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Mirrors > Home > HSE Home > Th. List > mdsldmd1i | Structured version Visualization version GIF version |
Description: Preservation of the dual modular pair property in the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mdslmd.1 | ⊢ 𝐴 ∈ Cℋ |
mdslmd.2 | ⊢ 𝐵 ∈ Cℋ |
mdslmd.3 | ⊢ 𝐶 ∈ Cℋ |
mdslmd.4 | ⊢ 𝐷 ∈ Cℋ |
Ref | Expression |
---|---|
mdsldmd1i | ⊢ (((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ* 𝐴) ∧ (𝐴 ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ (𝐴 ∨ℋ 𝐵))) → (𝐶 𝑀ℋ* 𝐷 ↔ (𝐶 ∩ 𝐵) 𝑀ℋ* (𝐷 ∩ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mdslmd.1 | . . . . 5 ⊢ 𝐴 ∈ Cℋ | |
2 | mdslmd.2 | . . . . 5 ⊢ 𝐵 ∈ Cℋ | |
3 | mddmd 29715 | . . . . 5 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ 𝐵 ↔ (⊥‘𝐴) 𝑀ℋ* (⊥‘𝐵))) | |
4 | 1, 2, 3 | mp2an 685 | . . . 4 ⊢ (𝐴 𝑀ℋ 𝐵 ↔ (⊥‘𝐴) 𝑀ℋ* (⊥‘𝐵)) |
5 | dmdmd 29714 | . . . . 5 ⊢ ((𝐵 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ) → (𝐵 𝑀ℋ* 𝐴 ↔ (⊥‘𝐵) 𝑀ℋ (⊥‘𝐴))) | |
6 | 2, 1, 5 | mp2an 685 | . . . 4 ⊢ (𝐵 𝑀ℋ* 𝐴 ↔ (⊥‘𝐵) 𝑀ℋ (⊥‘𝐴)) |
7 | 4, 6 | anbi12ci 623 | . . 3 ⊢ ((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ* 𝐴) ↔ ((⊥‘𝐵) 𝑀ℋ (⊥‘𝐴) ∧ (⊥‘𝐴) 𝑀ℋ* (⊥‘𝐵))) |
8 | mdslmd.3 | . . . . . . 7 ⊢ 𝐶 ∈ Cℋ | |
9 | mdslmd.4 | . . . . . . 7 ⊢ 𝐷 ∈ Cℋ | |
10 | 8, 9 | chincli 28874 | . . . . . 6 ⊢ (𝐶 ∩ 𝐷) ∈ Cℋ |
11 | 1, 10 | chsscon3i 28875 | . . . . 5 ⊢ (𝐴 ⊆ (𝐶 ∩ 𝐷) ↔ (⊥‘(𝐶 ∩ 𝐷)) ⊆ (⊥‘𝐴)) |
12 | 8, 9 | chdmm1i 28891 | . . . . . 6 ⊢ (⊥‘(𝐶 ∩ 𝐷)) = ((⊥‘𝐶) ∨ℋ (⊥‘𝐷)) |
13 | 12 | sseq1i 3854 | . . . . 5 ⊢ ((⊥‘(𝐶 ∩ 𝐷)) ⊆ (⊥‘𝐴) ↔ ((⊥‘𝐶) ∨ℋ (⊥‘𝐷)) ⊆ (⊥‘𝐴)) |
14 | 11, 13 | bitri 267 | . . . 4 ⊢ (𝐴 ⊆ (𝐶 ∩ 𝐷) ↔ ((⊥‘𝐶) ∨ℋ (⊥‘𝐷)) ⊆ (⊥‘𝐴)) |
15 | 8, 9 | chjcli 28871 | . . . . . 6 ⊢ (𝐶 ∨ℋ 𝐷) ∈ Cℋ |
16 | 1, 2 | chjcli 28871 | . . . . . 6 ⊢ (𝐴 ∨ℋ 𝐵) ∈ Cℋ |
17 | 15, 16 | chsscon3i 28875 | . . . . 5 ⊢ ((𝐶 ∨ℋ 𝐷) ⊆ (𝐴 ∨ℋ 𝐵) ↔ (⊥‘(𝐴 ∨ℋ 𝐵)) ⊆ (⊥‘(𝐶 ∨ℋ 𝐷))) |
18 | 1, 2 | chdmj1i 28895 | . . . . . . 7 ⊢ (⊥‘(𝐴 ∨ℋ 𝐵)) = ((⊥‘𝐴) ∩ (⊥‘𝐵)) |
19 | incom 4032 | . . . . . . 7 ⊢ ((⊥‘𝐴) ∩ (⊥‘𝐵)) = ((⊥‘𝐵) ∩ (⊥‘𝐴)) | |
20 | 18, 19 | eqtri 2849 | . . . . . 6 ⊢ (⊥‘(𝐴 ∨ℋ 𝐵)) = ((⊥‘𝐵) ∩ (⊥‘𝐴)) |
21 | 8, 9 | chdmj1i 28895 | . . . . . 6 ⊢ (⊥‘(𝐶 ∨ℋ 𝐷)) = ((⊥‘𝐶) ∩ (⊥‘𝐷)) |
22 | 20, 21 | sseq12i 3856 | . . . . 5 ⊢ ((⊥‘(𝐴 ∨ℋ 𝐵)) ⊆ (⊥‘(𝐶 ∨ℋ 𝐷)) ↔ ((⊥‘𝐵) ∩ (⊥‘𝐴)) ⊆ ((⊥‘𝐶) ∩ (⊥‘𝐷))) |
23 | 17, 22 | bitri 267 | . . . 4 ⊢ ((𝐶 ∨ℋ 𝐷) ⊆ (𝐴 ∨ℋ 𝐵) ↔ ((⊥‘𝐵) ∩ (⊥‘𝐴)) ⊆ ((⊥‘𝐶) ∩ (⊥‘𝐷))) |
24 | 14, 23 | anbi12ci 623 | . . 3 ⊢ ((𝐴 ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ (𝐴 ∨ℋ 𝐵)) ↔ (((⊥‘𝐵) ∩ (⊥‘𝐴)) ⊆ ((⊥‘𝐶) ∩ (⊥‘𝐷)) ∧ ((⊥‘𝐶) ∨ℋ (⊥‘𝐷)) ⊆ (⊥‘𝐴))) |
25 | 2 | choccli 28721 | . . . 4 ⊢ (⊥‘𝐵) ∈ Cℋ |
26 | 1 | choccli 28721 | . . . 4 ⊢ (⊥‘𝐴) ∈ Cℋ |
27 | 8 | choccli 28721 | . . . 4 ⊢ (⊥‘𝐶) ∈ Cℋ |
28 | 9 | choccli 28721 | . . . 4 ⊢ (⊥‘𝐷) ∈ Cℋ |
29 | 25, 26, 27, 28 | mdslmd2i 29744 | . . 3 ⊢ ((((⊥‘𝐵) 𝑀ℋ (⊥‘𝐴) ∧ (⊥‘𝐴) 𝑀ℋ* (⊥‘𝐵)) ∧ (((⊥‘𝐵) ∩ (⊥‘𝐴)) ⊆ ((⊥‘𝐶) ∩ (⊥‘𝐷)) ∧ ((⊥‘𝐶) ∨ℋ (⊥‘𝐷)) ⊆ (⊥‘𝐴))) → ((⊥‘𝐶) 𝑀ℋ (⊥‘𝐷) ↔ ((⊥‘𝐶) ∨ℋ (⊥‘𝐵)) 𝑀ℋ ((⊥‘𝐷) ∨ℋ (⊥‘𝐵)))) |
30 | 7, 24, 29 | syl2anb 593 | . 2 ⊢ (((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ* 𝐴) ∧ (𝐴 ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ (𝐴 ∨ℋ 𝐵))) → ((⊥‘𝐶) 𝑀ℋ (⊥‘𝐷) ↔ ((⊥‘𝐶) ∨ℋ (⊥‘𝐵)) 𝑀ℋ ((⊥‘𝐷) ∨ℋ (⊥‘𝐵)))) |
31 | dmdmd 29714 | . . 3 ⊢ ((𝐶 ∈ Cℋ ∧ 𝐷 ∈ Cℋ ) → (𝐶 𝑀ℋ* 𝐷 ↔ (⊥‘𝐶) 𝑀ℋ (⊥‘𝐷))) | |
32 | 8, 9, 31 | mp2an 685 | . 2 ⊢ (𝐶 𝑀ℋ* 𝐷 ↔ (⊥‘𝐶) 𝑀ℋ (⊥‘𝐷)) |
33 | 8, 2 | chincli 28874 | . . . 4 ⊢ (𝐶 ∩ 𝐵) ∈ Cℋ |
34 | 9, 2 | chincli 28874 | . . . 4 ⊢ (𝐷 ∩ 𝐵) ∈ Cℋ |
35 | dmdmd 29714 | . . . 4 ⊢ (((𝐶 ∩ 𝐵) ∈ Cℋ ∧ (𝐷 ∩ 𝐵) ∈ Cℋ ) → ((𝐶 ∩ 𝐵) 𝑀ℋ* (𝐷 ∩ 𝐵) ↔ (⊥‘(𝐶 ∩ 𝐵)) 𝑀ℋ (⊥‘(𝐷 ∩ 𝐵)))) | |
36 | 33, 34, 35 | mp2an 685 | . . 3 ⊢ ((𝐶 ∩ 𝐵) 𝑀ℋ* (𝐷 ∩ 𝐵) ↔ (⊥‘(𝐶 ∩ 𝐵)) 𝑀ℋ (⊥‘(𝐷 ∩ 𝐵))) |
37 | 8, 2 | chdmm1i 28891 | . . . 4 ⊢ (⊥‘(𝐶 ∩ 𝐵)) = ((⊥‘𝐶) ∨ℋ (⊥‘𝐵)) |
38 | 9, 2 | chdmm1i 28891 | . . . 4 ⊢ (⊥‘(𝐷 ∩ 𝐵)) = ((⊥‘𝐷) ∨ℋ (⊥‘𝐵)) |
39 | 37, 38 | breq12i 4882 | . . 3 ⊢ ((⊥‘(𝐶 ∩ 𝐵)) 𝑀ℋ (⊥‘(𝐷 ∩ 𝐵)) ↔ ((⊥‘𝐶) ∨ℋ (⊥‘𝐵)) 𝑀ℋ ((⊥‘𝐷) ∨ℋ (⊥‘𝐵))) |
40 | 36, 39 | bitri 267 | . 2 ⊢ ((𝐶 ∩ 𝐵) 𝑀ℋ* (𝐷 ∩ 𝐵) ↔ ((⊥‘𝐶) ∨ℋ (⊥‘𝐵)) 𝑀ℋ ((⊥‘𝐷) ∨ℋ (⊥‘𝐵))) |
41 | 30, 32, 40 | 3bitr4g 306 | 1 ⊢ (((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ* 𝐴) ∧ (𝐴 ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ (𝐴 ∨ℋ 𝐵))) → (𝐶 𝑀ℋ* 𝐷 ↔ (𝐶 ∩ 𝐵) 𝑀ℋ* (𝐷 ∩ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∈ wcel 2166 ∩ cin 3797 ⊆ wss 3798 class class class wbr 4873 ‘cfv 6123 (class class class)co 6905 Cℋ cch 28341 ⊥cort 28342 ∨ℋ chj 28345 𝑀ℋ cmd 28378 𝑀ℋ* cdmd 28379 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cc 9572 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 ax-addf 10331 ax-mulf 10332 ax-hilex 28411 ax-hfvadd 28412 ax-hvcom 28413 ax-hvass 28414 ax-hv0cl 28415 ax-hvaddid 28416 ax-hfvmul 28417 ax-hvmulid 28418 ax-hvmulass 28419 ax-hvdistr1 28420 ax-hvdistr2 28421 ax-hvmul0 28422 ax-hfi 28491 ax-his1 28494 ax-his2 28495 ax-his3 28496 ax-his4 28497 ax-hcompl 28614 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-iin 4743 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-of 7157 df-om 7327 df-1st 7428 df-2nd 7429 df-supp 7560 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-2o 7827 df-oadd 7830 df-omul 7831 df-er 8009 df-map 8124 df-pm 8125 df-ixp 8176 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-fsupp 8545 df-fi 8586 df-sup 8617 df-inf 8618 df-oi 8684 df-card 9078 df-acn 9081 df-cda 9305 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-dec 11822 df-uz 11969 df-q 12072 df-rp 12113 df-xneg 12232 df-xadd 12233 df-xmul 12234 df-ioo 12467 df-ico 12469 df-icc 12470 df-fz 12620 df-fzo 12761 df-fl 12888 df-seq 13096 df-exp 13155 df-hash 13411 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-clim 14596 df-rlim 14597 df-sum 14794 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-starv 16320 df-sca 16321 df-vsca 16322 df-ip 16323 df-tset 16324 df-ple 16325 df-ds 16327 df-unif 16328 df-hom 16329 df-cco 16330 df-rest 16436 df-topn 16437 df-0g 16455 df-gsum 16456 df-topgen 16457 df-pt 16458 df-prds 16461 df-xrs 16515 df-qtop 16520 df-imas 16521 df-xps 16523 df-mre 16599 df-mrc 16600 df-acs 16602 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-submnd 17689 df-mulg 17895 df-cntz 18100 df-cmn 18548 df-psmet 20098 df-xmet 20099 df-met 20100 df-bl 20101 df-mopn 20102 df-fbas 20103 df-fg 20104 df-cnfld 20107 df-top 21069 df-topon 21086 df-topsp 21108 df-bases 21121 df-cld 21194 df-ntr 21195 df-cls 21196 df-nei 21273 df-cn 21402 df-cnp 21403 df-lm 21404 df-haus 21490 df-tx 21736 df-hmeo 21929 df-fil 22020 df-fm 22112 df-flim 22113 df-flf 22114 df-xms 22495 df-ms 22496 df-tms 22497 df-cfil 23423 df-cau 23424 df-cmet 23425 df-grpo 27903 df-gid 27904 df-ginv 27905 df-gdiv 27906 df-ablo 27955 df-vc 27969 df-nv 28002 df-va 28005 df-ba 28006 df-sm 28007 df-0v 28008 df-vs 28009 df-nmcv 28010 df-ims 28011 df-dip 28111 df-ssp 28132 df-ph 28223 df-cbn 28274 df-hnorm 28380 df-hba 28381 df-hvsub 28383 df-hlim 28384 df-hcau 28385 df-sh 28619 df-ch 28633 df-oc 28664 df-ch0 28665 df-shs 28722 df-chj 28724 df-md 29694 df-dmd 29695 |
This theorem is referenced by: dmdcompli 29844 |
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