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| Mirrors > Home > HSE Home > Th. List > chjidm | Structured version Visualization version GIF version | ||
| Description: Idempotent law for Hilbert lattice join. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| chjidm | ⊢ (𝐴 ∈ Cℋ → (𝐴 ∨ℋ 𝐴) = 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | inidm 4217 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
| 2 | 1 | oveq2i 7429 | . 2 ⊢ (𝐴 ∨ℋ (𝐴 ∩ 𝐴)) = (𝐴 ∨ℋ 𝐴) |
| 3 | chabs1 31392 | . . 3 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ) → (𝐴 ∨ℋ (𝐴 ∩ 𝐴)) = 𝐴) | |
| 4 | 3 | anidms 565 | . 2 ⊢ (𝐴 ∈ Cℋ → (𝐴 ∨ℋ (𝐴 ∩ 𝐴)) = 𝐴) |
| 5 | 2, 4 | eqtr3id 2779 | 1 ⊢ (𝐴 ∈ Cℋ → (𝐴 ∨ℋ 𝐴) = 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∩ cin 3943 (class class class)co 7418 Cℋ cch 30805 ∨ℋ chj 30809 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cc 10459 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 ax-addf 11218 ax-mulf 11219 ax-hilex 30875 ax-hfvadd 30876 ax-hvcom 30877 ax-hvass 30878 ax-hv0cl 30879 ax-hvaddid 30880 ax-hfvmul 30881 ax-hvmulid 30882 ax-hvmulass 30883 ax-hvdistr1 30884 ax-hvdistr2 30885 ax-hvmul0 30886 ax-hfi 30955 ax-his1 30958 ax-his2 30959 ax-his3 30960 ax-his4 30961 ax-hcompl 31078 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6306 df-ord 6373 df-on 6374 df-lim 6375 df-suc 6376 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7374 df-ov 7421 df-oprab 7422 df-mpo 7423 df-of 7684 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8165 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-oadd 8490 df-omul 8491 df-er 8724 df-map 8846 df-pm 8847 df-ixp 8916 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-fsupp 9387 df-fi 9435 df-sup 9466 df-inf 9467 df-oi 9534 df-card 9963 df-acn 9966 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-ioo 13361 df-ico 13363 df-icc 13364 df-fz 13518 df-fzo 13661 df-fl 13791 df-seq 14001 df-exp 14061 df-hash 14324 df-cj 15080 df-re 15081 df-im 15082 df-sqrt 15216 df-abs 15217 df-clim 15466 df-rlim 15467 df-sum 15667 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17182 df-ress 17211 df-plusg 17247 df-mulr 17248 df-starv 17249 df-sca 17250 df-vsca 17251 df-ip 17252 df-tset 17253 df-ple 17254 df-ds 17256 df-unif 17257 df-hom 17258 df-cco 17259 df-rest 17405 df-topn 17406 df-0g 17424 df-gsum 17425 df-topgen 17426 df-pt 17427 df-prds 17430 df-xrs 17485 df-qtop 17490 df-imas 17491 df-xps 17493 df-mre 17567 df-mrc 17568 df-acs 17570 df-mgm 18601 df-sgrp 18680 df-mnd 18696 df-submnd 18742 df-mulg 19030 df-cntz 19277 df-cmn 19746 df-psmet 21285 df-xmet 21286 df-met 21287 df-bl 21288 df-mopn 21289 df-fbas 21290 df-fg 21291 df-cnfld 21294 df-top 22836 df-topon 22853 df-topsp 22875 df-bases 22889 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-cn 23171 df-cnp 23172 df-lm 23173 df-haus 23259 df-tx 23506 df-hmeo 23699 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-xms 24266 df-ms 24267 df-tms 24268 df-cfil 25223 df-cau 25224 df-cmet 25225 df-grpo 30369 df-gid 30370 df-ginv 30371 df-gdiv 30372 df-ablo 30421 df-vc 30435 df-nv 30468 df-va 30471 df-ba 30472 df-sm 30473 df-0v 30474 df-vs 30475 df-nmcv 30476 df-ims 30477 df-dip 30577 df-ssp 30598 df-ph 30689 df-cbn 30739 df-hnorm 30844 df-hba 30845 df-hvsub 30847 df-hlim 30848 df-hcau 30849 df-sh 31083 df-ch 31097 df-oc 31128 df-ch0 31129 df-shs 31184 df-chj 31186 |
| This theorem is referenced by: chjidmi 31397 atcv0eq 32255 atcv1 32256 atcvat3i 32272 |
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