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Mirrors > Home > HSE Home > Th. List > ledi | Structured version Visualization version GIF version |
Description: An ortholattice is distributive in one ordering direction. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ledi | ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐶)) ⊆ (𝐴 ∩ (𝐵 ∨ℋ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ineq1 4145 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) → (𝐴 ∩ 𝐵) = (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ 𝐵)) | |
2 | ineq1 4145 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) → (𝐴 ∩ 𝐶) = (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ 𝐶)) | |
3 | 1, 2 | oveq12d 7325 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) → ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐶)) = ((if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ 𝐵) ∨ℋ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ 𝐶))) |
4 | ineq1 4145 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) → (𝐴 ∩ (𝐵 ∨ℋ 𝐶)) = (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ (𝐵 ∨ℋ 𝐶))) | |
5 | 3, 4 | sseq12d 3959 | . 2 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) → (((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐶)) ⊆ (𝐴 ∩ (𝐵 ∨ℋ 𝐶)) ↔ ((if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ 𝐵) ∨ℋ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ 𝐶)) ⊆ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ (𝐵 ∨ℋ 𝐶)))) |
6 | ineq2 4146 | . . . 4 ⊢ (𝐵 = if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) → (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ 𝐵) = (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ if(𝐵 ∈ Cℋ , 𝐵, 0ℋ))) | |
7 | 6 | oveq1d 7322 | . . 3 ⊢ (𝐵 = if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) → ((if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ 𝐵) ∨ℋ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ 𝐶)) = ((if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ if(𝐵 ∈ Cℋ , 𝐵, 0ℋ)) ∨ℋ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ 𝐶))) |
8 | oveq1 7314 | . . . 4 ⊢ (𝐵 = if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) → (𝐵 ∨ℋ 𝐶) = (if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) ∨ℋ 𝐶)) | |
9 | 8 | ineq2d 4152 | . . 3 ⊢ (𝐵 = if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) → (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ (𝐵 ∨ℋ 𝐶)) = (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ (if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) ∨ℋ 𝐶))) |
10 | 7, 9 | sseq12d 3959 | . 2 ⊢ (𝐵 = if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) → (((if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ 𝐵) ∨ℋ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ 𝐶)) ⊆ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ (𝐵 ∨ℋ 𝐶)) ↔ ((if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ if(𝐵 ∈ Cℋ , 𝐵, 0ℋ)) ∨ℋ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ 𝐶)) ⊆ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ (if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) ∨ℋ 𝐶)))) |
11 | ineq2 4146 | . . . 4 ⊢ (𝐶 = if(𝐶 ∈ Cℋ , 𝐶, 0ℋ) → (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ 𝐶) = (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ if(𝐶 ∈ Cℋ , 𝐶, 0ℋ))) | |
12 | 11 | oveq2d 7323 | . . 3 ⊢ (𝐶 = if(𝐶 ∈ Cℋ , 𝐶, 0ℋ) → ((if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ if(𝐵 ∈ Cℋ , 𝐵, 0ℋ)) ∨ℋ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ 𝐶)) = ((if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ if(𝐵 ∈ Cℋ , 𝐵, 0ℋ)) ∨ℋ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ if(𝐶 ∈ Cℋ , 𝐶, 0ℋ)))) |
13 | oveq2 7315 | . . . 4 ⊢ (𝐶 = if(𝐶 ∈ Cℋ , 𝐶, 0ℋ) → (if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) ∨ℋ 𝐶) = (if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) ∨ℋ if(𝐶 ∈ Cℋ , 𝐶, 0ℋ))) | |
14 | 13 | ineq2d 4152 | . . 3 ⊢ (𝐶 = if(𝐶 ∈ Cℋ , 𝐶, 0ℋ) → (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ (if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) ∨ℋ 𝐶)) = (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ (if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) ∨ℋ if(𝐶 ∈ Cℋ , 𝐶, 0ℋ)))) |
15 | 12, 14 | sseq12d 3959 | . 2 ⊢ (𝐶 = if(𝐶 ∈ Cℋ , 𝐶, 0ℋ) → (((if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ if(𝐵 ∈ Cℋ , 𝐵, 0ℋ)) ∨ℋ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ 𝐶)) ⊆ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ (if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) ∨ℋ 𝐶)) ↔ ((if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ if(𝐵 ∈ Cℋ , 𝐵, 0ℋ)) ∨ℋ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ if(𝐶 ∈ Cℋ , 𝐶, 0ℋ))) ⊆ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ (if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) ∨ℋ if(𝐶 ∈ Cℋ , 𝐶, 0ℋ))))) |
16 | h0elch 29662 | . . . 4 ⊢ 0ℋ ∈ Cℋ | |
17 | 16 | elimel 4534 | . . 3 ⊢ if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∈ Cℋ |
18 | 16 | elimel 4534 | . . 3 ⊢ if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) ∈ Cℋ |
19 | 16 | elimel 4534 | . . 3 ⊢ if(𝐶 ∈ Cℋ , 𝐶, 0ℋ) ∈ Cℋ |
20 | 17, 18, 19 | ledii 29943 | . 2 ⊢ ((if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ if(𝐵 ∈ Cℋ , 𝐵, 0ℋ)) ∨ℋ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ if(𝐶 ∈ Cℋ , 𝐶, 0ℋ))) ⊆ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∩ (if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) ∨ℋ if(𝐶 ∈ Cℋ , 𝐶, 0ℋ))) |
21 | 5, 10, 15, 20 | dedth3h 4525 | 1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ 𝐶)) ⊆ (𝐴 ∩ (𝐵 ∨ℋ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1539 ∈ wcel 2104 ∩ cin 3891 ⊆ wss 3892 ifcif 4465 (class class class)co 7307 Cℋ cch 29336 ∨ℋ chj 29340 0ℋc0h 29342 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-inf2 9443 ax-cc 10237 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 ax-pre-sup 10995 ax-addf 10996 ax-mulf 10997 ax-hilex 29406 ax-hfvadd 29407 ax-hvcom 29408 ax-hvass 29409 ax-hv0cl 29410 ax-hvaddid 29411 ax-hfvmul 29412 ax-hvmulid 29413 ax-hvmulass 29414 ax-hvdistr1 29415 ax-hvdistr2 29416 ax-hvmul0 29417 ax-hfi 29486 ax-his1 29489 ax-his2 29490 ax-his3 29491 ax-his4 29492 ax-hcompl 29609 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 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 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 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 4566 df-pr 4568 df-tp 4570 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-iin 4934 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-se 5556 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-isom 6467 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-of 7565 df-om 7745 df-1st 7863 df-2nd 7864 df-supp 8009 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-2o 8329 df-oadd 8332 df-omul 8333 df-er 8529 df-map 8648 df-pm 8649 df-ixp 8717 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-fsupp 9173 df-fi 9214 df-sup 9245 df-inf 9246 df-oi 9313 df-card 9741 df-acn 9744 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-div 11679 df-nn 12020 df-2 12082 df-3 12083 df-4 12084 df-5 12085 df-6 12086 df-7 12087 df-8 12088 df-9 12089 df-n0 12280 df-z 12366 df-dec 12484 df-uz 12629 df-q 12735 df-rp 12777 df-xneg 12894 df-xadd 12895 df-xmul 12896 df-ioo 13129 df-ico 13131 df-icc 13132 df-fz 13286 df-fzo 13429 df-fl 13558 df-seq 13768 df-exp 13829 df-hash 14091 df-cj 14855 df-re 14856 df-im 14857 df-sqrt 14991 df-abs 14992 df-clim 15242 df-rlim 15243 df-sum 15443 df-struct 16893 df-sets 16910 df-slot 16928 df-ndx 16940 df-base 16958 df-ress 16987 df-plusg 17020 df-mulr 17021 df-starv 17022 df-sca 17023 df-vsca 17024 df-ip 17025 df-tset 17026 df-ple 17027 df-ds 17029 df-unif 17030 df-hom 17031 df-cco 17032 df-rest 17178 df-topn 17179 df-0g 17197 df-gsum 17198 df-topgen 17199 df-pt 17200 df-prds 17203 df-xrs 17258 df-qtop 17263 df-imas 17264 df-xps 17266 df-mre 17340 df-mrc 17341 df-acs 17343 df-mgm 18371 df-sgrp 18420 df-mnd 18431 df-submnd 18476 df-mulg 18746 df-cntz 18968 df-cmn 19433 df-psmet 20634 df-xmet 20635 df-met 20636 df-bl 20637 df-mopn 20638 df-fbas 20639 df-fg 20640 df-cnfld 20643 df-top 22088 df-topon 22105 df-topsp 22127 df-bases 22141 df-cld 22215 df-ntr 22216 df-cls 22217 df-nei 22294 df-cn 22423 df-cnp 22424 df-lm 22425 df-haus 22511 df-tx 22758 df-hmeo 22951 df-fil 23042 df-fm 23134 df-flim 23135 df-flf 23136 df-xms 23518 df-ms 23519 df-tms 23520 df-cfil 24464 df-cau 24465 df-cmet 24466 df-grpo 28900 df-gid 28901 df-ginv 28902 df-gdiv 28903 df-ablo 28952 df-vc 28966 df-nv 28999 df-va 29002 df-ba 29003 df-sm 29004 df-0v 29005 df-vs 29006 df-nmcv 29007 df-ims 29008 df-dip 29108 df-ssp 29129 df-ph 29220 df-cbn 29270 df-hnorm 29375 df-hba 29376 df-hvsub 29378 df-hlim 29379 df-hcau 29380 df-sh 29614 df-ch 29628 df-oc 29659 df-ch0 29660 df-shs 29715 df-chj 29717 |
This theorem is referenced by: fh1 30025 fh2 30026 |
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