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Mirrors > Home > HSE Home > Th. List > chpsscon1 | Structured version Visualization version GIF version |
Description: Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
chpsscon1 | ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ((⊥‘𝐴) ⊊ 𝐵 ↔ (⊥‘𝐵) ⊊ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | choccl 30251 | . . 3 ⊢ (𝐴 ∈ Cℋ → (⊥‘𝐴) ∈ Cℋ ) | |
2 | chpsscon3 30448 | . . 3 ⊢ (((⊥‘𝐴) ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ((⊥‘𝐴) ⊊ 𝐵 ↔ (⊥‘𝐵) ⊊ (⊥‘(⊥‘𝐴)))) | |
3 | 1, 2 | sylan 581 | . 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ((⊥‘𝐴) ⊊ 𝐵 ↔ (⊥‘𝐵) ⊊ (⊥‘(⊥‘𝐴)))) |
4 | ococ 30351 | . . . 4 ⊢ (𝐴 ∈ Cℋ → (⊥‘(⊥‘𝐴)) = 𝐴) | |
5 | 4 | adantr 482 | . . 3 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (⊥‘(⊥‘𝐴)) = 𝐴) |
6 | 5 | psseq2d 4054 | . 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ((⊥‘𝐵) ⊊ (⊥‘(⊥‘𝐴)) ↔ (⊥‘𝐵) ⊊ 𝐴)) |
7 | 3, 6 | bitrd 279 | 1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ((⊥‘𝐴) ⊊ 𝐵 ↔ (⊥‘𝐵) ⊊ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊊ wpss 3912 ‘cfv 6497 Cℋ cch 29874 ⊥cort 29875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9578 ax-cc 10372 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-pre-sup 11130 ax-addf 11131 ax-mulf 11132 ax-hilex 29944 ax-hfvadd 29945 ax-hvcom 29946 ax-hvass 29947 ax-hv0cl 29948 ax-hvaddid 29949 ax-hfvmul 29950 ax-hvmulid 29951 ax-hvmulass 29952 ax-hvdistr1 29953 ax-hvdistr2 29954 ax-hvmul0 29955 ax-hfi 30024 ax-his1 30027 ax-his2 30028 ax-his3 30029 ax-his4 30030 ax-hcompl 30147 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-oadd 8417 df-omul 8418 df-er 8649 df-map 8768 df-pm 8769 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9307 df-fi 9348 df-sup 9379 df-inf 9380 df-oi 9447 df-card 9876 df-acn 9879 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-7 12222 df-8 12223 df-9 12224 df-n0 12415 df-z 12501 df-dec 12620 df-uz 12765 df-q 12875 df-rp 12917 df-xneg 13034 df-xadd 13035 df-xmul 13036 df-ioo 13269 df-ico 13271 df-icc 13272 df-fz 13426 df-fzo 13569 df-fl 13698 df-seq 13908 df-exp 13969 df-hash 14232 df-cj 14985 df-re 14986 df-im 14987 df-sqrt 15121 df-abs 15122 df-clim 15371 df-rlim 15372 df-sum 15572 df-struct 17020 df-sets 17037 df-slot 17055 df-ndx 17067 df-base 17085 df-ress 17114 df-plusg 17147 df-mulr 17148 df-starv 17149 df-sca 17150 df-vsca 17151 df-ip 17152 df-tset 17153 df-ple 17154 df-ds 17156 df-unif 17157 df-hom 17158 df-cco 17159 df-rest 17305 df-topn 17306 df-0g 17324 df-gsum 17325 df-topgen 17326 df-pt 17327 df-prds 17330 df-xrs 17385 df-qtop 17390 df-imas 17391 df-xps 17393 df-mre 17467 df-mrc 17468 df-acs 17470 df-mgm 18498 df-sgrp 18547 df-mnd 18558 df-submnd 18603 df-mulg 18874 df-cntz 19098 df-cmn 19565 df-psmet 20791 df-xmet 20792 df-met 20793 df-bl 20794 df-mopn 20795 df-fbas 20796 df-fg 20797 df-cnfld 20800 df-top 22246 df-topon 22263 df-topsp 22285 df-bases 22299 df-cld 22373 df-ntr 22374 df-cls 22375 df-nei 22452 df-cn 22581 df-cnp 22582 df-lm 22583 df-haus 22669 df-tx 22916 df-hmeo 23109 df-fil 23200 df-fm 23292 df-flim 23293 df-flf 23294 df-xms 23676 df-ms 23677 df-tms 23678 df-cfil 24622 df-cau 24623 df-cmet 24624 df-grpo 29438 df-gid 29439 df-ginv 29440 df-gdiv 29441 df-ablo 29490 df-vc 29504 df-nv 29537 df-va 29540 df-ba 29541 df-sm 29542 df-0v 29543 df-vs 29544 df-nmcv 29545 df-ims 29546 df-dip 29646 df-ssp 29667 df-ph 29758 df-cbn 29808 df-hnorm 29913 df-hba 29914 df-hvsub 29916 df-hlim 29917 df-hcau 29918 df-sh 30152 df-ch 30166 df-oc 30197 df-ch0 30198 |
This theorem is referenced by: cvcon3 31229 |
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