Theorem pjpm 21642

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 2728	. . . . 5 ⊢ (ocv‘𝑊) = (ocv‘𝑊)
4		eqid 2728	. . . . 5 ⊢ (proj1‘𝑊) = (proj1‘𝑊)
5		pjpm.k	. . . . 5 ⊢ 𝐾 = (proj‘𝑊)
6	1, 2, 3, 4, 5	pjfval 21640	. . . 4 ⊢ 𝐾 = ((𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉)))
7		inss1 4229	. . . 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 4014	. . 3 ⊢ 𝐾 ⊆ (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥)))
9		funmpt 6591	. . 3 ⊢ Fun (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥)))
10		funss 6572	. . 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 2728	. . . . . 6 ⊢ (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))) = (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥)))
13		ovexd 7455	. . . . . 6 ⊢ (𝑥 ∈ 𝐿 → (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥)) ∈ V)
14	12, 13	fmpti 7122	. . . . 5 ⊢ (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))):𝐿⟶V
15		fssxp 6751	. . . . 5 ⊢ ((𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))):𝐿⟶V → (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))) ⊆ (𝐿 × V))
16		ssrin 4234	. . . . 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 4014	. . 3 ⊢ 𝐾 ⊆ ((𝐿 × V) ∩ (V × (𝑉 ↑m 𝑉)))
19		inxp 5834	. . . 4 ⊢ ((𝐿 × V) ∩ (V × (𝑉 ↑m 𝑉))) = ((𝐿 ∩ V) × (V ∩ (𝑉 ↑m 𝑉)))
20		inv1 4395	. . . . 5 ⊢ (𝐿 ∩ V) = 𝐿
21		incom 4201	. . . . . 6 ⊢ (V ∩ (𝑉 ↑m 𝑉)) = ((𝑉 ↑m 𝑉) ∩ V)
22		inv1 4395	. . . . . 6 ⊢ ((𝑉 ↑m 𝑉) ∩ V) = (𝑉 ↑m 𝑉)
23	21, 22	eqtri 2756	. . . . 5 ⊢ (V ∩ (𝑉 ↑m 𝑉)) = (𝑉 ↑m 𝑉)
24	20, 23	xpeq12i 5706	. . . 4 ⊢ ((𝐿 ∩ V) × (V ∩ (𝑉 ↑m 𝑉))) = (𝐿 × (𝑉 ↑m 𝑉))
25	19, 24	eqtri 2756	. . 3 ⊢ ((𝐿 × V) ∩ (V × (𝑉 ↑m 𝑉))) = (𝐿 × (𝑉 ↑m 𝑉))
26	18, 25	sseqtri 4016	. 2 ⊢ 𝐾 ⊆ (𝐿 × (𝑉 ↑m 𝑉))
27		ovex 7453	. . 3 ⊢ (𝑉 ↑m 𝑉) ∈ V
28	2	fvexi 6911	. . 3 ⊢ 𝐿 ∈ V
29	27, 28	elpm 8892	. 2 ⊢ (𝐾 ∈ ((𝑉 ↑m 𝑉) ↑pm 𝐿) ↔ (Fun 𝐾 ∧ 𝐾 ⊆ (𝐿 × (𝑉 ↑m 𝑉))))
30	11, 26, 29	mpbir2an 710	1 ⊢ 𝐾 ∈ ((𝑉 ↑m 𝑉) ↑pm 𝐿)

Syntax hints:   = wceq 1534   ∈ wcel 2099  Vcvv 3471   ∩ cin 3946   ⊆ wss 3947   ↦ cmpt 5231   × cxp 5676  Fun wfun 6542  ⟶wf 6544  ‘cfv 6548  (class class class)co 7420   ↑_m cmap 8845   ↑_pm cpm 8846  Basecbs 17180  proj₁cpj1 19590  LSubSpclss 20815  ocvcocv 21592  projcpj 21634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-pm 8848  df-pj 21637

This theorem is referenced by: (None)