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Mirrors > Home > HSE Home > Th. List > pjoml2 | Structured version Visualization version GIF version |
Description: Variation of orthomodular law. Definition in [Kalmbach] p. 22. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjoml2 | ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵) → (𝐴 ∨ℋ ((⊥‘𝐴) ∩ 𝐵)) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1 4002 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) → (𝐴 ⊆ 𝐵 ↔ if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ⊆ 𝐵)) | |
2 | id 22 | . . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) → 𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ)) | |
3 | fveq2 6884 | . . . . . . 7 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) → (⊥‘𝐴) = (⊥‘if(𝐴 ∈ Cℋ , 𝐴, 0ℋ))) | |
4 | 3 | ineq1d 4206 | . . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) → ((⊥‘𝐴) ∩ 𝐵) = ((⊥‘if(𝐴 ∈ Cℋ , 𝐴, 0ℋ)) ∩ 𝐵)) |
5 | 2, 4 | oveq12d 7422 | . . . . 5 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) → (𝐴 ∨ℋ ((⊥‘𝐴) ∩ 𝐵)) = (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∨ℋ ((⊥‘if(𝐴 ∈ Cℋ , 𝐴, 0ℋ)) ∩ 𝐵))) |
6 | 5 | eqeq1d 2728 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) → ((𝐴 ∨ℋ ((⊥‘𝐴) ∩ 𝐵)) = 𝐵 ↔ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∨ℋ ((⊥‘if(𝐴 ∈ Cℋ , 𝐴, 0ℋ)) ∩ 𝐵)) = 𝐵)) |
7 | 1, 6 | imbi12d 344 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) → ((𝐴 ⊆ 𝐵 → (𝐴 ∨ℋ ((⊥‘𝐴) ∩ 𝐵)) = 𝐵) ↔ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ⊆ 𝐵 → (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∨ℋ ((⊥‘if(𝐴 ∈ Cℋ , 𝐴, 0ℋ)) ∩ 𝐵)) = 𝐵))) |
8 | sseq2 4003 | . . . 4 ⊢ (𝐵 = if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) → (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ⊆ 𝐵 ↔ if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ⊆ if(𝐵 ∈ Cℋ , 𝐵, 0ℋ))) | |
9 | ineq2 4201 | . . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) → ((⊥‘if(𝐴 ∈ Cℋ , 𝐴, 0ℋ)) ∩ 𝐵) = ((⊥‘if(𝐴 ∈ Cℋ , 𝐴, 0ℋ)) ∩ if(𝐵 ∈ Cℋ , 𝐵, 0ℋ))) | |
10 | 9 | oveq2d 7420 | . . . . 5 ⊢ (𝐵 = if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) → (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∨ℋ ((⊥‘if(𝐴 ∈ Cℋ , 𝐴, 0ℋ)) ∩ 𝐵)) = (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∨ℋ ((⊥‘if(𝐴 ∈ Cℋ , 𝐴, 0ℋ)) ∩ if(𝐵 ∈ Cℋ , 𝐵, 0ℋ)))) |
11 | id 22 | . . . . 5 ⊢ (𝐵 = if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) → 𝐵 = if(𝐵 ∈ Cℋ , 𝐵, 0ℋ)) | |
12 | 10, 11 | eqeq12d 2742 | . . . 4 ⊢ (𝐵 = if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) → ((if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∨ℋ ((⊥‘if(𝐴 ∈ Cℋ , 𝐴, 0ℋ)) ∩ 𝐵)) = 𝐵 ↔ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∨ℋ ((⊥‘if(𝐴 ∈ Cℋ , 𝐴, 0ℋ)) ∩ if(𝐵 ∈ Cℋ , 𝐵, 0ℋ))) = if(𝐵 ∈ Cℋ , 𝐵, 0ℋ))) |
13 | 8, 12 | imbi12d 344 | . . 3 ⊢ (𝐵 = if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) → ((if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ⊆ 𝐵 → (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∨ℋ ((⊥‘if(𝐴 ∈ Cℋ , 𝐴, 0ℋ)) ∩ 𝐵)) = 𝐵) ↔ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ⊆ if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) → (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∨ℋ ((⊥‘if(𝐴 ∈ Cℋ , 𝐴, 0ℋ)) ∩ if(𝐵 ∈ Cℋ , 𝐵, 0ℋ))) = if(𝐵 ∈ Cℋ , 𝐵, 0ℋ)))) |
14 | h0elch 31013 | . . . . 5 ⊢ 0ℋ ∈ Cℋ | |
15 | 14 | elimel 4592 | . . . 4 ⊢ if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∈ Cℋ |
16 | 14 | elimel 4592 | . . . 4 ⊢ if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) ∈ Cℋ |
17 | 15, 16 | pjoml2i 31343 | . . 3 ⊢ (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ⊆ if(𝐵 ∈ Cℋ , 𝐵, 0ℋ) → (if(𝐴 ∈ Cℋ , 𝐴, 0ℋ) ∨ℋ ((⊥‘if(𝐴 ∈ Cℋ , 𝐴, 0ℋ)) ∩ if(𝐵 ∈ Cℋ , 𝐵, 0ℋ))) = if(𝐵 ∈ Cℋ , 𝐵, 0ℋ)) |
18 | 7, 13, 17 | dedth2h 4582 | . 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⊆ 𝐵 → (𝐴 ∨ℋ ((⊥‘𝐴) ∩ 𝐵)) = 𝐵)) |
19 | 18 | 3impia 1114 | 1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵) → (𝐴 ∨ℋ ((⊥‘𝐴) ∩ 𝐵)) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∩ cin 3942 ⊆ wss 3943 ifcif 4523 ‘cfv 6536 (class class class)co 7404 Cℋ cch 30687 ⊥cort 30688 ∨ℋ chj 30691 0ℋc0h 30693 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cc 10429 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189 ax-hilex 30757 ax-hfvadd 30758 ax-hvcom 30759 ax-hvass 30760 ax-hv0cl 30761 ax-hvaddid 30762 ax-hfvmul 30763 ax-hvmulid 30764 ax-hvmulass 30765 ax-hvdistr1 30766 ax-hvdistr2 30767 ax-hvmul0 30768 ax-hfi 30837 ax-his1 30840 ax-his2 30841 ax-his3 30842 ax-his4 30843 ax-hcompl 30960 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-oadd 8468 df-omul 8469 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-acn 9936 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-ioo 13331 df-ico 13333 df-icc 13334 df-fz 13488 df-fzo 13631 df-fl 13760 df-seq 13970 df-exp 14031 df-hash 14294 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15436 df-rlim 15437 df-sum 15637 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-pt 17397 df-prds 17400 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-mulg 18994 df-cntz 19231 df-cmn 19700 df-psmet 21228 df-xmet 21229 df-met 21230 df-bl 21231 df-mopn 21232 df-fbas 21233 df-fg 21234 df-cnfld 21237 df-top 22747 df-topon 22764 df-topsp 22786 df-bases 22800 df-cld 22874 df-ntr 22875 df-cls 22876 df-nei 22953 df-cn 23082 df-cnp 23083 df-lm 23084 df-haus 23170 df-tx 23417 df-hmeo 23610 df-fil 23701 df-fm 23793 df-flim 23794 df-flf 23795 df-xms 24177 df-ms 24178 df-tms 24179 df-cfil 25134 df-cau 25135 df-cmet 25136 df-grpo 30251 df-gid 30252 df-ginv 30253 df-gdiv 30254 df-ablo 30303 df-vc 30317 df-nv 30350 df-va 30353 df-ba 30354 df-sm 30355 df-0v 30356 df-vs 30357 df-nmcv 30358 df-ims 30359 df-dip 30459 df-ssp 30480 df-ph 30571 df-cbn 30621 df-hnorm 30726 df-hba 30727 df-hvsub 30729 df-hlim 30730 df-hcau 30731 df-sh 30965 df-ch 30979 df-oc 31010 df-ch0 31011 df-shs 31066 df-chj 31068 |
This theorem is referenced by: pjoml5 31371 chirredlem3 32150 |
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