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Theorem sshjval3 29058
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 28703 . . . . . 6 ℋ ∈ V
21elpw2 5239 . . . . 5 (𝐴 ∈ 𝒫 ℋ ↔ 𝐴 ⊆ ℋ)
31elpw2 5239 . . . . 5 (𝐵 ∈ 𝒫 ℋ ↔ 𝐵 ⊆ ℋ)
4 uniprg 4844 . . . . 5 ((𝐴 ∈ 𝒫 ℋ ∧ 𝐵 ∈ 𝒫 ℋ) → {𝐴, 𝐵} = (𝐴𝐵))
52, 3, 4syl2anbr 598 . . . 4 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → {𝐴, 𝐵} = (𝐴𝐵))
65fveq2d 6667 . . 3 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (⊥‘ {𝐴, 𝐵}) = (⊥‘(𝐴𝐵)))
76fveq2d 6667 . 2 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (⊥‘(⊥‘ {𝐴, 𝐵})) = (⊥‘(⊥‘(𝐴𝐵))))
8 prssi 4746 . . . 4 ((𝐴 ∈ 𝒫 ℋ ∧ 𝐵 ∈ 𝒫 ℋ) → {𝐴, 𝐵} ⊆ 𝒫 ℋ)
92, 3, 8syl2anbr 598 . . 3 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → {𝐴, 𝐵} ⊆ 𝒫 ℋ)
10 hsupval 29038 . . 3 ({𝐴, 𝐵} ⊆ 𝒫 ℋ → ( ‘{𝐴, 𝐵}) = (⊥‘(⊥‘ {𝐴, 𝐵})))
119, 10syl 17 . 2 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → ( ‘{𝐴, 𝐵}) = (⊥‘(⊥‘ {𝐴, 𝐵})))
12 sshjval 29054 . 2 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))
137, 11, 123eqtr4rd 2864 1 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 𝐵) = ( ‘{𝐴, 𝐵}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  cun 3931  wss 3933  𝒫 cpw 4535  {cpr 4559   cuni 4830  cfv 6348  (class class class)co 7145  chba 28623  cort 28634   chj 28637   chsup 28638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-hilex 28703
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-chj 29014  df-chsup 29015
This theorem is referenced by: (None)
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