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Mirrors > Home > HSE Home > Th. List > mdsl2bi | Structured version Visualization version GIF version |
Description: If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2. (Contributed by NM, 24-Dec-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mdsl.1 | ⊢ 𝐴 ∈ Cℋ |
mdsl.2 | ⊢ 𝐵 ∈ Cℋ |
Ref | Expression |
---|---|
mdsl2bi | ⊢ (𝐴 𝑀ℋ 𝐵 ↔ ∀𝑥 ∈ Cℋ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mdsl.1 | . . 3 ⊢ 𝐴 ∈ Cℋ | |
2 | mdsl.2 | . . 3 ⊢ 𝐵 ∈ Cℋ | |
3 | 1, 2 | mdsl2i 30102 | . 2 ⊢ (𝐴 𝑀ℋ 𝐵 ↔ ∀𝑥 ∈ Cℋ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨ℋ (𝐴 ∩ 𝐵)))) |
4 | 1, 2 | chincli 29240 | . . . . . . . . . . 11 ⊢ (𝐴 ∩ 𝐵) ∈ Cℋ |
5 | inss1 4208 | . . . . . . . . . . . 12 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
6 | chlej2 29291 | . . . . . . . . . . . 12 ⊢ ((((𝐴 ∩ 𝐵) ∈ Cℋ ∧ 𝐴 ∈ Cℋ ∧ 𝑥 ∈ Cℋ ) ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ⊆ (𝑥 ∨ℋ 𝐴)) | |
7 | 5, 6 | mpan2 689 | . . . . . . . . . . 11 ⊢ (((𝐴 ∩ 𝐵) ∈ Cℋ ∧ 𝐴 ∈ Cℋ ∧ 𝑥 ∈ Cℋ ) → (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ⊆ (𝑥 ∨ℋ 𝐴)) |
8 | 4, 1, 7 | mp3an12 1447 | . . . . . . . . . 10 ⊢ (𝑥 ∈ Cℋ → (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ⊆ (𝑥 ∨ℋ 𝐴)) |
9 | 8 | adantr 483 | . . . . . . . . 9 ⊢ ((𝑥 ∈ Cℋ ∧ 𝑥 ⊆ 𝐵) → (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ⊆ (𝑥 ∨ℋ 𝐴)) |
10 | simpr 487 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ Cℋ ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ 𝐵) | |
11 | inss2 4209 | . . . . . . . . . . 11 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
12 | 10, 11 | jctir 523 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ Cℋ ∧ 𝑥 ⊆ 𝐵) → (𝑥 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐵)) |
13 | chlub 29289 | . . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Cℋ ∧ (𝐴 ∩ 𝐵) ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ((𝑥 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐵) ↔ (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ⊆ 𝐵)) | |
14 | 4, 2, 13 | mp3an23 1449 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ Cℋ → ((𝑥 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐵) ↔ (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ⊆ 𝐵)) |
15 | 14 | adantr 483 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ Cℋ ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐵) ↔ (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ⊆ 𝐵)) |
16 | 12, 15 | mpbid 234 | . . . . . . . . 9 ⊢ ((𝑥 ∈ Cℋ ∧ 𝑥 ⊆ 𝐵) → (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ⊆ 𝐵) |
17 | 9, 16 | ssind 4212 | . . . . . . . 8 ⊢ ((𝑥 ∈ Cℋ ∧ 𝑥 ⊆ 𝐵) → (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ⊆ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵)) |
18 | 17 | biantrud 534 | . . . . . . 7 ⊢ ((𝑥 ∈ Cℋ ∧ 𝑥 ⊆ 𝐵) → (((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ↔ (((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ∧ (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ⊆ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵)))) |
19 | eqss 3985 | . . . . . . 7 ⊢ (((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ↔ (((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ∧ (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ⊆ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵))) | |
20 | 18, 19 | syl6bbr 291 | . . . . . 6 ⊢ ((𝑥 ∈ Cℋ ∧ 𝑥 ⊆ 𝐵) → (((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ↔ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)))) |
21 | 20 | ex 415 | . . . . 5 ⊢ (𝑥 ∈ Cℋ → (𝑥 ⊆ 𝐵 → (((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ↔ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵))))) |
22 | 21 | adantld 493 | . . . 4 ⊢ (𝑥 ∈ Cℋ → (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → (((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ↔ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵))))) |
23 | 22 | pm5.74d 275 | . . 3 ⊢ (𝑥 ∈ Cℋ → ((((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨ℋ (𝐴 ∩ 𝐵))) ↔ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵))))) |
24 | 23 | ralbiia 3167 | . 2 ⊢ (∀𝑥 ∈ Cℋ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨ℋ (𝐴 ∩ 𝐵))) ↔ ∀𝑥 ∈ Cℋ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)))) |
25 | 3, 24 | bitri 277 | 1 ⊢ (𝐴 𝑀ℋ 𝐵 ↔ ∀𝑥 ∈ Cℋ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ∀wral 3141 ∩ cin 3938 ⊆ wss 3939 class class class wbr 5069 (class class class)co 7159 Cℋ cch 28709 ∨ℋ chj 28713 𝑀ℋ cmd 28746 |
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 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cc 9860 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-addf 10619 ax-mulf 10620 ax-hilex 28779 ax-hfvadd 28780 ax-hvcom 28781 ax-hvass 28782 ax-hv0cl 28783 ax-hvaddid 28784 ax-hfvmul 28785 ax-hvmulid 28786 ax-hvmulass 28787 ax-hvdistr1 28788 ax-hvdistr2 28789 ax-hvmul0 28790 ax-hfi 28859 ax-his1 28862 ax-his2 28863 ax-his3 28864 ax-his4 28865 ax-hcompl 28982 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-omul 8110 df-er 8292 df-map 8411 df-pm 8412 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-fi 8878 df-sup 8909 df-inf 8910 df-oi 8977 df-card 9371 df-acn 9374 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-clim 14848 df-rlim 14849 df-sum 15046 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-hom 16592 df-cco 16593 df-rest 16699 df-topn 16700 df-0g 16718 df-gsum 16719 df-topgen 16720 df-pt 16721 df-prds 16724 df-xrs 16778 df-qtop 16783 df-imas 16784 df-xps 16786 df-mre 16860 df-mrc 16861 df-acs 16863 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-mulg 18228 df-cntz 18450 df-cmn 18911 df-psmet 20540 df-xmet 20541 df-met 20542 df-bl 20543 df-mopn 20544 df-fbas 20545 df-fg 20546 df-cnfld 20549 df-top 21505 df-topon 21522 df-topsp 21544 df-bases 21557 df-cld 21630 df-ntr 21631 df-cls 21632 df-nei 21709 df-cn 21838 df-cnp 21839 df-lm 21840 df-haus 21926 df-tx 22173 df-hmeo 22366 df-fil 22457 df-fm 22549 df-flim 22550 df-flf 22551 df-xms 22933 df-ms 22934 df-tms 22935 df-cfil 23861 df-cau 23862 df-cmet 23863 df-grpo 28273 df-gid 28274 df-ginv 28275 df-gdiv 28276 df-ablo 28325 df-vc 28339 df-nv 28372 df-va 28375 df-ba 28376 df-sm 28377 df-0v 28378 df-vs 28379 df-nmcv 28380 df-ims 28381 df-dip 28481 df-ssp 28502 df-ph 28593 df-cbn 28643 df-hnorm 28748 df-hba 28749 df-hvsub 28751 df-hlim 28752 df-hcau 28753 df-sh 28987 df-ch 29001 df-oc 29032 df-ch0 29033 df-shs 29088 df-chj 29090 df-md 30060 |
This theorem is referenced by: csmdsymi 30114 |
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