Theorem sshjval3 29137

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 28782	. . . . . 6 ⊢ ℋ ∈ V
2	1	elpw2 5212	. . . . 5 ⊢ (𝐴 ∈ 𝒫 ℋ ↔ 𝐴 ⊆ ℋ)
3	1	elpw2 5212	. . . . 5 ⊢ (𝐵 ∈ 𝒫 ℋ ↔ 𝐵 ⊆ ℋ)
4		uniprg 4818	. . . . 5 ⊢ ((𝐴 ∈ 𝒫 ℋ ∧ 𝐵 ∈ 𝒫 ℋ) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
5	2, 3, 4	syl2anbr 601	. . . 4 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
6	5	fveq2d 6649	. . 3 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (⊥‘∪ {𝐴, 𝐵}) = (⊥‘(𝐴 ∪ 𝐵)))
7	6	fveq2d 6649	. 2 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (⊥‘(⊥‘∪ {𝐴, 𝐵})) = (⊥‘(⊥‘(𝐴 ∪ 𝐵))))
8		prssi 4714	. . . 4 ⊢ ((𝐴 ∈ 𝒫 ℋ ∧ 𝐵 ∈ 𝒫 ℋ) → {𝐴, 𝐵} ⊆ 𝒫 ℋ)
9	2, 3, 8	syl2anbr 601	. . 3 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → {𝐴, 𝐵} ⊆ 𝒫 ℋ)
10		hsupval 29117	. . 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 29133	. 2 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵))))
13	7, 11, 12	3eqtr4rd 2844	1 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ∨ℋ 𝐵) = ( ∨ℋ ‘{𝐴, 𝐵}))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ∪ cun 3879   ⊆ wss 3881  𝒫 cpw 4497  {cpr 4527  ∪ cuni 4800  ‘cfv 6324  (class class class)co 7135   ℋchba 28702  ⊥cort 28713   ∨_ℋ chj 28716   ∨_ℋ chsup 28717

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-hilex 28782

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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-chj 29093  df-chsup 29094

This theorem is referenced by: (None)