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Mirrors > Home > HSE Home > Th. List > atcvati | Structured version Visualization version GIF version |
Description: A nonzero Hilbert lattice element less than the join of two atoms is an atom. (Contributed by NM, 28-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
atoml.1 | ⊢ 𝐴 ∈ Cℋ |
Ref | Expression |
---|---|
atcvati | ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((𝐴 ≠ 0ℋ ∧ 𝐴 ⊊ (𝐵 ∨ℋ 𝐶)) → 𝐴 ∈ HAtoms)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | atoml.1 | . . . 4 ⊢ 𝐴 ∈ Cℋ | |
2 | 1 | atcvatlem 29933 | . . 3 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ (𝐴 ≠ 0ℋ ∧ 𝐴 ⊊ (𝐵 ∨ℋ 𝐶))) → (¬ 𝐵 ⊆ 𝐴 → 𝐴 ∈ HAtoms)) |
3 | atelch 29892 | . . . . . . . . 9 ⊢ (𝐶 ∈ HAtoms → 𝐶 ∈ Cℋ ) | |
4 | atelch 29892 | . . . . . . . . 9 ⊢ (𝐵 ∈ HAtoms → 𝐵 ∈ Cℋ ) | |
5 | chjcom 29054 | . . . . . . . . 9 ⊢ ((𝐶 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐶 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐶)) | |
6 | 3, 4, 5 | syl2an 586 | . . . . . . . 8 ⊢ ((𝐶 ∈ HAtoms ∧ 𝐵 ∈ HAtoms) → (𝐶 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐶)) |
7 | 6 | psseq2d 3956 | . . . . . . 7 ⊢ ((𝐶 ∈ HAtoms ∧ 𝐵 ∈ HAtoms) → (𝐴 ⊊ (𝐶 ∨ℋ 𝐵) ↔ 𝐴 ⊊ (𝐵 ∨ℋ 𝐶))) |
8 | 7 | anbi2d 619 | . . . . . 6 ⊢ ((𝐶 ∈ HAtoms ∧ 𝐵 ∈ HAtoms) → ((𝐴 ≠ 0ℋ ∧ 𝐴 ⊊ (𝐶 ∨ℋ 𝐵)) ↔ (𝐴 ≠ 0ℋ ∧ 𝐴 ⊊ (𝐵 ∨ℋ 𝐶)))) |
9 | 1 | atcvatlem 29933 | . . . . . . 7 ⊢ (((𝐶 ∈ HAtoms ∧ 𝐵 ∈ HAtoms) ∧ (𝐴 ≠ 0ℋ ∧ 𝐴 ⊊ (𝐶 ∨ℋ 𝐵))) → (¬ 𝐶 ⊆ 𝐴 → 𝐴 ∈ HAtoms)) |
10 | 9 | ex 405 | . . . . . 6 ⊢ ((𝐶 ∈ HAtoms ∧ 𝐵 ∈ HAtoms) → ((𝐴 ≠ 0ℋ ∧ 𝐴 ⊊ (𝐶 ∨ℋ 𝐵)) → (¬ 𝐶 ⊆ 𝐴 → 𝐴 ∈ HAtoms))) |
11 | 8, 10 | sylbird 252 | . . . . 5 ⊢ ((𝐶 ∈ HAtoms ∧ 𝐵 ∈ HAtoms) → ((𝐴 ≠ 0ℋ ∧ 𝐴 ⊊ (𝐵 ∨ℋ 𝐶)) → (¬ 𝐶 ⊆ 𝐴 → 𝐴 ∈ HAtoms))) |
12 | 11 | ancoms 451 | . . . 4 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((𝐴 ≠ 0ℋ ∧ 𝐴 ⊊ (𝐵 ∨ℋ 𝐶)) → (¬ 𝐶 ⊆ 𝐴 → 𝐴 ∈ HAtoms))) |
13 | 12 | imp 398 | . . 3 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ (𝐴 ≠ 0ℋ ∧ 𝐴 ⊊ (𝐵 ∨ℋ 𝐶))) → (¬ 𝐶 ⊆ 𝐴 → 𝐴 ∈ HAtoms)) |
14 | chlub 29057 | . . . . . . . . . . 11 ⊢ ((𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ) → ((𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) ↔ (𝐵 ∨ℋ 𝐶) ⊆ 𝐴)) | |
15 | 14 | 3comr 1105 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ((𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) ↔ (𝐵 ∨ℋ 𝐶) ⊆ 𝐴)) |
16 | ssnpss 3966 | . . . . . . . . . 10 ⊢ ((𝐵 ∨ℋ 𝐶) ⊆ 𝐴 → ¬ 𝐴 ⊊ (𝐵 ∨ℋ 𝐶)) | |
17 | 15, 16 | syl6bi 245 | . . . . . . . . 9 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ((𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → ¬ 𝐴 ⊊ (𝐵 ∨ℋ 𝐶))) |
18 | 17 | con2d 132 | . . . . . . . 8 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⊊ (𝐵 ∨ℋ 𝐶) → ¬ (𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴))) |
19 | ianor 964 | . . . . . . . 8 ⊢ (¬ (𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) ↔ (¬ 𝐵 ⊆ 𝐴 ∨ ¬ 𝐶 ⊆ 𝐴)) | |
20 | 18, 19 | syl6ib 243 | . . . . . . 7 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⊊ (𝐵 ∨ℋ 𝐶) → (¬ 𝐵 ⊆ 𝐴 ∨ ¬ 𝐶 ⊆ 𝐴))) |
21 | 1, 20 | mp3an1 1427 | . . . . . 6 ⊢ ((𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⊊ (𝐵 ∨ℋ 𝐶) → (¬ 𝐵 ⊆ 𝐴 ∨ ¬ 𝐶 ⊆ 𝐴))) |
22 | 4, 3, 21 | syl2an 586 | . . . . 5 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (𝐴 ⊊ (𝐵 ∨ℋ 𝐶) → (¬ 𝐵 ⊆ 𝐴 ∨ ¬ 𝐶 ⊆ 𝐴))) |
23 | 22 | imp 398 | . . . 4 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ 𝐴 ⊊ (𝐵 ∨ℋ 𝐶)) → (¬ 𝐵 ⊆ 𝐴 ∨ ¬ 𝐶 ⊆ 𝐴)) |
24 | 23 | adantrl 703 | . . 3 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ (𝐴 ≠ 0ℋ ∧ 𝐴 ⊊ (𝐵 ∨ℋ 𝐶))) → (¬ 𝐵 ⊆ 𝐴 ∨ ¬ 𝐶 ⊆ 𝐴)) |
25 | 2, 13, 24 | mpjaod 846 | . 2 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ (𝐴 ≠ 0ℋ ∧ 𝐴 ⊊ (𝐵 ∨ℋ 𝐶))) → 𝐴 ∈ HAtoms) |
26 | 25 | ex 405 | 1 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((𝐴 ≠ 0ℋ ∧ 𝐴 ⊊ (𝐵 ∨ℋ 𝐶)) → 𝐴 ∈ HAtoms)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 ∨ wo 833 ∧ w3a 1068 = wceq 1507 ∈ wcel 2048 ≠ wne 2961 ⊆ wss 3825 ⊊ wpss 3826 (class class class)co 6970 Cℋ cch 28475 ∨ℋ chj 28479 0ℋc0h 28481 HAtomscat 28511 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-inf2 8890 ax-cc 9647 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-pre-sup 10405 ax-addf 10406 ax-mulf 10407 ax-hilex 28545 ax-hfvadd 28546 ax-hvcom 28547 ax-hvass 28548 ax-hv0cl 28549 ax-hvaddid 28550 ax-hfvmul 28551 ax-hvmulid 28552 ax-hvmulass 28553 ax-hvdistr1 28554 ax-hvdistr2 28555 ax-hvmul0 28556 ax-hfi 28625 ax-his1 28628 ax-his2 28629 ax-his3 28630 ax-his4 28631 ax-hcompl 28748 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-se 5360 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-of 7221 df-om 7391 df-1st 7494 df-2nd 7495 df-supp 7627 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-2o 7898 df-oadd 7901 df-omul 7902 df-er 8081 df-map 8200 df-pm 8201 df-ixp 8252 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-fsupp 8621 df-fi 8662 df-sup 8693 df-inf 8694 df-oi 8761 df-card 9154 df-acn 9157 df-cda 9380 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-9 11503 df-n0 11701 df-z 11787 df-dec 11905 df-uz 12052 df-q 12156 df-rp 12198 df-xneg 12317 df-xadd 12318 df-xmul 12319 df-ioo 12551 df-ico 12553 df-icc 12554 df-fz 12702 df-fzo 12843 df-fl 12970 df-seq 13178 df-exp 13238 df-hash 13499 df-cj 14309 df-re 14310 df-im 14311 df-sqrt 14445 df-abs 14446 df-clim 14696 df-rlim 14697 df-sum 14894 df-struct 16331 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-ress 16337 df-plusg 16424 df-mulr 16425 df-starv 16426 df-sca 16427 df-vsca 16428 df-ip 16429 df-tset 16430 df-ple 16431 df-ds 16433 df-unif 16434 df-hom 16435 df-cco 16436 df-rest 16542 df-topn 16543 df-0g 16561 df-gsum 16562 df-topgen 16563 df-pt 16564 df-prds 16567 df-xrs 16621 df-qtop 16626 df-imas 16627 df-xps 16629 df-mre 16705 df-mrc 16706 df-acs 16708 df-mgm 17700 df-sgrp 17742 df-mnd 17753 df-submnd 17794 df-mulg 18002 df-cntz 18208 df-cmn 18658 df-psmet 20229 df-xmet 20230 df-met 20231 df-bl 20232 df-mopn 20233 df-fbas 20234 df-fg 20235 df-cnfld 20238 df-top 21196 df-topon 21213 df-topsp 21235 df-bases 21248 df-cld 21321 df-ntr 21322 df-cls 21323 df-nei 21400 df-cn 21529 df-cnp 21530 df-lm 21531 df-haus 21617 df-tx 21864 df-hmeo 22057 df-fil 22148 df-fm 22240 df-flim 22241 df-flf 22242 df-xms 22623 df-ms 22624 df-tms 22625 df-cfil 23551 df-cau 23552 df-cmet 23553 df-grpo 28037 df-gid 28038 df-ginv 28039 df-gdiv 28040 df-ablo 28089 df-vc 28103 df-nv 28136 df-va 28139 df-ba 28140 df-sm 28141 df-0v 28142 df-vs 28143 df-nmcv 28144 df-ims 28145 df-dip 28245 df-ssp 28266 df-ph 28357 df-cbn 28408 df-hnorm 28514 df-hba 28515 df-hvsub 28517 df-hlim 28518 df-hcau 28519 df-sh 28753 df-ch 28767 df-oc 28798 df-ch0 28799 df-shs 28856 df-span 28857 df-chj 28858 df-chsup 28859 df-pjh 28943 df-cv 29827 df-at 29886 |
This theorem is referenced by: atcvat2i 29935 |
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