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Theorem sshjval3 27597
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. (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 𝐵) = ( ‘{𝐴, 𝐵}))

Proof of Theorem sshjval3
StepHypRef Expression
1 ax-hilex 27240 . . . . . 6 ℋ ∈ V
21elpw2 4755 . . . . 5 (𝐴 ∈ 𝒫 ℋ ↔ 𝐴 ⊆ ℋ)
31elpw2 4755 . . . . 5 (𝐵 ∈ 𝒫 ℋ ↔ 𝐵 ⊆ ℋ)
4 uniprg 4386 . . . . 5 ((𝐴 ∈ 𝒫 ℋ ∧ 𝐵 ∈ 𝒫 ℋ) → {𝐴, 𝐵} = (𝐴𝐵))
52, 3, 4syl2anbr 496 . . . 4 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → {𝐴, 𝐵} = (𝐴𝐵))
65fveq2d 6107 . . 3 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (⊥‘ {𝐴, 𝐵}) = (⊥‘(𝐴𝐵)))
76fveq2d 6107 . 2 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (⊥‘(⊥‘ {𝐴, 𝐵})) = (⊥‘(⊥‘(𝐴𝐵))))
8 prssi 4293 . . . 4 ((𝐴 ∈ 𝒫 ℋ ∧ 𝐵 ∈ 𝒫 ℋ) → {𝐴, 𝐵} ⊆ 𝒫 ℋ)
92, 3, 8syl2anbr 496 . . 3 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → {𝐴, 𝐵} ⊆ 𝒫 ℋ)
10 hsupval 27577 . . 3 ({𝐴, 𝐵} ⊆ 𝒫 ℋ → ( ‘{𝐴, 𝐵}) = (⊥‘(⊥‘ {𝐴, 𝐵})))
119, 10syl 17 . 2 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → ( ‘{𝐴, 𝐵}) = (⊥‘(⊥‘ {𝐴, 𝐵})))
12 sshjval 27593 . 2 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))
137, 11, 123eqtr4rd 2655 1 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 𝐵) = ( ‘{𝐴, 𝐵}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  cun 3538  wss 3540  𝒫 cpw 4108  {cpr 4127   cuni 4372  cfv 5804  (class class class)co 6549  chil 27160  cort 27171   chj 27174   chsup 27175
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-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-hilex 27240
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-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-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-chj 27553  df-chsup 27554
This theorem is referenced by: (None)
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