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 ∈ ((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 2737	. . . . 5 ⊢ (ocv‘𝑊) = (ocv‘𝑊)
4		eqid 2737	. . . . 5 ⊢ (proj1‘𝑊) = (proj1‘𝑊)
5		pjpm.k	. . . . 5 ⊢ 𝐾 = (proj‘𝑊)
6	1, 2, 3, 4, 5	pjfval 21726	. . . 4 ⊢ 𝐾 = ((𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉)))
7		inss1 4237	. . . 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 4030	. . 3 ⊢ 𝐾 ⊆ (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥)))
9		funmpt 6604	. . 3 ⊢ Fun (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥)))
10		funss 6585	. . 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 2737	. . . . . 6 ⊢ (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))) = (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥)))
13		ovexd 7466	. . . . . 6 ⊢ (𝑥 ∈ 𝐿 → (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥)) ∈ V)
14	12, 13	fmpti 7132	. . . . 5 ⊢ (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))):𝐿⟶V
15		fssxp 6763	. . . . 5 ⊢ ((𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))):𝐿⟶V → (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))) ⊆ (𝐿 × V))
16		ssrin 4242	. . . . 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 4030	. . 3 ⊢ 𝐾 ⊆ ((𝐿 × V) ∩ (V × (𝑉 ↑m 𝑉)))
19		inxp 5842	. . . 4 ⊢ ((𝐿 × V) ∩ (V × (𝑉 ↑m 𝑉))) = ((𝐿 ∩ V) × (V ∩ (𝑉 ↑m 𝑉)))
20		inv1 4398	. . . . 5 ⊢ (𝐿 ∩ V) = 𝐿
21		incom 4209	. . . . . 6 ⊢ (V ∩ (𝑉 ↑m 𝑉)) = ((𝑉 ↑m 𝑉) ∩ V)
22		inv1 4398	. . . . . 6 ⊢ ((𝑉 ↑m 𝑉) ∩ V) = (𝑉 ↑m 𝑉)
23	21, 22	eqtri 2765	. . . . 5 ⊢ (V ∩ (𝑉 ↑m 𝑉)) = (𝑉 ↑m 𝑉)
24	20, 23	xpeq12i 5713	. . . 4 ⊢ ((𝐿 ∩ V) × (V ∩ (𝑉 ↑m 𝑉))) = (𝐿 × (𝑉 ↑m 𝑉))
25	19, 24	eqtri 2765	. . 3 ⊢ ((𝐿 × V) ∩ (V × (𝑉 ↑m 𝑉))) = (𝐿 × (𝑉 ↑m 𝑉))
26	18, 25	sseqtri 4032	. 2 ⊢ 𝐾 ⊆ (𝐿 × (𝑉 ↑m 𝑉))
27		ovex 7464	. . 3 ⊢ (𝑉 ↑m 𝑉) ∈ V
28	2	fvexi 6920	. . 3 ⊢ 𝐿 ∈ V
29	27, 28	elpm 8913	. 2 ⊢ (𝐾 ∈ ((𝑉 ↑m 𝑉) ↑pm 𝐿) ↔ (Fun 𝐾 ∧ 𝐾 ⊆ (𝐿 × (𝑉 ↑m 𝑉))))
30	11, 26, 29	mpbir2an 711	1 ⊢ 𝐾 ∈ ((𝑉 ↑m 𝑉) ↑pm 𝐿)

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)