Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > HSE Home > Th. List > chabs1 | Structured version Visualization version GIF version |
Description: Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
chabs1 | ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ∨ℋ (𝐴 ∩ 𝐵)) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3948 | . . . 4 ⊢ 𝐴 ⊆ 𝐴 | |
2 | inss1 4168 | . . . 4 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
3 | 1, 2 | pm3.2i 471 | . . 3 ⊢ (𝐴 ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) |
4 | simpl 483 | . . . 4 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → 𝐴 ∈ Cℋ ) | |
5 | chincl 29870 | . . . 4 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ∩ 𝐵) ∈ Cℋ ) | |
6 | chlub 29880 | . . . 4 ⊢ ((𝐴 ∈ Cℋ ∧ (𝐴 ∩ 𝐵) ∈ Cℋ ∧ 𝐴 ∈ Cℋ ) → ((𝐴 ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) ↔ (𝐴 ∨ℋ (𝐴 ∩ 𝐵)) ⊆ 𝐴)) | |
7 | 4, 5, 4, 6 | syl3anc 1370 | . . 3 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ((𝐴 ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) ↔ (𝐴 ∨ℋ (𝐴 ∩ 𝐵)) ⊆ 𝐴)) |
8 | 3, 7 | mpbii 232 | . 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ∨ℋ (𝐴 ∩ 𝐵)) ⊆ 𝐴) |
9 | chub1 29878 | . . 3 ⊢ ((𝐴 ∈ Cℋ ∧ (𝐴 ∩ 𝐵) ∈ Cℋ ) → 𝐴 ⊆ (𝐴 ∨ℋ (𝐴 ∩ 𝐵))) | |
10 | 5, 9 | syldan 591 | . 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → 𝐴 ⊆ (𝐴 ∨ℋ (𝐴 ∩ 𝐵))) |
11 | 8, 10 | eqssd 3943 | 1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ∨ℋ (𝐴 ∩ 𝐵)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ∩ cin 3891 ⊆ wss 3892 (class class class)co 7272 Cℋ cch 29300 ∨ℋ chj 29304 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-inf2 9387 ax-cc 10202 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 ax-pre-sup 10960 ax-addf 10961 ax-mulf 10962 ax-hilex 29370 ax-hfvadd 29371 ax-hvcom 29372 ax-hvass 29373 ax-hv0cl 29374 ax-hvaddid 29375 ax-hfvmul 29376 ax-hvmulid 29377 ax-hvmulass 29378 ax-hvdistr1 29379 ax-hvdistr2 29380 ax-hvmul0 29381 ax-hfi 29450 ax-his1 29453 ax-his2 29454 ax-his3 29455 ax-his4 29456 ax-hcompl 29573 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-of 7528 df-om 7708 df-1st 7825 df-2nd 7826 df-supp 7970 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-1o 8289 df-2o 8290 df-oadd 8293 df-omul 8294 df-er 8490 df-map 8609 df-pm 8610 df-ixp 8678 df-en 8726 df-dom 8727 df-sdom 8728 df-fin 8729 df-fsupp 9117 df-fi 9158 df-sup 9189 df-inf 9190 df-oi 9257 df-card 9708 df-acn 9711 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-div 11644 df-nn 11985 df-2 12047 df-3 12048 df-4 12049 df-5 12050 df-6 12051 df-7 12052 df-8 12053 df-9 12054 df-n0 12245 df-z 12331 df-dec 12449 df-uz 12594 df-q 12700 df-rp 12742 df-xneg 12859 df-xadd 12860 df-xmul 12861 df-ioo 13094 df-ico 13096 df-icc 13097 df-fz 13251 df-fzo 13394 df-fl 13523 df-seq 13733 df-exp 13794 df-hash 14056 df-cj 14821 df-re 14822 df-im 14823 df-sqrt 14957 df-abs 14958 df-clim 15208 df-rlim 15209 df-sum 15409 df-struct 16859 df-sets 16876 df-slot 16894 df-ndx 16906 df-base 16924 df-ress 16953 df-plusg 16986 df-mulr 16987 df-starv 16988 df-sca 16989 df-vsca 16990 df-ip 16991 df-tset 16992 df-ple 16993 df-ds 16995 df-unif 16996 df-hom 16997 df-cco 16998 df-rest 17144 df-topn 17145 df-0g 17163 df-gsum 17164 df-topgen 17165 df-pt 17166 df-prds 17169 df-xrs 17224 df-qtop 17229 df-imas 17230 df-xps 17232 df-mre 17306 df-mrc 17307 df-acs 17309 df-mgm 18337 df-sgrp 18386 df-mnd 18397 df-submnd 18442 df-mulg 18712 df-cntz 18934 df-cmn 19399 df-psmet 20600 df-xmet 20601 df-met 20602 df-bl 20603 df-mopn 20604 df-fbas 20605 df-fg 20606 df-cnfld 20609 df-top 22054 df-topon 22071 df-topsp 22093 df-bases 22107 df-cld 22181 df-ntr 22182 df-cls 22183 df-nei 22260 df-cn 22389 df-cnp 22390 df-lm 22391 df-haus 22477 df-tx 22724 df-hmeo 22917 df-fil 23008 df-fm 23100 df-flim 23101 df-flf 23102 df-xms 23484 df-ms 23485 df-tms 23486 df-cfil 24430 df-cau 24431 df-cmet 24432 df-grpo 28864 df-gid 28865 df-ginv 28866 df-gdiv 28867 df-ablo 28916 df-vc 28930 df-nv 28963 df-va 28966 df-ba 28967 df-sm 28968 df-0v 28969 df-vs 28970 df-nmcv 28971 df-ims 28972 df-dip 29072 df-ssp 29093 df-ph 29184 df-cbn 29234 df-hnorm 29339 df-hba 29340 df-hvsub 29342 df-hlim 29343 df-hcau 29344 df-sh 29578 df-ch 29592 df-oc 29623 df-ch0 29624 df-shs 29679 df-chj 29681 |
This theorem is referenced by: chabs1i 29889 chjidm 29891 |
Copyright terms: Public domain | W3C validator |