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Mirrors > Home > HSE Home > Th. List > sshjval3 | Structured version Visualization version GIF version |
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 Cℋ. (Contributed by NM, 2-Mar-2004.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sshjval3 | ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ∨ℋ 𝐵) = ( ∨ℋ ‘{𝐴, 𝐵})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-hilex 27856 | . . . . . 6 ⊢ ℋ ∈ V | |
2 | 1 | elpw2 4828 | . . . . 5 ⊢ (𝐴 ∈ 𝒫 ℋ ↔ 𝐴 ⊆ ℋ) |
3 | 1 | elpw2 4828 | . . . . 5 ⊢ (𝐵 ∈ 𝒫 ℋ ↔ 𝐵 ⊆ ℋ) |
4 | uniprg 4450 | . . . . 5 ⊢ ((𝐴 ∈ 𝒫 ℋ ∧ 𝐵 ∈ 𝒫 ℋ) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) | |
5 | 2, 3, 4 | syl2anbr 497 | . . . 4 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) |
6 | 5 | fveq2d 6195 | . . 3 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (⊥‘∪ {𝐴, 𝐵}) = (⊥‘(𝐴 ∪ 𝐵))) |
7 | 6 | fveq2d 6195 | . 2 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (⊥‘(⊥‘∪ {𝐴, 𝐵})) = (⊥‘(⊥‘(𝐴 ∪ 𝐵)))) |
8 | prssi 4353 | . . . 4 ⊢ ((𝐴 ∈ 𝒫 ℋ ∧ 𝐵 ∈ 𝒫 ℋ) → {𝐴, 𝐵} ⊆ 𝒫 ℋ) | |
9 | 2, 3, 8 | syl2anbr 497 | . . 3 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → {𝐴, 𝐵} ⊆ 𝒫 ℋ) |
10 | hsupval 28193 | . . 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 28209 | . 2 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵)))) | |
13 | 7, 11, 12 | 3eqtr4rd 2667 | 1 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ∨ℋ 𝐵) = ( ∨ℋ ‘{𝐴, 𝐵})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∪ cun 3572 ⊆ wss 3574 𝒫 cpw 4158 {cpr 4179 ∪ cuni 4436 ‘cfv 5888 (class class class)co 6650 ℋchil 27776 ⊥cort 27787 ∨ℋ chj 27790 ∨ℋ chsup 27791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-hilex 27856 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-chj 28169 df-chsup 28170 |
This theorem is referenced by: (None) |
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