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Theorem sshjval3 28764
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 28407 . . . . . 6 ℋ ∈ V
21elpw2 5052 . . . . 5 (𝐴 ∈ 𝒫 ℋ ↔ 𝐴 ⊆ ℋ)
31elpw2 5052 . . . . 5 (𝐵 ∈ 𝒫 ℋ ↔ 𝐵 ⊆ ℋ)
4 uniprg 4674 . . . . 5 ((𝐴 ∈ 𝒫 ℋ ∧ 𝐵 ∈ 𝒫 ℋ) → {𝐴, 𝐵} = (𝐴𝐵))
52, 3, 4syl2anbr 592 . . . 4 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → {𝐴, 𝐵} = (𝐴𝐵))
65fveq2d 6441 . . 3 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (⊥‘ {𝐴, 𝐵}) = (⊥‘(𝐴𝐵)))
76fveq2d 6441 . 2 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (⊥‘(⊥‘ {𝐴, 𝐵})) = (⊥‘(⊥‘(𝐴𝐵))))
8 prssi 4572 . . . 4 ((𝐴 ∈ 𝒫 ℋ ∧ 𝐵 ∈ 𝒫 ℋ) → {𝐴, 𝐵} ⊆ 𝒫 ℋ)
92, 3, 8syl2anbr 592 . . 3 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → {𝐴, 𝐵} ⊆ 𝒫 ℋ)
10 hsupval 28744 . . 3 ({𝐴, 𝐵} ⊆ 𝒫 ℋ → ( ‘{𝐴, 𝐵}) = (⊥‘(⊥‘ {𝐴, 𝐵})))
119, 10syl 17 . 2 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → ( ‘{𝐴, 𝐵}) = (⊥‘(⊥‘ {𝐴, 𝐵})))
12 sshjval 28760 . 2 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))
137, 11, 123eqtr4rd 2872 1 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 𝐵) = ( ‘{𝐴, 𝐵}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wcel 2164  cun 3796  wss 3798  𝒫 cpw 4380  {cpr 4401   cuni 4660  cfv 6127  (class class class)co 6910  chba 28327  cort 28338   chj 28341   chsup 28342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-hilex 28407
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-iota 6090  df-fun 6129  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-chj 28720  df-chsup 28721
This theorem is referenced by: (None)
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