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Theorem sshjval3 31383
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 31028 . . . . . 6 ℋ ∈ V
21elpw2 5340 . . . . 5 (𝐴 ∈ 𝒫 ℋ ↔ 𝐴 ⊆ ℋ)
31elpw2 5340 . . . . 5 (𝐵 ∈ 𝒫 ℋ ↔ 𝐵 ⊆ ℋ)
4 uniprg 4928 . . . . 5 ((𝐴 ∈ 𝒫 ℋ ∧ 𝐵 ∈ 𝒫 ℋ) → {𝐴, 𝐵} = (𝐴𝐵))
52, 3, 4syl2anbr 599 . . . 4 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → {𝐴, 𝐵} = (𝐴𝐵))
65fveq2d 6911 . . 3 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (⊥‘ {𝐴, 𝐵}) = (⊥‘(𝐴𝐵)))
76fveq2d 6911 . 2 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (⊥‘(⊥‘ {𝐴, 𝐵})) = (⊥‘(⊥‘(𝐴𝐵))))
8 prssi 4826 . . . 4 ((𝐴 ∈ 𝒫 ℋ ∧ 𝐵 ∈ 𝒫 ℋ) → {𝐴, 𝐵} ⊆ 𝒫 ℋ)
92, 3, 8syl2anbr 599 . . 3 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → {𝐴, 𝐵} ⊆ 𝒫 ℋ)
10 hsupval 31363 . . 3 ({𝐴, 𝐵} ⊆ 𝒫 ℋ → ( ‘{𝐴, 𝐵}) = (⊥‘(⊥‘ {𝐴, 𝐵})))
119, 10syl 17 . 2 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → ( ‘{𝐴, 𝐵}) = (⊥‘(⊥‘ {𝐴, 𝐵})))
12 sshjval 31379 . 2 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))
137, 11, 123eqtr4rd 2786 1 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 𝐵) = ( ‘{𝐴, 𝐵}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cun 3961  wss 3963  𝒫 cpw 4605  {cpr 4633   cuni 4912  cfv 6563  (class class class)co 7431  chba 30948  cort 30959   chj 30962   chsup 30963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-hilex 31028
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-chj 31339  df-chsup 31340
This theorem is referenced by: (None)
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