝐿) ↔ (Fun 𝐾 ∧ 𝐾 ⊆ (𝐿 × (𝑉 ↑𝑚 𝑉))))
31	11, 26, 30	mpbir2an 955	1 ⊢ 𝐾 ∈ ((𝑉 ↑𝑚 𝑉) ↑pm 𝐿)

Syntax hints:   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574   ↦ cmpt 4729   × cxp 5112  Fun wfun 5882  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857   ↑_pm cpm 7858  Basecbs 15857  proj₁cpj1 18050  LSubSpclss 18932  ocvcocv 20004  projcpj 20044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-pm 7860  df-pj 20047

This theorem is referenced by: (None)