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Mirrors > Home > HSE Home > Th. List > qlax1i | Structured version Visualization version GIF version |
Description: One of the equations showing Cℋ is an ortholattice. (This corresponds to axiom "ax-1" in the Quantum Logic Explorer.) (Contributed by NM, 4-Aug-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
qlax1.1 | ⊢ 𝐴 ∈ Cℋ |
Ref | Expression |
---|---|
qlax1i | ⊢ 𝐴 = (⊥‘(⊥‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qlax1.1 | . . 3 ⊢ 𝐴 ∈ Cℋ | |
2 | 1 | ococi 29166 | . 2 ⊢ (⊥‘(⊥‘𝐴)) = 𝐴 |
3 | 2 | eqcomi 2830 | 1 ⊢ 𝐴 = (⊥‘(⊥‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ‘cfv 6341 Cℋ cch 28690 ⊥cort 28691 |
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-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-om 7567 df-1st 7675 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-oadd 8092 df-omul 8093 df-er 8275 df-map 8394 df-pm 8395 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 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-n0 11885 df-z 11969 df-uz 12231 df-q 12336 df-rp 12377 df-xneg 12494 df-xadd 12495 df-xmul 12496 df-ico 12731 df-icc 12732 df-fz 12883 df-fl 13152 df-seq 13360 df-exp 13420 df-cj 14443 df-re 14444 df-im 14445 df-sqrt 14579 df-abs 14580 df-clim 14830 df-rlim 14831 df-rest 16679 df-topgen 16700 df-psmet 20520 df-xmet 20521 df-met 20522 df-bl 20523 df-mopn 20524 df-fbas 20525 df-fg 20526 df-top 21485 df-topon 21502 df-bases 21537 df-cld 21610 df-ntr 21611 df-cls 21612 df-nei 21689 df-lm 21820 df-haus 21906 df-fil 22437 df-fm 22529 df-flim 22530 df-flf 22531 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-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 |
This theorem is referenced by: (None) |
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