Theorem sshjval3 31316

Description: Value of join for subsets of Hilbert space in terms of supremum: the join is the supremum of its two arguments. Based on the definition of join in [Beran] p. 3. For later convenience we prove a general version that works for any subset of Hilbert space, not just the elements of the lattice C_ℋ. (Contributed by NM, 2-Mar-2004.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
sshjval3	⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ∨ℋ 𝐵) = ( ∨ℋ ‘{𝐴, 𝐵}))

Proof of Theorem sshjval3

Step	Hyp	Ref	Expression
1		ax-hilex 30961	. . . . . 6 ⊢ ℋ ∈ V
2	1	elpw2 5276	. . . . 5 ⊢ (𝐴 ∈ 𝒫 ℋ ↔ 𝐴 ⊆ ℋ)
3	1	elpw2 5276	. . . . 5 ⊢ (𝐵 ∈ 𝒫 ℋ ↔ 𝐵 ⊆ ℋ)
4		uniprg 4877	. . . . 5 ⊢ ((𝐴 ∈ 𝒫 ℋ ∧ 𝐵 ∈ 𝒫 ℋ) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
5	2, 3, 4	syl2anbr 599	. . . 4 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
6	5	fveq2d 6830	. . 3 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (⊥‘∪ {𝐴, 𝐵}) = (⊥‘(𝐴 ∪ 𝐵)))
7	6	fveq2d 6830	. 2 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (⊥‘(⊥‘∪ {𝐴, 𝐵})) = (⊥‘(⊥‘(𝐴 ∪ 𝐵))))
8		prssi 4775	. . . 4 ⊢ ((𝐴 ∈ 𝒫 ℋ ∧ 𝐵 ∈ 𝒫 ℋ) → {𝐴, 𝐵} ⊆ 𝒫 ℋ)
9	2, 3, 8	syl2anbr 599	. . 3 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → {𝐴, 𝐵} ⊆ 𝒫 ℋ)
10		hsupval 31296	. . 3 ⊢ ({𝐴, 𝐵} ⊆ 𝒫 ℋ → ( ∨ℋ ‘{𝐴, 𝐵}) = (⊥‘(⊥‘∪ {𝐴, 𝐵})))
11	9, 10	syl 17	. 2 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → ( ∨ℋ ‘{𝐴, 𝐵}) = (⊥‘(⊥‘∪ {𝐴, 𝐵})))
12		sshjval 31312	. 2 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵))))
13	7, 11, 12	3eqtr4rd 2775	1 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ∨ℋ 𝐵) = ( ∨ℋ ‘{𝐴, 𝐵}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∪ cun 3903   ⊆ wss 3905  𝒫 cpw 4553  {cpr 4581  ∪ cuni 4861  ‘cfv 6486  (class class class)co 7353   ℋchba 30881  ⊥cort 30892   ∨_ℋ chj 30895   ∨_ℋ chsup 30896

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-hilex 30961

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-chj 31272  df-chsup 31273

This theorem is referenced by: (None)