Nearby theorems
|
|
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > HSE Home > Th. List > mdslmd2i | Structured version Visualization version GIF version | ||
| Description: Preservation of the modular pair property in the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2 (join version). (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 |
|---|---|
| mdslmd2i | ⊢ (((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ* 𝐴) ∧ ((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵)) → (𝐶 𝑀ℋ 𝐷 ↔ (𝐶 ∨ℋ 𝐴) 𝑀ℋ (𝐷 ∨ℋ 𝐴))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mdslmd.3 | . . . . . . . 8 ⊢ 𝐶 ∈ Cℋ | |
| 2 | mdslmd.4 | . . . . . . . 8 ⊢ 𝐷 ∈ Cℋ | |
| 3 | 1, 2 | chjcli 28444 | . . . . . . 7 ⊢ (𝐶 ∨ℋ 𝐷) ∈ Cℋ |
| 4 | mdslmd.2 | . . . . . . 7 ⊢ 𝐵 ∈ Cℋ | |
| 5 | mdslmd.1 | . . . . . . 7 ⊢ 𝐴 ∈ Cℋ | |
| 6 | 3, 4, 5 | chlej1i 28460 | . . . . . 6 ⊢ ((𝐶 ∨ℋ 𝐷) ⊆ 𝐵 → ((𝐶 ∨ℋ 𝐷) ∨ℋ 𝐴) ⊆ (𝐵 ∨ℋ 𝐴)) |
| 7 | 1, 2, 5 | chjjdiri 28511 | . . . . . 6 ⊢ ((𝐶 ∨ℋ 𝐷) ∨ℋ 𝐴) = ((𝐶 ∨ℋ 𝐴) ∨ℋ (𝐷 ∨ℋ 𝐴)) |
| 8 | 4, 5 | chjcomi 28455 | . . . . . 6 ⊢ (𝐵 ∨ℋ 𝐴) = (𝐴 ∨ℋ 𝐵) |
| 9 | 6, 7, 8 | 3sstr3g 3678 | . . . . 5 ⊢ ((𝐶 ∨ℋ 𝐷) ⊆ 𝐵 → ((𝐶 ∨ℋ 𝐴) ∨ℋ (𝐷 ∨ℋ 𝐴)) ⊆ (𝐴 ∨ℋ 𝐵)) |
| 10 | 9 | adantl 481 | . . . 4 ⊢ (((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵) → ((𝐶 ∨ℋ 𝐴) ∨ℋ (𝐷 ∨ℋ 𝐴)) ⊆ (𝐴 ∨ℋ 𝐵)) |
| 11 | 5, 1 | chub2i 28457 | . . . . 5 ⊢ 𝐴 ⊆ (𝐶 ∨ℋ 𝐴) |
| 12 | 5, 2 | chub2i 28457 | . . . . 5 ⊢ 𝐴 ⊆ (𝐷 ∨ℋ 𝐴) |
| 13 | 11, 12 | ssini 3869 | . . . 4 ⊢ 𝐴 ⊆ ((𝐶 ∨ℋ 𝐴) ∩ (𝐷 ∨ℋ 𝐴)) |
| 14 | 10, 13 | jctil 559 | . . 3 ⊢ (((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵) → (𝐴 ⊆ ((𝐶 ∨ℋ 𝐴) ∩ (𝐷 ∨ℋ 𝐴)) ∧ ((𝐶 ∨ℋ 𝐴) ∨ℋ (𝐷 ∨ℋ 𝐴)) ⊆ (𝐴 ∨ℋ 𝐵))) |
| 15 | 1, 5 | chjcli 28444 | . . . 4 ⊢ (𝐶 ∨ℋ 𝐴) ∈ Cℋ |
| 16 | 2, 5 | chjcli 28444 | . . . 4 ⊢ (𝐷 ∨ℋ 𝐴) ∈ Cℋ |
| 17 | 5, 4, 15, 16 | mdslmd1i 29316 | . . 3 ⊢ (((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ* 𝐴) ∧ (𝐴 ⊆ ((𝐶 ∨ℋ 𝐴) ∩ (𝐷 ∨ℋ 𝐴)) ∧ ((𝐶 ∨ℋ 𝐴) ∨ℋ (𝐷 ∨ℋ 𝐴)) ⊆ (𝐴 ∨ℋ 𝐵))) → ((𝐶 ∨ℋ 𝐴) 𝑀ℋ (𝐷 ∨ℋ 𝐴) ↔ ((𝐶 ∨ℋ 𝐴) ∩ 𝐵) 𝑀ℋ ((𝐷 ∨ℋ 𝐴) ∩ 𝐵))) |
| 18 | 14, 17 | sylan2 490 | . 2 ⊢ (((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ* 𝐴) ∧ ((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵)) → ((𝐶 ∨ℋ 𝐴) 𝑀ℋ (𝐷 ∨ℋ 𝐴) ↔ ((𝐶 ∨ℋ 𝐴) ∩ 𝐵) 𝑀ℋ ((𝐷 ∨ℋ 𝐴) ∩ 𝐵))) |
| 19 | id 22 | . . . . . 6 ⊢ (𝐴 𝑀ℋ 𝐵 → 𝐴 𝑀ℋ 𝐵) | |
| 20 | inss1 3866 | . . . . . . 7 ⊢ (𝐶 ∩ 𝐷) ⊆ 𝐶 | |
| 21 | sstr 3644 | . . . . . . 7 ⊢ (((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝐶) → (𝐴 ∩ 𝐵) ⊆ 𝐶) | |
| 22 | 20, 21 | mpan2 707 | . . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) → (𝐴 ∩ 𝐵) ⊆ 𝐶) |
| 23 | 1, 2 | chub1i 28456 | . . . . . . 7 ⊢ 𝐶 ⊆ (𝐶 ∨ℋ 𝐷) |
| 24 | sstr 3644 | . . . . . . 7 ⊢ ((𝐶 ⊆ (𝐶 ∨ℋ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵) → 𝐶 ⊆ 𝐵) | |
| 25 | 23, 24 | mpan 706 | . . . . . 6 ⊢ ((𝐶 ∨ℋ 𝐷) ⊆ 𝐵 → 𝐶 ⊆ 𝐵) |
| 26 | 5, 4, 1 | 3pm3.2i 1259 | . . . . . . 7 ⊢ (𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) |
| 27 | mdsl3 29303 | . . . . . . 7 ⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ (𝐴 𝑀ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵)) → ((𝐶 ∨ℋ 𝐴) ∩ 𝐵) = 𝐶) | |
| 28 | 26, 27 | mpan 706 | . . . . . 6 ⊢ ((𝐴 𝑀ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) → ((𝐶 ∨ℋ 𝐴) ∩ 𝐵) = 𝐶) |
| 29 | 19, 22, 25, 28 | syl3an 1408 | . . . . 5 ⊢ ((𝐴 𝑀ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵) → ((𝐶 ∨ℋ 𝐴) ∩ 𝐵) = 𝐶) |
| 30 | inss2 3867 | . . . . . . 7 ⊢ (𝐶 ∩ 𝐷) ⊆ 𝐷 | |
| 31 | sstr 3644 | . . . . . . 7 ⊢ (((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝐷) → (𝐴 ∩ 𝐵) ⊆ 𝐷) | |
| 32 | 30, 31 | mpan2 707 | . . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) → (𝐴 ∩ 𝐵) ⊆ 𝐷) |
| 33 | 2, 1 | chub2i 28457 | . . . . . . 7 ⊢ 𝐷 ⊆ (𝐶 ∨ℋ 𝐷) |
| 34 | sstr 3644 | . . . . . . 7 ⊢ ((𝐷 ⊆ (𝐶 ∨ℋ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵) → 𝐷 ⊆ 𝐵) | |
| 35 | 33, 34 | mpan 706 | . . . . . 6 ⊢ ((𝐶 ∨ℋ 𝐷) ⊆ 𝐵 → 𝐷 ⊆ 𝐵) |
| 36 | 5, 4, 2 | 3pm3.2i 1259 | . . . . . . 7 ⊢ (𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐷 ∈ Cℋ ) |
| 37 | mdsl3 29303 | . . . . . . 7 ⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐷 ∈ Cℋ ) ∧ (𝐴 𝑀ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐵)) → ((𝐷 ∨ℋ 𝐴) ∩ 𝐵) = 𝐷) | |
| 38 | 36, 37 | mpan 706 | . . . . . 6 ⊢ ((𝐴 𝑀ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐵) → ((𝐷 ∨ℋ 𝐴) ∩ 𝐵) = 𝐷) |
| 39 | 19, 32, 35, 38 | syl3an 1408 | . . . . 5 ⊢ ((𝐴 𝑀ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵) → ((𝐷 ∨ℋ 𝐴) ∩ 𝐵) = 𝐷) |
| 40 | 29, 39 | breq12d 4698 | . . . 4 ⊢ ((𝐴 𝑀ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵) → (((𝐶 ∨ℋ 𝐴) ∩ 𝐵) 𝑀ℋ ((𝐷 ∨ℋ 𝐴) ∩ 𝐵) ↔ 𝐶 𝑀ℋ 𝐷)) |
| 41 | 40 | 3expb 1285 | . . 3 ⊢ ((𝐴 𝑀ℋ 𝐵 ∧ ((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵)) → (((𝐶 ∨ℋ 𝐴) ∩ 𝐵) 𝑀ℋ ((𝐷 ∨ℋ 𝐴) ∩ 𝐵) ↔ 𝐶 𝑀ℋ 𝐷)) |
| 42 | 41 | adantlr 751 | . 2 ⊢ (((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ* 𝐴) ∧ ((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵)) → (((𝐶 ∨ℋ 𝐴) ∩ 𝐵) 𝑀ℋ ((𝐷 ∨ℋ 𝐴) ∩ 𝐵) ↔ 𝐶 𝑀ℋ 𝐷)) |
| 43 | 18, 42 | bitr2d 269 | 1 ⊢ (((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ* 𝐴) ∧ ((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵)) → (𝐶 𝑀ℋ 𝐷 ↔ (𝐶 ∨ℋ 𝐴) 𝑀ℋ (𝐷 ∨ℋ 𝐴))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ∩ cin 3606 ⊆ wss 3607 class class class wbr 4685 (class class class)co 6690 Cℋ cch 27914 ∨ℋ chj 27918 𝑀ℋ cmd 27951 𝑀ℋ* cdmd 27952 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cc 9295 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-addf 10053 ax-mulf 10054 ax-hilex 27984 ax-hfvadd 27985 ax-hvcom 27986 ax-hvass 27987 ax-hv0cl 27988 ax-hvaddid 27989 ax-hfvmul 27990 ax-hvmulid 27991 ax-hvmulass 27992 ax-hvdistr1 27993 ax-hvdistr2 27994 ax-hvmul0 27995 ax-hfi 28064 ax-his1 28067 ax-his2 28068 ax-his3 28069 ax-his4 28070 ax-hcompl 28187 |
| This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-omul 7610 df-er 7787 df-map 7901 df-pm 7902 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-fi 8358 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-acn 8806 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-q 11827 df-rp 11871 df-xneg 11984 df-xadd 11985 df-xmul 11986 df-ioo 12217 df-ico 12219 df-icc 12220 df-fz 12365 df-fzo 12505 df-fl 12633 df-seq 12842 df-exp 12901 df-hash 13158 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-clim 14263 df-rlim 14264 df-sum 14461 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-starv 16003 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-hom 16013 df-cco 16014 df-rest 16130 df-topn 16131 df-0g 16149 df-gsum 16150 df-topgen 16151 df-pt 16152 df-prds 16155 df-xrs 16209 df-qtop 16214 df-imas 16215 df-xps 16217 df-mre 16293 df-mrc 16294 df-acs 16296 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-submnd 17383 df-mulg 17588 df-cntz 17796 df-cmn 18241 df-psmet 19786 df-xmet 19787 df-met 19788 df-bl 19789 df-mopn 19790 df-fbas 19791 df-fg 19792 df-cnfld 19795 df-top 20747 df-topon 20764 df-topsp 20785 df-bases 20798 df-cld 20871 df-ntr 20872 df-cls 20873 df-nei 20950 df-cn 21079 df-cnp 21080 df-lm 21081 df-haus 21167 df-tx 21413 df-hmeo 21606 df-fil 21697 df-fm 21789 df-flim 21790 df-flf 21791 df-xms 22172 df-ms 22173 df-tms 22174 df-cfil 23099 df-cau 23100 df-cmet 23101 df-grpo 27475 df-gid 27476 df-ginv 27477 df-gdiv 27478 df-ablo 27527 df-vc 27542 df-nv 27575 df-va 27578 df-ba 27579 df-sm 27580 df-0v 27581 df-vs 27582 df-nmcv 27583 df-ims 27584 df-dip 27684 df-ssp 27705 df-ph 27796 df-cbn 27847 df-hnorm 27953 df-hba 27954 df-hvsub 27956 df-hlim 27957 df-hcau 27958 df-sh 28192 df-ch 28206 df-oc 28237 df-ch0 28238 df-shs 28295 df-chj 28297 df-md 29267 df-dmd 29268 |
| This theorem is referenced by: mdsldmd1i 29318 |
| Copyright terms: Public domain | W3C validator |