Theorem pjpm 20830

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 2820	. . . . 5 ⊢ (ocv‘𝑊) = (ocv‘𝑊)
4		eqid 2820	. . . . 5 ⊢ (proj1‘𝑊) = (proj1‘𝑊)
5		pjpm.k	. . . . 5 ⊢ 𝐾 = (proj‘𝑊)
6	1, 2, 3, 4, 5	pjfval 20828	. . . 4 ⊢ 𝐾 = ((𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉)))
7		inss1 4188	. . . 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 3984	. . 3 ⊢ 𝐾 ⊆ (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥)))
9		funmpt 6374	. . 3 ⊢ Fun (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥)))
10		funss 6355	. . 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 2820	. . . . . 6 ⊢ (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))) = (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥)))
13		ovexd 7172	. . . . . 6 ⊢ (𝑥 ∈ 𝐿 → (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥)) ∈ V)
14	12, 13	fmpti 6857	. . . . 5 ⊢ (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))):𝐿⟶V
15		fssxp 6515	. . . . 5 ⊢ ((𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))):𝐿⟶V → (𝑥 ∈ 𝐿 ↦ (𝑥(proj1‘𝑊)((ocv‘𝑊)‘𝑥))) ⊆ (𝐿 × V))
16		ssrin 4193	. . . . 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 3984	. . 3 ⊢ 𝐾 ⊆ ((𝐿 × V) ∩ (V × (𝑉 ↑m 𝑉)))
19		inxp 5684	. . . 4 ⊢ ((𝐿 × V) ∩ (V × (𝑉 ↑m 𝑉))) = ((𝐿 ∩ V) × (V ∩ (𝑉 ↑m 𝑉)))
20		inv1 4329	. . . . 5 ⊢ (𝐿 ∩ V) = 𝐿
21		incom 4161	. . . . . 6 ⊢ (V ∩ (𝑉 ↑m 𝑉)) = ((𝑉 ↑m 𝑉) ∩ V)
22		inv1 4329	. . . . . 6 ⊢ ((𝑉 ↑m 𝑉) ∩ V) = (𝑉 ↑m 𝑉)
23	21, 22	eqtri 2843	. . . . 5 ⊢ (V ∩ (𝑉 ↑m 𝑉)) = (𝑉 ↑m 𝑉)
24	20, 23	xpeq12i 5564	. . . 4 ⊢ ((𝐿 ∩ V) × (V ∩ (𝑉 ↑m 𝑉))) = (𝐿 × (𝑉 ↑m 𝑉))
25	19, 24	eqtri 2843	. . 3 ⊢ ((𝐿 × V) ∩ (V × (𝑉 ↑m 𝑉))) = (𝐿 × (𝑉 ↑m 𝑉))
26	18, 25	sseqtri 3986	. 2 ⊢ 𝐾 ⊆ (𝐿 × (𝑉 ↑m 𝑉))
27		ovex 7170	. . 3 ⊢ (𝑉 ↑m 𝑉) ∈ V
28	2	fvexi 6665	. . 3 ⊢ 𝐿 ∈ V
29	27, 28	elpm 8418	. 2 ⊢ (𝐾 ∈ ((𝑉 ↑m 𝑉) ↑pm 𝐿) ↔ (Fun 𝐾 ∧ 𝐾 ⊆ (𝐿 × (𝑉 ↑m 𝑉))))
30	11, 26, 29	mpbir2an 709	1 ⊢ 𝐾 ∈ ((𝑉 ↑m 𝑉) ↑pm 𝐿)

Syntax hints:   = wceq 1537   ∈ wcel 2114  Vcvv 3481   ∩ cin 3918   ⊆ wss 3919   ↦ cmpt 5127   × cxp 5534  Fun wfun 6330  ⟶wf 6332  ‘cfv 6336  (class class class)co 7137   ↑_m cmap 8387   ↑_pm cpm 8388  Basecbs 16461  proj₁cpj1 18738  LSubSpclss 19681  ocvcocv 20782  projcpj 20822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7442

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3012  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3483  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-fv 6344  df-ov 7140  df-oprab 7141  df-mpo 7142  df-pm 8390  df-pj 20825

This theorem is referenced by: (None)