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Mirrors > Home > HSE Home > Th. List > elat2 | Structured version Visualization version GIF version |
Description: Expanded membership relation for the set of atoms, i.e. the predicate "is an atom (of the Hilbert lattice)." An atom is a nonzero element of a lattice such that anything less than it is zero, i.e. it is the smallest nonzero element of the lattice. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elat2 | ⊢ (𝐴 ∈ HAtoms ↔ (𝐴 ∈ Cℋ ∧ (𝐴 ≠ 0ℋ ∧ ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐴 → (𝑥 = 𝐴 ∨ 𝑥 = 0ℋ))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ela 32148 | . 2 ⊢ (𝐴 ∈ HAtoms ↔ (𝐴 ∈ Cℋ ∧ 0ℋ ⋖ℋ 𝐴)) | |
2 | h0elch 31064 | . . . . 5 ⊢ 0ℋ ∈ Cℋ | |
3 | cvbr2 32092 | . . . . 5 ⊢ ((0ℋ ∈ Cℋ ∧ 𝐴 ∈ Cℋ ) → (0ℋ ⋖ℋ 𝐴 ↔ (0ℋ ⊊ 𝐴 ∧ ∀𝑥 ∈ Cℋ ((0ℋ ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐴) → 𝑥 = 𝐴)))) | |
4 | 2, 3 | mpan 689 | . . . 4 ⊢ (𝐴 ∈ Cℋ → (0ℋ ⋖ℋ 𝐴 ↔ (0ℋ ⊊ 𝐴 ∧ ∀𝑥 ∈ Cℋ ((0ℋ ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐴) → 𝑥 = 𝐴)))) |
5 | ch0pss 31254 | . . . . 5 ⊢ (𝐴 ∈ Cℋ → (0ℋ ⊊ 𝐴 ↔ 𝐴 ≠ 0ℋ)) | |
6 | ch0pss 31254 | . . . . . . . . . 10 ⊢ (𝑥 ∈ Cℋ → (0ℋ ⊊ 𝑥 ↔ 𝑥 ≠ 0ℋ)) | |
7 | 6 | imbi1d 341 | . . . . . . . . 9 ⊢ (𝑥 ∈ Cℋ → ((0ℋ ⊊ 𝑥 → 𝑥 = 𝐴) ↔ (𝑥 ≠ 0ℋ → 𝑥 = 𝐴))) |
8 | 7 | imbi2d 340 | . . . . . . . 8 ⊢ (𝑥 ∈ Cℋ → ((𝑥 ⊆ 𝐴 → (0ℋ ⊊ 𝑥 → 𝑥 = 𝐴)) ↔ (𝑥 ⊆ 𝐴 → (𝑥 ≠ 0ℋ → 𝑥 = 𝐴)))) |
9 | impexp 450 | . . . . . . . . 9 ⊢ (((0ℋ ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐴) → 𝑥 = 𝐴) ↔ (0ℋ ⊊ 𝑥 → (𝑥 ⊆ 𝐴 → 𝑥 = 𝐴))) | |
10 | bi2.04 387 | . . . . . . . . 9 ⊢ ((0ℋ ⊊ 𝑥 → (𝑥 ⊆ 𝐴 → 𝑥 = 𝐴)) ↔ (𝑥 ⊆ 𝐴 → (0ℋ ⊊ 𝑥 → 𝑥 = 𝐴))) | |
11 | 9, 10 | bitri 275 | . . . . . . . 8 ⊢ (((0ℋ ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐴) → 𝑥 = 𝐴) ↔ (𝑥 ⊆ 𝐴 → (0ℋ ⊊ 𝑥 → 𝑥 = 𝐴))) |
12 | orcom 869 | . . . . . . . . . 10 ⊢ ((𝑥 = 𝐴 ∨ 𝑥 = 0ℋ) ↔ (𝑥 = 0ℋ ∨ 𝑥 = 𝐴)) | |
13 | neor 3031 | . . . . . . . . . 10 ⊢ ((𝑥 = 0ℋ ∨ 𝑥 = 𝐴) ↔ (𝑥 ≠ 0ℋ → 𝑥 = 𝐴)) | |
14 | 12, 13 | bitri 275 | . . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∨ 𝑥 = 0ℋ) ↔ (𝑥 ≠ 0ℋ → 𝑥 = 𝐴)) |
15 | 14 | imbi2i 336 | . . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 → (𝑥 = 𝐴 ∨ 𝑥 = 0ℋ)) ↔ (𝑥 ⊆ 𝐴 → (𝑥 ≠ 0ℋ → 𝑥 = 𝐴))) |
16 | 8, 11, 15 | 3bitr4g 314 | . . . . . . 7 ⊢ (𝑥 ∈ Cℋ → (((0ℋ ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐴) → 𝑥 = 𝐴) ↔ (𝑥 ⊆ 𝐴 → (𝑥 = 𝐴 ∨ 𝑥 = 0ℋ)))) |
17 | 16 | ralbiia 3088 | . . . . . 6 ⊢ (∀𝑥 ∈ Cℋ ((0ℋ ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐴) → 𝑥 = 𝐴) ↔ ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐴 → (𝑥 = 𝐴 ∨ 𝑥 = 0ℋ))) |
18 | 17 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ Cℋ → (∀𝑥 ∈ Cℋ ((0ℋ ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐴) → 𝑥 = 𝐴) ↔ ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐴 → (𝑥 = 𝐴 ∨ 𝑥 = 0ℋ)))) |
19 | 5, 18 | anbi12d 631 | . . . 4 ⊢ (𝐴 ∈ Cℋ → ((0ℋ ⊊ 𝐴 ∧ ∀𝑥 ∈ Cℋ ((0ℋ ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐴) → 𝑥 = 𝐴)) ↔ (𝐴 ≠ 0ℋ ∧ ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐴 → (𝑥 = 𝐴 ∨ 𝑥 = 0ℋ))))) |
20 | 4, 19 | bitr2d 280 | . . 3 ⊢ (𝐴 ∈ Cℋ → ((𝐴 ≠ 0ℋ ∧ ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐴 → (𝑥 = 𝐴 ∨ 𝑥 = 0ℋ))) ↔ 0ℋ ⋖ℋ 𝐴)) |
21 | 20 | pm5.32i 574 | . 2 ⊢ ((𝐴 ∈ Cℋ ∧ (𝐴 ≠ 0ℋ ∧ ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐴 → (𝑥 = 𝐴 ∨ 𝑥 = 0ℋ)))) ↔ (𝐴 ∈ Cℋ ∧ 0ℋ ⋖ℋ 𝐴)) |
22 | 1, 21 | bitr4i 278 | 1 ⊢ (𝐴 ∈ HAtoms ↔ (𝐴 ∈ Cℋ ∧ (𝐴 ≠ 0ℋ ∧ ∀𝑥 ∈ Cℋ (𝑥 ⊆ 𝐴 → (𝑥 = 𝐴 ∨ 𝑥 = 0ℋ))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ∀wral 3058 ⊆ wss 3947 ⊊ wpss 3948 class class class wbr 5148 Cℋ cch 30738 0ℋc0h 30744 ⋖ℋ ccv 30773 HAtomscat 30774 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217 ax-mulf 11218 ax-hilex 30808 ax-hfvadd 30809 ax-hvcom 30810 ax-hvass 30811 ax-hv0cl 30812 ax-hvaddid 30813 ax-hfvmul 30814 ax-hvmulid 30815 ax-hvmulass 30816 ax-hvdistr1 30817 ax-hvdistr2 30818 ax-hvmul0 30819 ax-hfi 30888 ax-his1 30891 ax-his2 30892 ax-his3 30893 ax-his4 30894 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-map 8846 df-pm 8847 df-en 8964 df-dom 8965 df-sdom 8966 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-n0 12503 df-z 12589 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-icc 13363 df-seq 13999 df-exp 14059 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-topgen 17424 df-psmet 21270 df-xmet 21271 df-met 21272 df-bl 21273 df-mopn 21274 df-top 22795 df-topon 22812 df-bases 22848 df-lm 23132 df-haus 23218 df-grpo 30302 df-gid 30303 df-ginv 30304 df-gdiv 30305 df-ablo 30354 df-vc 30368 df-nv 30401 df-va 30404 df-ba 30405 df-sm 30406 df-0v 30407 df-vs 30408 df-nmcv 30409 df-ims 30410 df-hnorm 30777 df-hvsub 30780 df-hlim 30781 df-sh 31016 df-ch 31030 df-ch0 31062 df-cv 32088 df-at 32147 |
This theorem is referenced by: atne0 32154 atss 32155 h1da 32158 atom1d 32162 |
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