Theorem sshjval3 31956

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 31601	. . . . . 6 ⊢ ℋ ∈ V
2	1	elpw2 5295	. . . . 5 ⊢ (𝐴 ∈ 𝒫 ℋ ↔ 𝐴 ⊆ ℋ)
3	1	elpw2 5295	. . . . 5 ⊢ (𝐵 ∈ 𝒫 ℋ ↔ 𝐵 ⊆ ℋ)
4		uniprg 4882	. . . . 5 ⊢ ((𝐴 ∈ 𝒫 ℋ ∧ 𝐵 ∈ 𝒫 ℋ) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
5	2, 3, 4	syl2anbr 611	. . . 4 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
6	5	fveq2d 6889	. . 3 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (⊥‘∪ {𝐴, 𝐵}) = (⊥‘(𝐴 ∪ 𝐵)))
7	6	fveq2d 6889	. 2 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (⊥‘(⊥‘∪ {𝐴, 𝐵})) = (⊥‘(⊥‘(𝐴 ∪ 𝐵))))
8		prssi 4781	. . . 4 ⊢ ((𝐴 ∈ 𝒫 ℋ ∧ 𝐵 ∈ 𝒫 ℋ) → {𝐴, 𝐵} ⊆ 𝒫 ℋ)
9	2, 3, 8	syl2anbr 611	. . 3 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → {𝐴, 𝐵} ⊆ 𝒫 ℋ)
10		hsupval 31936	. . 3 ⊢ ({𝐴, 𝐵} ⊆ 𝒫 ℋ → ( ∨ℋ ‘{𝐴, 𝐵}) = (⊥‘(⊥‘∪ {𝐴, 𝐵})))
11	9, 10	syl 18	. 2 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → ( ∨ℋ ‘{𝐴, 𝐵}) = (⊥‘(⊥‘∪ {𝐴, 𝐵})))
12		sshjval 31952	. 2 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵))))
13	7, 11, 12	3eqtr4rd 2806	1 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ∨ℋ 𝐵) = ( ∨ℋ ‘{𝐴, 𝐵}))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895   ∪ cun 4149  𝒫 cpw 4556  {cpr 4585  ∪ cuni 4866  ‘cfv 6538  (class class class)co 7420   ℋchba 31521  ⊥cort 31532   ∨_ℋ chj 31535   ∨_ℋ chsup 31536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-hilex 31601

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-chj 31912  df-chsup 31913

This theorem is used by: (None)