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Mirrors > Home > HSE Home > Th. List > cm2mi | Structured version Visualization version GIF version |
Description: A lattice element that commutes with two others also commutes with their meet. Theorem 4.2 of [Beran] p. 49. (Contributed by NM, 11-May-2009.) (New usage is discouraged.) |
Ref | Expression |
---|---|
fh1.1 | ⊢ 𝐴 ∈ Cℋ |
fh1.2 | ⊢ 𝐵 ∈ Cℋ |
fh1.3 | ⊢ 𝐶 ∈ Cℋ |
fh1.4 | ⊢ 𝐴 𝐶ℋ 𝐵 |
fh1.5 | ⊢ 𝐴 𝐶ℋ 𝐶 |
Ref | Expression |
---|---|
cm2mi | ⊢ 𝐴 𝐶ℋ (𝐵 ∩ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fh1.1 | . . . 4 ⊢ 𝐴 ∈ Cℋ | |
2 | fh1.2 | . . . . 5 ⊢ 𝐵 ∈ Cℋ | |
3 | 2 | choccli 28496 | . . . 4 ⊢ (⊥‘𝐵) ∈ Cℋ |
4 | fh1.3 | . . . . 5 ⊢ 𝐶 ∈ Cℋ | |
5 | 4 | choccli 28496 | . . . 4 ⊢ (⊥‘𝐶) ∈ Cℋ |
6 | fh1.4 | . . . . 5 ⊢ 𝐴 𝐶ℋ 𝐵 | |
7 | 1, 2, 6 | cmcm2ii 28787 | . . . 4 ⊢ 𝐴 𝐶ℋ (⊥‘𝐵) |
8 | fh1.5 | . . . . 5 ⊢ 𝐴 𝐶ℋ 𝐶 | |
9 | 1, 4, 8 | cmcm2ii 28787 | . . . 4 ⊢ 𝐴 𝐶ℋ (⊥‘𝐶) |
10 | 1, 3, 5, 7, 9 | cm2ji 28814 | . . 3 ⊢ 𝐴 𝐶ℋ ((⊥‘𝐵) ∨ℋ (⊥‘𝐶)) |
11 | 2, 4 | chdmm1i 28666 | . . 3 ⊢ (⊥‘(𝐵 ∩ 𝐶)) = ((⊥‘𝐵) ∨ℋ (⊥‘𝐶)) |
12 | 10, 11 | breqtrri 4831 | . 2 ⊢ 𝐴 𝐶ℋ (⊥‘(𝐵 ∩ 𝐶)) |
13 | 2, 4 | chincli 28649 | . . 3 ⊢ (𝐵 ∩ 𝐶) ∈ Cℋ |
14 | 1, 13 | cmcm2i 28782 | . 2 ⊢ (𝐴 𝐶ℋ (𝐵 ∩ 𝐶) ↔ 𝐴 𝐶ℋ (⊥‘(𝐵 ∩ 𝐶))) |
15 | 12, 14 | mpbir 221 | 1 ⊢ 𝐴 𝐶ℋ (𝐵 ∩ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2139 ∩ cin 3714 class class class wbr 4804 ‘cfv 6049 (class class class)co 6814 Cℋ cch 28116 ⊥cort 28117 ∨ℋ chj 28120 𝐶ℋ ccm 28123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-inf2 8713 ax-cc 9469 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 ax-pre-sup 10226 ax-addf 10227 ax-mulf 10228 ax-hilex 28186 ax-hfvadd 28187 ax-hvcom 28188 ax-hvass 28189 ax-hv0cl 28190 ax-hvaddid 28191 ax-hfvmul 28192 ax-hvmulid 28193 ax-hvmulass 28194 ax-hvdistr1 28195 ax-hvdistr2 28196 ax-hvmul0 28197 ax-hfi 28266 ax-his1 28269 ax-his2 28270 ax-his3 28271 ax-his4 28272 ax-hcompl 28389 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-of 7063 df-om 7232 df-1st 7334 df-2nd 7335 df-supp 7465 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-2o 7731 df-oadd 7734 df-omul 7735 df-er 7913 df-map 8027 df-pm 8028 df-ixp 8077 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-fsupp 8443 df-fi 8484 df-sup 8515 df-inf 8516 df-oi 8582 df-card 8975 df-acn 8978 df-cda 9202 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-div 10897 df-nn 11233 df-2 11291 df-3 11292 df-4 11293 df-5 11294 df-6 11295 df-7 11296 df-8 11297 df-9 11298 df-n0 11505 df-z 11590 df-dec 11706 df-uz 11900 df-q 12002 df-rp 12046 df-xneg 12159 df-xadd 12160 df-xmul 12161 df-ioo 12392 df-ico 12394 df-icc 12395 df-fz 12540 df-fzo 12680 df-fl 12807 df-seq 13016 df-exp 13075 df-hash 13332 df-cj 14058 df-re 14059 df-im 14060 df-sqrt 14194 df-abs 14195 df-clim 14438 df-rlim 14439 df-sum 14636 df-struct 16081 df-ndx 16082 df-slot 16083 df-base 16085 df-sets 16086 df-ress 16087 df-plusg 16176 df-mulr 16177 df-starv 16178 df-sca 16179 df-vsca 16180 df-ip 16181 df-tset 16182 df-ple 16183 df-ds 16186 df-unif 16187 df-hom 16188 df-cco 16189 df-rest 16305 df-topn 16306 df-0g 16324 df-gsum 16325 df-topgen 16326 df-pt 16327 df-prds 16330 df-xrs 16384 df-qtop 16389 df-imas 16390 df-xps 16392 df-mre 16468 df-mrc 16469 df-acs 16471 df-mgm 17463 df-sgrp 17505 df-mnd 17516 df-submnd 17557 df-mulg 17762 df-cntz 17970 df-cmn 18415 df-psmet 19960 df-xmet 19961 df-met 19962 df-bl 19963 df-mopn 19964 df-fbas 19965 df-fg 19966 df-cnfld 19969 df-top 20921 df-topon 20938 df-topsp 20959 df-bases 20972 df-cld 21045 df-ntr 21046 df-cls 21047 df-nei 21124 df-cn 21253 df-cnp 21254 df-lm 21255 df-haus 21341 df-tx 21587 df-hmeo 21780 df-fil 21871 df-fm 21963 df-flim 21964 df-flf 21965 df-xms 22346 df-ms 22347 df-tms 22348 df-cfil 23273 df-cau 23274 df-cmet 23275 df-grpo 27677 df-gid 27678 df-ginv 27679 df-gdiv 27680 df-ablo 27729 df-vc 27744 df-nv 27777 df-va 27780 df-ba 27781 df-sm 27782 df-0v 27783 df-vs 27784 df-nmcv 27785 df-ims 27786 df-dip 27886 df-ssp 27907 df-ph 27998 df-cbn 28049 df-hnorm 28155 df-hba 28156 df-hvsub 28158 df-hlim 28159 df-hcau 28160 df-sh 28394 df-ch 28408 df-oc 28439 df-ch0 28440 df-shs 28497 df-chj 28499 df-cm 28772 |
This theorem is referenced by: mayetes3i 28918 |
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