Theorem sshjval3 31412

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_ℋ. (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 ∨ℋ 𝐵) = ( ∨ℋ ‘{𝐴, 𝐵}))

Proof of Theorem sshjval3

Step	Hyp	Ref	Expression
1		ax-hilex 31057	. . . . . 6 ⊢ ℋ ∈ V
2	1	elpw2 5280	. . . . 5 ⊢ (𝐴 ∈ 𝒫 ℋ ↔ 𝐴 ⊆ ℋ)
3	1	elpw2 5280	. . . . 5 ⊢ (𝐵 ∈ 𝒫 ℋ ↔ 𝐵 ⊆ ℋ)
4		uniprg 4880	. . . . 5 ⊢ ((𝐴 ∈ 𝒫 ℋ ∧ 𝐵 ∈ 𝒫 ℋ) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
5	2, 3, 4	syl2anbr 600	. . . 4 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
6	5	fveq2d 6839	. . 3 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (⊥‘∪ {𝐴, 𝐵}) = (⊥‘(𝐴 ∪ 𝐵)))
7	6	fveq2d 6839	. 2 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (⊥‘(⊥‘∪ {𝐴, 𝐵})) = (⊥‘(⊥‘(𝐴 ∪ 𝐵))))
8		prssi 4778	. . . 4 ⊢ ((𝐴 ∈ 𝒫 ℋ ∧ 𝐵 ∈ 𝒫 ℋ) → {𝐴, 𝐵} ⊆ 𝒫 ℋ)
9	2, 3, 8	syl2anbr 600	. . 3 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → {𝐴, 𝐵} ⊆ 𝒫 ℋ)
10		hsupval 31392	. . 3 ⊢ ({𝐴, 𝐵} ⊆ 𝒫 ℋ → ( ∨ℋ ‘{𝐴, 𝐵}) = (⊥‘(⊥‘∪ {𝐴, 𝐵})))
11	9, 10	syl 17	. 2 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → ( ∨ℋ ‘{𝐴, 𝐵}) = (⊥‘(⊥‘∪ {𝐴, 𝐵})))
12		sshjval 31408	. 2 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵))))
13	7, 11, 12	3eqtr4rd 2783	1 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ∨ℋ 𝐵) = ( ∨ℋ ‘{𝐴, 𝐵}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∪ cun 3900   ⊆ wss 3902  𝒫 cpw 4555  {cpr 4583  ∪ cuni 4864  ‘cfv 6493  (class class class)co 7360   ℋchba 30977  ⊥cort 30988   ∨_ℋ chj 30991   ∨_ℋ chsup 30992

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-hilex 31057

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6449  df-fun 6495  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-chj 31368  df-chsup 31369

This theorem is referenced by: (None)