Theorem pjpm 19871

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 2610	. . . . 5 ⊢ (ocv‘𝑊) = (ocv‘𝑊)
4		eqid 2610	. . . . 5 ⊢ (proj1‘𝑊) = (proj1‘𝑊)
5		pjpm.k	. . . . 5 ⊢ 𝐾 = (proj‘𝑊)
6	1, 2, 3, 4, 5	pjfval 19869	. . . 4 ⊢ 𝐾 = ((𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))) ∩ (V × (𝑉 ↑𝑚 𝑉)))
7		inss1 3795	. . . 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 3598	. . 3 ⊢ 𝐾 ⊆ (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥)))
9		funmpt 5840	. . 3 ⊢ Fun (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥)))
10		funss 5822	. . 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 2610	. . . . . 6 ⊢ (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))) = (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥)))
13		ovex 6577	. . . . . . 7 ⊢ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥)) ∈ V
14	13	a1i 11	. . . . . 6 ⊢ (𝑥 ∈ 𝐿 → (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥)) ∈ V)
15	12, 14	fmpti 6291	. . . . 5 ⊢ (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))):𝐿⟶V
16		fssxp 5973	. . . . 5 ⊢ ((𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))):𝐿⟶V → (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))) ⊆ (𝐿 × V))
17		ssrin 3800	. . . . 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))))
18	15, 16, 17	mp2b 10	. . . 4 ⊢ ((𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))) ∩ (V × (𝑉 ↑𝑚 𝑉))) ⊆ ((𝐿 × V) ∩ (V × (𝑉 ↑𝑚 𝑉)))
19	6, 18	eqsstri 3598	. . 3 ⊢ 𝐾 ⊆ ((𝐿 × V) ∩ (V × (𝑉 ↑𝑚 𝑉)))
20		inxp 5176	. . . 4 ⊢ ((𝐿 × V) ∩ (V × (𝑉 ↑𝑚 𝑉))) = ((𝐿 ∩ V) × (V ∩ (𝑉 ↑𝑚 𝑉)))
21		inv1 3922	. . . . 5 ⊢ (𝐿 ∩ V) = 𝐿
22		incom 3767	. . . . . 6 ⊢ (V ∩ (𝑉 ↑𝑚 𝑉)) = ((𝑉 ↑𝑚 𝑉) ∩ V)
23		inv1 3922	. . . . . 6 ⊢ ((𝑉 ↑𝑚 𝑉) ∩ V) = (𝑉 ↑𝑚 𝑉)
24	22, 23	eqtri 2632	. . . . 5 ⊢ (V ∩ (𝑉 ↑𝑚 𝑉)) = (𝑉 ↑𝑚 𝑉)
25	21, 24	xpeq12i 5061	. . . 4 ⊢ ((𝐿 ∩ V) × (V ∩ (𝑉 ↑𝑚 𝑉))) = (𝐿 × (𝑉 ↑𝑚 𝑉))
26	20, 25	eqtri 2632	. . 3 ⊢ ((𝐿 × V) ∩ (V × (𝑉 ↑𝑚 𝑉))) = (𝐿 × (𝑉 ↑𝑚 𝑉))
27	19, 26	sseqtri 3600	. 2 ⊢ 𝐾 ⊆ (𝐿 × (𝑉 ↑𝑚 𝑉))
28		ovex 6577	. . 3 ⊢ (𝑉 ↑𝑚 𝑉) ∈ V
29		fvex 6113	. . . 4 ⊢ (LSubSp‘𝑊) ∈ V
30	2, 29	eqeltri 2684	. . 3 ⊢ 𝐿 ∈ V
31	28, 30	elpm 7774	. 2 ⊢ (𝐾 ∈ ((𝑉 ↑𝑚 𝑉) ↑pm 𝐿) ↔ (Fun 𝐾 ∧ 𝐾 ⊆ (𝐿 × (𝑉 ↑𝑚 𝑉))))
32	11, 27, 31	mpbir2an 957	1 ⊢ 𝐾 ∈ ((𝑉 ↑𝑚 𝑉) ↑pm 𝐿)

Syntax hints:   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540   ↦ cmpt 4643   × cxp 5036  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑_𝑚 cmap 7744   ↑_pm cpm 7745  Basecbs 15695  proj₁cpj1 17873  LSubSpclss 18753  ocvcocv 19823  projcpj 19863

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-pm 7747  df-pj 19866

This theorem is referenced by: (None)