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Theorem sshjval3 29140
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 28785 . . . . . 6 ℋ ∈ V
21elpw2 5234 . . . . 5 (𝐴 ∈ 𝒫 ℋ ↔ 𝐴 ⊆ ℋ)
31elpw2 5234 . . . . 5 (𝐵 ∈ 𝒫 ℋ ↔ 𝐵 ⊆ ℋ)
4 uniprg 4842 . . . . 5 ((𝐴 ∈ 𝒫 ℋ ∧ 𝐵 ∈ 𝒫 ℋ) → {𝐴, 𝐵} = (𝐴𝐵))
52, 3, 4syl2anbr 601 . . . 4 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → {𝐴, 𝐵} = (𝐴𝐵))
65fveq2d 6665 . . 3 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (⊥‘ {𝐴, 𝐵}) = (⊥‘(𝐴𝐵)))
76fveq2d 6665 . 2 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (⊥‘(⊥‘ {𝐴, 𝐵})) = (⊥‘(⊥‘(𝐴𝐵))))
8 prssi 4738 . . . 4 ((𝐴 ∈ 𝒫 ℋ ∧ 𝐵 ∈ 𝒫 ℋ) → {𝐴, 𝐵} ⊆ 𝒫 ℋ)
92, 3, 8syl2anbr 601 . . 3 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → {𝐴, 𝐵} ⊆ 𝒫 ℋ)
10 hsupval 29120 . . 3 ({𝐴, 𝐵} ⊆ 𝒫 ℋ → ( ‘{𝐴, 𝐵}) = (⊥‘(⊥‘ {𝐴, 𝐵})))
119, 10syl 17 . 2 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → ( ‘{𝐴, 𝐵}) = (⊥‘(⊥‘ {𝐴, 𝐵})))
12 sshjval 29136 . 2 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))
137, 11, 123eqtr4rd 2870 1 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 𝐵) = ( ‘{𝐴, 𝐵}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  cun 3917  wss 3919  𝒫 cpw 4522  {cpr 4552   cuni 4824  cfv 6343  (class class class)co 7149  chba 28705  cort 28716   chj 28719   chsup 28720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-hilex 28785
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-chj 29096  df-chsup 29097
This theorem is referenced by: (None)
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