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Mirrors > Home > HSE Home > Th. List > chdmj1i | Structured version Visualization version GIF version |
Description: De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ch0le.1 | ⊢ 𝐴 ∈ Cℋ |
chjcl.2 | ⊢ 𝐵 ∈ Cℋ |
Ref | Expression |
---|---|
chdmj1i | ⊢ (⊥‘(𝐴 ∨ℋ 𝐵)) = ((⊥‘𝐴) ∩ (⊥‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ch0le.1 | . . . 4 ⊢ 𝐴 ∈ Cℋ | |
2 | chjcl.2 | . . . 4 ⊢ 𝐵 ∈ Cℋ | |
3 | 1, 2 | chdmm4i 28678 | . . 3 ⊢ (⊥‘((⊥‘𝐴) ∩ (⊥‘𝐵))) = (𝐴 ∨ℋ 𝐵) |
4 | 3 | fveq2i 6336 | . 2 ⊢ (⊥‘(⊥‘((⊥‘𝐴) ∩ (⊥‘𝐵)))) = (⊥‘(𝐴 ∨ℋ 𝐵)) |
5 | 1 | choccli 28505 | . . . 4 ⊢ (⊥‘𝐴) ∈ Cℋ |
6 | 2 | choccli 28505 | . . . 4 ⊢ (⊥‘𝐵) ∈ Cℋ |
7 | 5, 6 | chincli 28658 | . . 3 ⊢ ((⊥‘𝐴) ∩ (⊥‘𝐵)) ∈ Cℋ |
8 | 7 | pjococi 28635 | . 2 ⊢ (⊥‘(⊥‘((⊥‘𝐴) ∩ (⊥‘𝐵)))) = ((⊥‘𝐴) ∩ (⊥‘𝐵)) |
9 | 4, 8 | eqtr3i 2795 | 1 ⊢ (⊥‘(𝐴 ∨ℋ 𝐵)) = ((⊥‘𝐴) ∩ (⊥‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1631 ∈ wcel 2145 ∩ cin 3722 ‘cfv 6030 (class class class)co 6795 Cℋ cch 28125 ⊥cort 28126 ∨ℋ chj 28129 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7099 ax-inf2 8705 ax-cc 9462 ax-cnex 10197 ax-resscn 10198 ax-1cn 10199 ax-icn 10200 ax-addcl 10201 ax-addrcl 10202 ax-mulcl 10203 ax-mulrcl 10204 ax-mulcom 10205 ax-addass 10206 ax-mulass 10207 ax-distr 10208 ax-i2m1 10209 ax-1ne0 10210 ax-1rid 10211 ax-rnegex 10212 ax-rrecex 10213 ax-cnre 10214 ax-pre-lttri 10215 ax-pre-lttrn 10216 ax-pre-ltadd 10217 ax-pre-mulgt0 10218 ax-pre-sup 10219 ax-addf 10220 ax-mulf 10221 ax-hilex 28195 ax-hfvadd 28196 ax-hvcom 28197 ax-hvass 28198 ax-hv0cl 28199 ax-hvaddid 28200 ax-hfvmul 28201 ax-hvmulid 28202 ax-hvmulass 28203 ax-hvdistr1 28204 ax-hvdistr2 28205 ax-hvmul0 28206 ax-hfi 28275 ax-his1 28278 ax-his2 28279 ax-his3 28280 ax-his4 28281 ax-hcompl 28398 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-iin 4658 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-se 5210 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-isom 6039 df-riota 6756 df-ov 6798 df-oprab 6799 df-mpt2 6800 df-of 7047 df-om 7216 df-1st 7318 df-2nd 7319 df-supp 7450 df-wrecs 7562 df-recs 7624 df-rdg 7662 df-1o 7716 df-2o 7717 df-oadd 7720 df-omul 7721 df-er 7899 df-map 8014 df-pm 8015 df-ixp 8066 df-en 8113 df-dom 8114 df-sdom 8115 df-fin 8116 df-fsupp 8435 df-fi 8476 df-sup 8507 df-inf 8508 df-oi 8574 df-card 8968 df-acn 8971 df-cda 9195 df-pnf 10281 df-mnf 10282 df-xr 10283 df-ltxr 10284 df-le 10285 df-sub 10473 df-neg 10474 df-div 10890 df-nn 11226 df-2 11284 df-3 11285 df-4 11286 df-5 11287 df-6 11288 df-7 11289 df-8 11290 df-9 11291 df-n0 11499 df-z 11584 df-dec 11700 df-uz 11893 df-q 11996 df-rp 12035 df-xneg 12150 df-xadd 12151 df-xmul 12152 df-ioo 12383 df-ico 12385 df-icc 12386 df-fz 12533 df-fzo 12673 df-fl 12800 df-seq 13008 df-exp 13067 df-hash 13321 df-cj 14046 df-re 14047 df-im 14048 df-sqrt 14182 df-abs 14183 df-clim 14426 df-rlim 14427 df-sum 14624 df-struct 16065 df-ndx 16066 df-slot 16067 df-base 16069 df-sets 16070 df-ress 16071 df-plusg 16161 df-mulr 16162 df-starv 16163 df-sca 16164 df-vsca 16165 df-ip 16166 df-tset 16167 df-ple 16168 df-ds 16171 df-unif 16172 df-hom 16173 df-cco 16174 df-rest 16290 df-topn 16291 df-0g 16309 df-gsum 16310 df-topgen 16311 df-pt 16312 df-prds 16315 df-xrs 16369 df-qtop 16374 df-imas 16375 df-xps 16377 df-mre 16453 df-mrc 16454 df-acs 16456 df-mgm 17449 df-sgrp 17491 df-mnd 17502 df-submnd 17543 df-mulg 17748 df-cntz 17956 df-cmn 18401 df-psmet 19952 df-xmet 19953 df-met 19954 df-bl 19955 df-mopn 19956 df-fbas 19957 df-fg 19958 df-cnfld 19961 df-top 20918 df-topon 20935 df-topsp 20957 df-bases 20970 df-cld 21043 df-ntr 21044 df-cls 21045 df-nei 21122 df-cn 21251 df-cnp 21252 df-lm 21253 df-haus 21339 df-tx 21585 df-hmeo 21778 df-fil 21869 df-fm 21961 df-flim 21962 df-flf 21963 df-xms 22344 df-ms 22345 df-tms 22346 df-cfil 23271 df-cau 23272 df-cmet 23273 df-grpo 27686 df-gid 27687 df-ginv 27688 df-gdiv 27689 df-ablo 27738 df-vc 27753 df-nv 27786 df-va 27789 df-ba 27790 df-sm 27791 df-0v 27792 df-vs 27793 df-nmcv 27794 df-ims 27795 df-dip 27895 df-ssp 27916 df-ph 28007 df-cbn 28058 df-hnorm 28164 df-hba 28165 df-hvsub 28167 df-hlim 28168 df-hcau 28169 df-sh 28403 df-ch 28417 df-oc 28448 df-ch0 28449 df-shs 28506 df-chj 28508 |
This theorem is referenced by: chdmj2i 28680 chdmj3i 28681 chjassi 28684 pjoml2i 28783 cmbr2i 28794 fh3i 28821 fh4i 28822 mayetes3i 28927 mdsldmd1i 29529 cvexchi 29567 |
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