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Mirrors > Home > HSE Home > Th. List > chjassi | Structured version Visualization version GIF version |
Description: Associative law for Hilbert lattice join. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ch0le.1 | ⊢ 𝐴 ∈ Cℋ |
chjcl.2 | ⊢ 𝐵 ∈ Cℋ |
chjass.3 | ⊢ 𝐶 ∈ Cℋ |
Ref | Expression |
---|---|
chjassi | ⊢ ((𝐴 ∨ℋ 𝐵) ∨ℋ 𝐶) = (𝐴 ∨ℋ (𝐵 ∨ℋ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inass 4134 | . . . 4 ⊢ (((⊥‘𝐴) ∩ (⊥‘𝐵)) ∩ (⊥‘𝐶)) = ((⊥‘𝐴) ∩ ((⊥‘𝐵) ∩ (⊥‘𝐶))) | |
2 | ch0le.1 | . . . . . 6 ⊢ 𝐴 ∈ Cℋ | |
3 | chjcl.2 | . . . . . 6 ⊢ 𝐵 ∈ Cℋ | |
4 | 2, 3 | chdmj1i 29562 | . . . . 5 ⊢ (⊥‘(𝐴 ∨ℋ 𝐵)) = ((⊥‘𝐴) ∩ (⊥‘𝐵)) |
5 | 4 | ineq1i 4123 | . . . 4 ⊢ ((⊥‘(𝐴 ∨ℋ 𝐵)) ∩ (⊥‘𝐶)) = (((⊥‘𝐴) ∩ (⊥‘𝐵)) ∩ (⊥‘𝐶)) |
6 | chjass.3 | . . . . . 6 ⊢ 𝐶 ∈ Cℋ | |
7 | 3, 6 | chdmj1i 29562 | . . . . 5 ⊢ (⊥‘(𝐵 ∨ℋ 𝐶)) = ((⊥‘𝐵) ∩ (⊥‘𝐶)) |
8 | 7 | ineq2i 4124 | . . . 4 ⊢ ((⊥‘𝐴) ∩ (⊥‘(𝐵 ∨ℋ 𝐶))) = ((⊥‘𝐴) ∩ ((⊥‘𝐵) ∩ (⊥‘𝐶))) |
9 | 1, 5, 8 | 3eqtr4i 2775 | . . 3 ⊢ ((⊥‘(𝐴 ∨ℋ 𝐵)) ∩ (⊥‘𝐶)) = ((⊥‘𝐴) ∩ (⊥‘(𝐵 ∨ℋ 𝐶))) |
10 | 9 | fveq2i 6720 | . 2 ⊢ (⊥‘((⊥‘(𝐴 ∨ℋ 𝐵)) ∩ (⊥‘𝐶))) = (⊥‘((⊥‘𝐴) ∩ (⊥‘(𝐵 ∨ℋ 𝐶)))) |
11 | 2, 3 | chjcli 29538 | . . 3 ⊢ (𝐴 ∨ℋ 𝐵) ∈ Cℋ |
12 | 11, 6 | chdmm4i 29561 | . 2 ⊢ (⊥‘((⊥‘(𝐴 ∨ℋ 𝐵)) ∩ (⊥‘𝐶))) = ((𝐴 ∨ℋ 𝐵) ∨ℋ 𝐶) |
13 | 3, 6 | chjcli 29538 | . . 3 ⊢ (𝐵 ∨ℋ 𝐶) ∈ Cℋ |
14 | 2, 13 | chdmm4i 29561 | . 2 ⊢ (⊥‘((⊥‘𝐴) ∩ (⊥‘(𝐵 ∨ℋ 𝐶)))) = (𝐴 ∨ℋ (𝐵 ∨ℋ 𝐶)) |
15 | 10, 12, 14 | 3eqtr3i 2773 | 1 ⊢ ((𝐴 ∨ℋ 𝐵) ∨ℋ 𝐶) = (𝐴 ∨ℋ (𝐵 ∨ℋ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2110 ∩ cin 3865 ‘cfv 6380 (class class class)co 7213 Cℋ cch 29010 ⊥cort 29011 ∨ℋ chj 29014 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cc 10049 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-addf 10808 ax-mulf 10809 ax-hilex 29080 ax-hfvadd 29081 ax-hvcom 29082 ax-hvass 29083 ax-hv0cl 29084 ax-hvaddid 29085 ax-hfvmul 29086 ax-hvmulid 29087 ax-hvmulass 29088 ax-hvdistr1 29089 ax-hvdistr2 29090 ax-hvmul0 29091 ax-hfi 29160 ax-his1 29163 ax-his2 29164 ax-his3 29165 ax-his4 29166 ax-hcompl 29283 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-oadd 8206 df-omul 8207 df-er 8391 df-map 8510 df-pm 8511 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-card 9555 df-acn 9558 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-ioo 12939 df-ico 12941 df-icc 12942 df-fz 13096 df-fzo 13239 df-fl 13367 df-seq 13575 df-exp 13636 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-clim 15049 df-rlim 15050 df-sum 15250 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-hom 16826 df-cco 16827 df-rest 16927 df-topn 16928 df-0g 16946 df-gsum 16947 df-topgen 16948 df-pt 16949 df-prds 16952 df-xrs 17007 df-qtop 17012 df-imas 17013 df-xps 17015 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-mulg 18489 df-cntz 18711 df-cmn 19172 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-fbas 20360 df-fg 20361 df-cnfld 20364 df-top 21791 df-topon 21808 df-topsp 21830 df-bases 21843 df-cld 21916 df-ntr 21917 df-cls 21918 df-nei 21995 df-cn 22124 df-cnp 22125 df-lm 22126 df-haus 22212 df-tx 22459 df-hmeo 22652 df-fil 22743 df-fm 22835 df-flim 22836 df-flf 22837 df-xms 23218 df-ms 23219 df-tms 23220 df-cfil 24152 df-cau 24153 df-cmet 24154 df-grpo 28574 df-gid 28575 df-ginv 28576 df-gdiv 28577 df-ablo 28626 df-vc 28640 df-nv 28673 df-va 28676 df-ba 28677 df-sm 28678 df-0v 28679 df-vs 28680 df-nmcv 28681 df-ims 28682 df-dip 28782 df-ssp 28803 df-ph 28894 df-cbn 28944 df-hnorm 29049 df-hba 29050 df-hvsub 29052 df-hlim 29053 df-hcau 29054 df-sh 29288 df-ch 29302 df-oc 29333 df-ch0 29334 df-shs 29389 df-chj 29391 |
This theorem is referenced by: chj12i 29603 chj4i 29604 chjass 29614 qlax3i 29710 mayetes3i 29810 atabs2i 30483 |
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