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Mirrors > Home > HSE Home > Th. List > atmd2 | Structured version Visualization version GIF version |
Description: Two Hilbert lattice elements have the dual modular pair property if the second is an atom. Part of Exercise 6 of [Kalmbach] p. 103. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
atmd2 | ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms) → 𝐴 𝑀ℋ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvp 31162 | . . 3 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms) → ((𝐴 ∩ 𝐵) = 0ℋ ↔ 𝐴 ⋖ℋ (𝐴 ∨ℋ 𝐵))) | |
2 | atelch 31131 | . . . 4 ⊢ (𝐵 ∈ HAtoms → 𝐵 ∈ Cℋ ) | |
3 | cvexch 31161 | . . . . 5 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ((𝐴 ∩ 𝐵) ⋖ℋ 𝐵 ↔ 𝐴 ⋖ℋ (𝐴 ∨ℋ 𝐵))) | |
4 | cvmd 31123 | . . . . . 6 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ (𝐴 ∩ 𝐵) ⋖ℋ 𝐵) → 𝐴 𝑀ℋ 𝐵) | |
5 | 4 | 3expia 1121 | . . . . 5 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ((𝐴 ∩ 𝐵) ⋖ℋ 𝐵 → 𝐴 𝑀ℋ 𝐵)) |
6 | 3, 5 | sylbird 259 | . . . 4 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ (𝐴 ∨ℋ 𝐵) → 𝐴 𝑀ℋ 𝐵)) |
7 | 2, 6 | sylan2 593 | . . 3 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms) → (𝐴 ⋖ℋ (𝐴 ∨ℋ 𝐵) → 𝐴 𝑀ℋ 𝐵)) |
8 | 1, 7 | sylbid 239 | . 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms) → ((𝐴 ∩ 𝐵) = 0ℋ → 𝐴 𝑀ℋ 𝐵)) |
9 | atnssm0 31163 | . . . 4 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms) → (¬ 𝐵 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐵) = 0ℋ)) | |
10 | 9 | con1bid 355 | . . 3 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms) → (¬ (𝐴 ∩ 𝐵) = 0ℋ ↔ 𝐵 ⊆ 𝐴)) |
11 | ssmd2 31099 | . . . . . 6 ⊢ ((𝐵 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ∧ 𝐵 ⊆ 𝐴) → 𝐴 𝑀ℋ 𝐵) | |
12 | 11 | 3com12 1123 | . . . . 5 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐵 ⊆ 𝐴) → 𝐴 𝑀ℋ 𝐵) |
13 | 2, 12 | syl3an2 1164 | . . . 4 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ∧ 𝐵 ⊆ 𝐴) → 𝐴 𝑀ℋ 𝐵) |
14 | 13 | 3expia 1121 | . . 3 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms) → (𝐵 ⊆ 𝐴 → 𝐴 𝑀ℋ 𝐵)) |
15 | 10, 14 | sylbid 239 | . 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms) → (¬ (𝐴 ∩ 𝐵) = 0ℋ → 𝐴 𝑀ℋ 𝐵)) |
16 | 8, 15 | pm2.61d 179 | 1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms) → 𝐴 𝑀ℋ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∩ cin 3907 ⊆ wss 3908 class class class wbr 5103 (class class class)co 7351 Cℋ cch 29716 ∨ℋ chj 29720 0ℋc0h 29722 ⋖ℋ ccv 29751 HAtomscat 29752 𝑀ℋ cmd 29753 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-inf2 9535 ax-cc 10329 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088 ax-mulf 11089 ax-hilex 29786 ax-hfvadd 29787 ax-hvcom 29788 ax-hvass 29789 ax-hv0cl 29790 ax-hvaddid 29791 ax-hfvmul 29792 ax-hvmulid 29793 ax-hvmulass 29794 ax-hvdistr1 29795 ax-hvdistr2 29796 ax-hvmul0 29797 ax-hfi 29866 ax-his1 29869 ax-his2 29870 ax-his3 29871 ax-his4 29872 ax-hcompl 29989 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-of 7609 df-om 7795 df-1st 7913 df-2nd 7914 df-supp 8085 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-2o 8405 df-oadd 8408 df-omul 8409 df-er 8606 df-map 8725 df-pm 8726 df-ixp 8794 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-fsupp 9264 df-fi 9305 df-sup 9336 df-inf 9337 df-oi 9404 df-card 9833 df-acn 9836 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-dec 12577 df-uz 12722 df-q 12828 df-rp 12870 df-xneg 12987 df-xadd 12988 df-xmul 12989 df-ioo 13222 df-ico 13224 df-icc 13225 df-fz 13379 df-fzo 13522 df-fl 13651 df-seq 13861 df-exp 13922 df-hash 14185 df-cj 14938 df-re 14939 df-im 14940 df-sqrt 15074 df-abs 15075 df-clim 15324 df-rlim 15325 df-sum 15525 df-struct 16973 df-sets 16990 df-slot 17008 df-ndx 17020 df-base 17038 df-ress 17067 df-plusg 17100 df-mulr 17101 df-starv 17102 df-sca 17103 df-vsca 17104 df-ip 17105 df-tset 17106 df-ple 17107 df-ds 17109 df-unif 17110 df-hom 17111 df-cco 17112 df-rest 17258 df-topn 17259 df-0g 17277 df-gsum 17278 df-topgen 17279 df-pt 17280 df-prds 17283 df-xrs 17338 df-qtop 17343 df-imas 17344 df-xps 17346 df-mre 17420 df-mrc 17421 df-acs 17423 df-mgm 18451 df-sgrp 18500 df-mnd 18511 df-submnd 18556 df-mulg 18826 df-cntz 19050 df-cmn 19517 df-psmet 20735 df-xmet 20736 df-met 20737 df-bl 20738 df-mopn 20739 df-fbas 20740 df-fg 20741 df-cnfld 20744 df-top 22189 df-topon 22206 df-topsp 22228 df-bases 22242 df-cld 22316 df-ntr 22317 df-cls 22318 df-nei 22395 df-cn 22524 df-cnp 22525 df-lm 22526 df-haus 22612 df-tx 22859 df-hmeo 23052 df-fil 23143 df-fm 23235 df-flim 23236 df-flf 23237 df-xms 23619 df-ms 23620 df-tms 23621 df-cfil 24565 df-cau 24566 df-cmet 24567 df-grpo 29280 df-gid 29281 df-ginv 29282 df-gdiv 29283 df-ablo 29332 df-vc 29346 df-nv 29379 df-va 29382 df-ba 29383 df-sm 29384 df-0v 29385 df-vs 29386 df-nmcv 29387 df-ims 29388 df-dip 29488 df-ssp 29509 df-ph 29600 df-cbn 29650 df-hnorm 29755 df-hba 29756 df-hvsub 29758 df-hlim 29759 df-hcau 29760 df-sh 29994 df-ch 30008 df-oc 30039 df-ch0 30040 df-shs 30095 df-span 30096 df-chj 30097 df-chsup 30098 df-pjh 30182 df-cv 31066 df-md 31067 df-at 31125 |
This theorem is referenced by: (None) |
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