Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > HSE Home > Th. List > mdsymi | Structured version Visualization version GIF version |
Description: M-symmetry of the Hilbert lattice. Lemma 5 of [Maeda] p. 168. (Contributed by NM, 3-Jul-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mdsym.1 | ⊢ 𝐴 ∈ Cℋ |
mdsym.2 | ⊢ 𝐵 ∈ Cℋ |
Ref | Expression |
---|---|
mdsymi | ⊢ (𝐴 𝑀ℋ 𝐵 ↔ 𝐵 𝑀ℋ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mdsym.2 | . . . . 5 ⊢ 𝐵 ∈ Cℋ | |
2 | 1 | choccli 29068 | . . . 4 ⊢ (⊥‘𝐵) ∈ Cℋ |
3 | mdsym.1 | . . . . 5 ⊢ 𝐴 ∈ Cℋ | |
4 | 3 | choccli 29068 | . . . 4 ⊢ (⊥‘𝐴) ∈ Cℋ |
5 | eqid 2821 | . . . 4 ⊢ ((⊥‘𝐵) ∨ℋ 𝑥) = ((⊥‘𝐵) ∨ℋ 𝑥) | |
6 | 2, 4, 5 | mdsymlem8 30171 | . . 3 ⊢ (((⊥‘𝐵) ≠ 0ℋ ∧ (⊥‘𝐴) ≠ 0ℋ) → ((⊥‘𝐴) 𝑀ℋ* (⊥‘𝐵) ↔ (⊥‘𝐵) 𝑀ℋ* (⊥‘𝐴))) |
7 | mddmd 30062 | . . . 4 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ 𝐵 ↔ (⊥‘𝐴) 𝑀ℋ* (⊥‘𝐵))) | |
8 | 3, 1, 7 | mp2an 690 | . . 3 ⊢ (𝐴 𝑀ℋ 𝐵 ↔ (⊥‘𝐴) 𝑀ℋ* (⊥‘𝐵)) |
9 | mddmd 30062 | . . . 4 ⊢ ((𝐵 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ) → (𝐵 𝑀ℋ 𝐴 ↔ (⊥‘𝐵) 𝑀ℋ* (⊥‘𝐴))) | |
10 | 1, 3, 9 | mp2an 690 | . . 3 ⊢ (𝐵 𝑀ℋ 𝐴 ↔ (⊥‘𝐵) 𝑀ℋ* (⊥‘𝐴)) |
11 | 6, 8, 10 | 3bitr4g 316 | . 2 ⊢ (((⊥‘𝐵) ≠ 0ℋ ∧ (⊥‘𝐴) ≠ 0ℋ) → (𝐴 𝑀ℋ 𝐵 ↔ 𝐵 𝑀ℋ 𝐴)) |
12 | 3 | chssii 28992 | . . . 4 ⊢ 𝐴 ⊆ ℋ |
13 | fveq2 6656 | . . . . 5 ⊢ ((⊥‘𝐵) = 0ℋ → (⊥‘(⊥‘𝐵)) = (⊥‘0ℋ)) | |
14 | 1 | pjococi 29198 | . . . . 5 ⊢ (⊥‘(⊥‘𝐵)) = 𝐵 |
15 | choc0 29087 | . . . . 5 ⊢ (⊥‘0ℋ) = ℋ | |
16 | 13, 14, 15 | 3eqtr3g 2879 | . . . 4 ⊢ ((⊥‘𝐵) = 0ℋ → 𝐵 = ℋ) |
17 | 12, 16 | sseqtrrid 4008 | . . 3 ⊢ ((⊥‘𝐵) = 0ℋ → 𝐴 ⊆ 𝐵) |
18 | ssmd1 30072 | . . . . 5 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵) → 𝐴 𝑀ℋ 𝐵) | |
19 | 3, 1, 18 | mp3an12 1447 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 𝑀ℋ 𝐵) |
20 | ssmd2 30073 | . . . . 5 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵) → 𝐵 𝑀ℋ 𝐴) | |
21 | 3, 1, 20 | mp3an12 1447 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → 𝐵 𝑀ℋ 𝐴) |
22 | 19, 21 | jca 514 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ 𝐴)) |
23 | pm5.1 821 | . . 3 ⊢ ((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ 𝐴) → (𝐴 𝑀ℋ 𝐵 ↔ 𝐵 𝑀ℋ 𝐴)) | |
24 | 17, 22, 23 | 3syl 18 | . 2 ⊢ ((⊥‘𝐵) = 0ℋ → (𝐴 𝑀ℋ 𝐵 ↔ 𝐵 𝑀ℋ 𝐴)) |
25 | 1 | chssii 28992 | . . . 4 ⊢ 𝐵 ⊆ ℋ |
26 | fveq2 6656 | . . . . 5 ⊢ ((⊥‘𝐴) = 0ℋ → (⊥‘(⊥‘𝐴)) = (⊥‘0ℋ)) | |
27 | 3 | pjococi 29198 | . . . . 5 ⊢ (⊥‘(⊥‘𝐴)) = 𝐴 |
28 | 26, 27, 15 | 3eqtr3g 2879 | . . . 4 ⊢ ((⊥‘𝐴) = 0ℋ → 𝐴 = ℋ) |
29 | 25, 28 | sseqtrrid 4008 | . . 3 ⊢ ((⊥‘𝐴) = 0ℋ → 𝐵 ⊆ 𝐴) |
30 | ssmd2 30073 | . . . . 5 ⊢ ((𝐵 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ∧ 𝐵 ⊆ 𝐴) → 𝐴 𝑀ℋ 𝐵) | |
31 | 1, 3, 30 | mp3an12 1447 | . . . 4 ⊢ (𝐵 ⊆ 𝐴 → 𝐴 𝑀ℋ 𝐵) |
32 | ssmd1 30072 | . . . . 5 ⊢ ((𝐵 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ∧ 𝐵 ⊆ 𝐴) → 𝐵 𝑀ℋ 𝐴) | |
33 | 1, 3, 32 | mp3an12 1447 | . . . 4 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 𝑀ℋ 𝐴) |
34 | 31, 33 | jca 514 | . . 3 ⊢ (𝐵 ⊆ 𝐴 → (𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ 𝐴)) |
35 | 29, 34, 23 | 3syl 18 | . 2 ⊢ ((⊥‘𝐴) = 0ℋ → (𝐴 𝑀ℋ 𝐵 ↔ 𝐵 𝑀ℋ 𝐴)) |
36 | 11, 24, 35 | pm2.61iine 3107 | 1 ⊢ (𝐴 𝑀ℋ 𝐵 ↔ 𝐵 𝑀ℋ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ⊆ wss 3924 class class class wbr 5052 ‘cfv 6341 (class class class)co 7142 ℋchba 28680 Cℋ cch 28690 ⊥cort 28691 ∨ℋ chj 28694 0ℋc0h 28696 𝑀ℋ cmd 28727 𝑀ℋ* cdmd 28728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-inf2 9090 ax-cc 9843 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 ax-pre-sup 10601 ax-addf 10602 ax-mulf 10603 ax-hilex 28760 ax-hfvadd 28761 ax-hvcom 28762 ax-hvass 28763 ax-hv0cl 28764 ax-hvaddid 28765 ax-hfvmul 28766 ax-hvmulid 28767 ax-hvmulass 28768 ax-hvdistr1 28769 ax-hvdistr2 28770 ax-hvmul0 28771 ax-hfi 28840 ax-his1 28843 ax-his2 28844 ax-his3 28845 ax-his4 28846 ax-hcompl 28963 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-iin 4908 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-se 5501 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-isom 6350 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-of 7395 df-om 7567 df-1st 7675 df-2nd 7676 df-supp 7817 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-2o 8089 df-oadd 8092 df-omul 8093 df-er 8275 df-map 8394 df-pm 8395 df-ixp 8448 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-fsupp 8820 df-fi 8861 df-sup 8892 df-inf 8893 df-oi 8960 df-card 9354 df-acn 9357 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-div 11284 df-nn 11625 df-2 11687 df-3 11688 df-4 11689 df-5 11690 df-6 11691 df-7 11692 df-8 11693 df-9 11694 df-n0 11885 df-z 11969 df-dec 12086 df-uz 12231 df-q 12336 df-rp 12377 df-xneg 12494 df-xadd 12495 df-xmul 12496 df-ioo 12729 df-ico 12731 df-icc 12732 df-fz 12883 df-fzo 13024 df-fl 13152 df-seq 13360 df-exp 13420 df-hash 13681 df-cj 14443 df-re 14444 df-im 14445 df-sqrt 14579 df-abs 14580 df-clim 14830 df-rlim 14831 df-sum 15028 df-struct 16468 df-ndx 16469 df-slot 16470 df-base 16472 df-sets 16473 df-ress 16474 df-plusg 16561 df-mulr 16562 df-starv 16563 df-sca 16564 df-vsca 16565 df-ip 16566 df-tset 16567 df-ple 16568 df-ds 16570 df-unif 16571 df-hom 16572 df-cco 16573 df-rest 16679 df-topn 16680 df-0g 16698 df-gsum 16699 df-topgen 16700 df-pt 16701 df-prds 16704 df-xrs 16758 df-qtop 16763 df-imas 16764 df-xps 16766 df-mre 16840 df-mrc 16841 df-acs 16843 df-mgm 17835 df-sgrp 17884 df-mnd 17895 df-submnd 17940 df-mulg 18208 df-cntz 18430 df-cmn 18891 df-psmet 20520 df-xmet 20521 df-met 20522 df-bl 20523 df-mopn 20524 df-fbas 20525 df-fg 20526 df-cnfld 20529 df-top 21485 df-topon 21502 df-topsp 21524 df-bases 21537 df-cld 21610 df-ntr 21611 df-cls 21612 df-nei 21689 df-cn 21818 df-cnp 21819 df-lm 21820 df-haus 21906 df-tx 22153 df-hmeo 22346 df-fil 22437 df-fm 22529 df-flim 22530 df-flf 22531 df-xms 22913 df-ms 22914 df-tms 22915 df-cfil 23841 df-cau 23842 df-cmet 23843 df-grpo 28254 df-gid 28255 df-ginv 28256 df-gdiv 28257 df-ablo 28306 df-vc 28320 df-nv 28353 df-va 28356 df-ba 28357 df-sm 28358 df-0v 28359 df-vs 28360 df-nmcv 28361 df-ims 28362 df-dip 28462 df-ssp 28483 df-ph 28574 df-cbn 28624 df-hnorm 28729 df-hba 28730 df-hvsub 28732 df-hlim 28733 df-hcau 28734 df-sh 28968 df-ch 28982 df-oc 29013 df-ch0 29014 df-shs 29069 df-span 29070 df-chj 29071 df-chsup 29072 df-pjh 29156 df-cv 30040 df-md 30041 df-dmd 30042 df-at 30099 |
This theorem is referenced by: mdsym 30173 |
Copyright terms: Public domain | W3C validator |