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Mirrors > Home > HSE Home > Th. List > pjoml4i | Structured version Visualization version GIF version |
Description: Variation of orthomodular law. (Contributed by NM, 6-Dec-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjoml2.1 | ⊢ 𝐴 ∈ Cℋ |
pjoml2.2 | ⊢ 𝐵 ∈ Cℋ |
Ref | Expression |
---|---|
pjoml4i | ⊢ (𝐴 ∨ℋ (𝐵 ∩ ((⊥‘𝐴) ∨ℋ (⊥‘𝐵)))) = (𝐴 ∨ℋ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss1 4228 | . . 3 ⊢ (𝐵 ∩ ((⊥‘𝐴) ∨ℋ (⊥‘𝐵))) ⊆ 𝐵 | |
2 | pjoml2.2 | . . . . 5 ⊢ 𝐵 ∈ Cℋ | |
3 | pjoml2.1 | . . . . . . 7 ⊢ 𝐴 ∈ Cℋ | |
4 | 3 | choccli 30828 | . . . . . 6 ⊢ (⊥‘𝐴) ∈ Cℋ |
5 | 2 | choccli 30828 | . . . . . 6 ⊢ (⊥‘𝐵) ∈ Cℋ |
6 | 4, 5 | chjcli 30978 | . . . . 5 ⊢ ((⊥‘𝐴) ∨ℋ (⊥‘𝐵)) ∈ Cℋ |
7 | 2, 6 | chincli 30981 | . . . 4 ⊢ (𝐵 ∩ ((⊥‘𝐴) ∨ℋ (⊥‘𝐵))) ∈ Cℋ |
8 | 7, 2, 3 | chlej2i 30995 | . . 3 ⊢ ((𝐵 ∩ ((⊥‘𝐴) ∨ℋ (⊥‘𝐵))) ⊆ 𝐵 → (𝐴 ∨ℋ (𝐵 ∩ ((⊥‘𝐴) ∨ℋ (⊥‘𝐵)))) ⊆ (𝐴 ∨ℋ 𝐵)) |
9 | 1, 8 | ax-mp 5 | . 2 ⊢ (𝐴 ∨ℋ (𝐵 ∩ ((⊥‘𝐴) ∨ℋ (⊥‘𝐵)))) ⊆ (𝐴 ∨ℋ 𝐵) |
10 | 3, 7 | chub1i 30990 | . . 3 ⊢ 𝐴 ⊆ (𝐴 ∨ℋ (𝐵 ∩ ((⊥‘𝐴) ∨ℋ (⊥‘𝐵)))) |
11 | 3, 2 | chdmm1i 30998 | . . . . . . . 8 ⊢ (⊥‘(𝐴 ∩ 𝐵)) = ((⊥‘𝐴) ∨ℋ (⊥‘𝐵)) |
12 | 11 | ineq1i 4208 | . . . . . . 7 ⊢ ((⊥‘(𝐴 ∩ 𝐵)) ∩ 𝐵) = (((⊥‘𝐴) ∨ℋ (⊥‘𝐵)) ∩ 𝐵) |
13 | incom 4201 | . . . . . . 7 ⊢ (((⊥‘𝐴) ∨ℋ (⊥‘𝐵)) ∩ 𝐵) = (𝐵 ∩ ((⊥‘𝐴) ∨ℋ (⊥‘𝐵))) | |
14 | 12, 13 | eqtri 2759 | . . . . . 6 ⊢ ((⊥‘(𝐴 ∩ 𝐵)) ∩ 𝐵) = (𝐵 ∩ ((⊥‘𝐴) ∨ℋ (⊥‘𝐵))) |
15 | 14 | oveq2i 7423 | . . . . 5 ⊢ ((𝐴 ∩ 𝐵) ∨ℋ ((⊥‘(𝐴 ∩ 𝐵)) ∩ 𝐵)) = ((𝐴 ∩ 𝐵) ∨ℋ (𝐵 ∩ ((⊥‘𝐴) ∨ℋ (⊥‘𝐵)))) |
16 | inss2 4229 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
17 | 3, 2 | chincli 30981 | . . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ∈ Cℋ |
18 | 17, 2 | pjoml2i 31106 | . . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐵 → ((𝐴 ∩ 𝐵) ∨ℋ ((⊥‘(𝐴 ∩ 𝐵)) ∩ 𝐵)) = 𝐵) |
19 | 16, 18 | ax-mp 5 | . . . . 5 ⊢ ((𝐴 ∩ 𝐵) ∨ℋ ((⊥‘(𝐴 ∩ 𝐵)) ∩ 𝐵)) = 𝐵 |
20 | 15, 19 | eqtr3i 2761 | . . . 4 ⊢ ((𝐴 ∩ 𝐵) ∨ℋ (𝐵 ∩ ((⊥‘𝐴) ∨ℋ (⊥‘𝐵)))) = 𝐵 |
21 | inss1 4228 | . . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
22 | 17, 3, 7 | chlej1i 30994 | . . . . 5 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 → ((𝐴 ∩ 𝐵) ∨ℋ (𝐵 ∩ ((⊥‘𝐴) ∨ℋ (⊥‘𝐵)))) ⊆ (𝐴 ∨ℋ (𝐵 ∩ ((⊥‘𝐴) ∨ℋ (⊥‘𝐵))))) |
23 | 21, 22 | ax-mp 5 | . . . 4 ⊢ ((𝐴 ∩ 𝐵) ∨ℋ (𝐵 ∩ ((⊥‘𝐴) ∨ℋ (⊥‘𝐵)))) ⊆ (𝐴 ∨ℋ (𝐵 ∩ ((⊥‘𝐴) ∨ℋ (⊥‘𝐵)))) |
24 | 20, 23 | eqsstrri 4017 | . . 3 ⊢ 𝐵 ⊆ (𝐴 ∨ℋ (𝐵 ∩ ((⊥‘𝐴) ∨ℋ (⊥‘𝐵)))) |
25 | 3, 7 | chjcli 30978 | . . . 4 ⊢ (𝐴 ∨ℋ (𝐵 ∩ ((⊥‘𝐴) ∨ℋ (⊥‘𝐵)))) ∈ Cℋ |
26 | 3, 2, 25 | chlubii 30993 | . . 3 ⊢ ((𝐴 ⊆ (𝐴 ∨ℋ (𝐵 ∩ ((⊥‘𝐴) ∨ℋ (⊥‘𝐵)))) ∧ 𝐵 ⊆ (𝐴 ∨ℋ (𝐵 ∩ ((⊥‘𝐴) ∨ℋ (⊥‘𝐵))))) → (𝐴 ∨ℋ 𝐵) ⊆ (𝐴 ∨ℋ (𝐵 ∩ ((⊥‘𝐴) ∨ℋ (⊥‘𝐵))))) |
27 | 10, 24, 26 | mp2an 689 | . 2 ⊢ (𝐴 ∨ℋ 𝐵) ⊆ (𝐴 ∨ℋ (𝐵 ∩ ((⊥‘𝐴) ∨ℋ (⊥‘𝐵)))) |
28 | 9, 27 | eqssi 3998 | 1 ⊢ (𝐴 ∨ℋ (𝐵 ∩ ((⊥‘𝐴) ∨ℋ (⊥‘𝐵)))) = (𝐴 ∨ℋ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∈ wcel 2105 ∩ cin 3947 ⊆ wss 3948 ‘cfv 6543 (class class class)co 7412 Cℋ cch 30450 ⊥cort 30451 ∨ℋ chj 30454 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728 ax-inf2 9639 ax-cc 10433 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191 ax-addf 11192 ax-mulf 11193 ax-hilex 30520 ax-hfvadd 30521 ax-hvcom 30522 ax-hvass 30523 ax-hv0cl 30524 ax-hvaddid 30525 ax-hfvmul 30526 ax-hvmulid 30527 ax-hvmulass 30528 ax-hvdistr1 30529 ax-hvdistr2 30530 ax-hvmul0 30531 ax-hfi 30600 ax-his1 30603 ax-his2 30604 ax-his3 30605 ax-his4 30606 ax-hcompl 30723 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7673 df-om 7859 df-1st 7978 df-2nd 7979 df-supp 8150 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-2o 8470 df-oadd 8473 df-omul 8474 df-er 8706 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9365 df-fi 9409 df-sup 9440 df-inf 9441 df-oi 9508 df-card 9937 df-acn 9940 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-ioo 13333 df-ico 13335 df-icc 13336 df-fz 13490 df-fzo 13633 df-fl 13762 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-clim 15437 df-rlim 15438 df-sum 15638 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18707 df-mulg 18988 df-cntz 19223 df-cmn 19692 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-fbas 21142 df-fg 21143 df-cnfld 21146 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-cld 22744 df-ntr 22745 df-cls 22746 df-nei 22823 df-cn 22952 df-cnp 22953 df-lm 22954 df-haus 23040 df-tx 23287 df-hmeo 23480 df-fil 23571 df-fm 23663 df-flim 23664 df-flf 23665 df-xms 24047 df-ms 24048 df-tms 24049 df-cfil 25004 df-cau 25005 df-cmet 25006 df-grpo 30014 df-gid 30015 df-ginv 30016 df-gdiv 30017 df-ablo 30066 df-vc 30080 df-nv 30113 df-va 30116 df-ba 30117 df-sm 30118 df-0v 30119 df-vs 30120 df-nmcv 30121 df-ims 30122 df-dip 30222 df-ssp 30243 df-ph 30334 df-cbn 30384 df-hnorm 30489 df-hba 30490 df-hvsub 30492 df-hlim 30493 df-hcau 30494 df-sh 30728 df-ch 30742 df-oc 30773 df-ch0 30774 df-shs 30829 df-chj 30831 |
This theorem is referenced by: osumcor2i 31165 |
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