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Mirrors > Home > HSE Home > Th. List > mddmdin0i | Structured version Visualization version GIF version |
Description: If dual modular implies modular whenever meet is zero, then dual modular implies modular for arbitrary lattice elements. This theorem is needed for the remark after Lemma 7 of [Holland] p. 1524 to hold. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mddmdin0.1 | ⊢ 𝐴 ∈ Cℋ |
mddmdin0.2 | ⊢ 𝐵 ∈ Cℋ |
mddmdin0.3 | ⊢ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ ((𝑥 𝑀ℋ* 𝑦 ∧ (𝑥 ∩ 𝑦) = 0ℋ) → 𝑥 𝑀ℋ 𝑦) |
Ref | Expression |
---|---|
mddmdin0i | ⊢ (𝐴 𝑀ℋ* 𝐵 → 𝐴 𝑀ℋ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inindir 4161 | . . . 4 ⊢ ((𝐴 ∩ 𝐵) ∩ (⊥‘(𝐴 ∩ 𝐵))) = ((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) ∩ (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵)))) | |
2 | mddmdin0.1 | . . . . . 6 ⊢ 𝐴 ∈ Cℋ | |
3 | mddmdin0.2 | . . . . . 6 ⊢ 𝐵 ∈ Cℋ | |
4 | 2, 3 | chincli 29830 | . . . . 5 ⊢ (𝐴 ∩ 𝐵) ∈ Cℋ |
5 | 4 | chocini 29824 | . . . 4 ⊢ ((𝐴 ∩ 𝐵) ∩ (⊥‘(𝐴 ∩ 𝐵))) = 0ℋ |
6 | 1, 5 | eqtr3i 2768 | . . 3 ⊢ ((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) ∩ (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵)))) = 0ℋ |
7 | mddmdin0.3 | . . . 4 ⊢ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ ((𝑥 𝑀ℋ* 𝑦 ∧ (𝑥 ∩ 𝑦) = 0ℋ) → 𝑥 𝑀ℋ 𝑦) | |
8 | 4 | choccli 29677 | . . . . . 6 ⊢ (⊥‘(𝐴 ∩ 𝐵)) ∈ Cℋ |
9 | 2, 8 | chincli 29830 | . . . . 5 ⊢ (𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) ∈ Cℋ |
10 | 3, 8 | chincli 29830 | . . . . 5 ⊢ (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵))) ∈ Cℋ |
11 | breq1 5076 | . . . . . . . 8 ⊢ (𝑥 = (𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) → (𝑥 𝑀ℋ* 𝑦 ↔ (𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ* 𝑦)) | |
12 | ineq1 4139 | . . . . . . . . 9 ⊢ (𝑥 = (𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) → (𝑥 ∩ 𝑦) = ((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) ∩ 𝑦)) | |
13 | 12 | eqeq1d 2740 | . . . . . . . 8 ⊢ (𝑥 = (𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) → ((𝑥 ∩ 𝑦) = 0ℋ ↔ ((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) ∩ 𝑦) = 0ℋ)) |
14 | 11, 13 | anbi12d 631 | . . . . . . 7 ⊢ (𝑥 = (𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) → ((𝑥 𝑀ℋ* 𝑦 ∧ (𝑥 ∩ 𝑦) = 0ℋ) ↔ ((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ* 𝑦 ∧ ((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) ∩ 𝑦) = 0ℋ))) |
15 | breq1 5076 | . . . . . . 7 ⊢ (𝑥 = (𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) → (𝑥 𝑀ℋ 𝑦 ↔ (𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ 𝑦)) | |
16 | 14, 15 | imbi12d 345 | . . . . . 6 ⊢ (𝑥 = (𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) → (((𝑥 𝑀ℋ* 𝑦 ∧ (𝑥 ∩ 𝑦) = 0ℋ) → 𝑥 𝑀ℋ 𝑦) ↔ (((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ* 𝑦 ∧ ((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) ∩ 𝑦) = 0ℋ) → (𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ 𝑦))) |
17 | breq2 5077 | . . . . . . . 8 ⊢ (𝑦 = (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵))) → ((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ* 𝑦 ↔ (𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ* (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵))))) | |
18 | ineq2 4140 | . . . . . . . . 9 ⊢ (𝑦 = (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵))) → ((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) ∩ 𝑦) = ((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) ∩ (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵))))) | |
19 | 18 | eqeq1d 2740 | . . . . . . . 8 ⊢ (𝑦 = (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵))) → (((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) ∩ 𝑦) = 0ℋ ↔ ((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) ∩ (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵)))) = 0ℋ)) |
20 | 17, 19 | anbi12d 631 | . . . . . . 7 ⊢ (𝑦 = (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵))) → (((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ* 𝑦 ∧ ((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) ∩ 𝑦) = 0ℋ) ↔ ((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ* (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵))) ∧ ((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) ∩ (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵)))) = 0ℋ))) |
21 | breq2 5077 | . . . . . . 7 ⊢ (𝑦 = (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵))) → ((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ 𝑦 ↔ (𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵))))) | |
22 | 20, 21 | imbi12d 345 | . . . . . 6 ⊢ (𝑦 = (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵))) → ((((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ* 𝑦 ∧ ((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) ∩ 𝑦) = 0ℋ) → (𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ 𝑦) ↔ (((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ* (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵))) ∧ ((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) ∩ (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵)))) = 0ℋ) → (𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵)))))) |
23 | 16, 22 | rspc2v 3569 | . . . . 5 ⊢ (((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) ∈ Cℋ ∧ (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵))) ∈ Cℋ ) → (∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ ((𝑥 𝑀ℋ* 𝑦 ∧ (𝑥 ∩ 𝑦) = 0ℋ) → 𝑥 𝑀ℋ 𝑦) → (((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ* (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵))) ∧ ((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) ∩ (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵)))) = 0ℋ) → (𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵)))))) |
24 | 9, 10, 23 | mp2an 689 | . . . 4 ⊢ (∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ ((𝑥 𝑀ℋ* 𝑦 ∧ (𝑥 ∩ 𝑦) = 0ℋ) → 𝑥 𝑀ℋ 𝑦) → (((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ* (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵))) ∧ ((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) ∩ (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵)))) = 0ℋ) → (𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵))))) |
25 | 7, 24 | ax-mp 5 | . . 3 ⊢ (((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ* (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵))) ∧ ((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) ∩ (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵)))) = 0ℋ) → (𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵)))) |
26 | 6, 25 | mpan2 688 | . 2 ⊢ ((𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ* (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵))) → (𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵)))) |
27 | 2, 3 | dmdcompli 30800 | . 2 ⊢ (𝐴 𝑀ℋ* 𝐵 ↔ (𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ* (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵)))) |
28 | 2, 3 | mdcompli 30799 | . 2 ⊢ (𝐴 𝑀ℋ 𝐵 ↔ (𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵)))) |
29 | 26, 27, 28 | 3imtr4i 292 | 1 ⊢ (𝐴 𝑀ℋ* 𝐵 → 𝐴 𝑀ℋ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∩ cin 3885 class class class wbr 5073 ‘cfv 6426 Cℋ cch 29299 ⊥cort 29300 0ℋc0h 29305 𝑀ℋ cmd 29336 𝑀ℋ* cdmd 29337 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-inf2 9386 ax-cc 10201 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 ax-pre-sup 10959 ax-addf 10960 ax-mulf 10961 ax-hilex 29369 ax-hfvadd 29370 ax-hvcom 29371 ax-hvass 29372 ax-hv0cl 29373 ax-hvaddid 29374 ax-hfvmul 29375 ax-hvmulid 29376 ax-hvmulass 29377 ax-hvdistr1 29378 ax-hvdistr2 29379 ax-hvmul0 29380 ax-hfi 29449 ax-his1 29452 ax-his2 29453 ax-his3 29454 ax-his4 29455 ax-hcompl 29572 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-se 5540 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-isom 6435 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-of 7523 df-om 7703 df-1st 7820 df-2nd 7821 df-supp 7965 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-2o 8285 df-oadd 8288 df-omul 8289 df-er 8485 df-map 8604 df-pm 8605 df-ixp 8673 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-fsupp 9116 df-fi 9157 df-sup 9188 df-inf 9189 df-oi 9256 df-card 9707 df-acn 9710 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-div 11643 df-nn 11984 df-2 12046 df-3 12047 df-4 12048 df-5 12049 df-6 12050 df-7 12051 df-8 12052 df-9 12053 df-n0 12244 df-z 12330 df-dec 12448 df-uz 12593 df-q 12699 df-rp 12741 df-xneg 12858 df-xadd 12859 df-xmul 12860 df-ioo 13093 df-ico 13095 df-icc 13096 df-fz 13250 df-fzo 13393 df-fl 13522 df-seq 13732 df-exp 13793 df-hash 14055 df-cj 14820 df-re 14821 df-im 14822 df-sqrt 14956 df-abs 14957 df-clim 15207 df-rlim 15208 df-sum 15408 df-struct 16858 df-sets 16875 df-slot 16893 df-ndx 16905 df-base 16923 df-ress 16952 df-plusg 16985 df-mulr 16986 df-starv 16987 df-sca 16988 df-vsca 16989 df-ip 16990 df-tset 16991 df-ple 16992 df-ds 16994 df-unif 16995 df-hom 16996 df-cco 16997 df-rest 17143 df-topn 17144 df-0g 17162 df-gsum 17163 df-topgen 17164 df-pt 17165 df-prds 17168 df-xrs 17223 df-qtop 17228 df-imas 17229 df-xps 17231 df-mre 17305 df-mrc 17306 df-acs 17308 df-mgm 18336 df-sgrp 18385 df-mnd 18396 df-submnd 18441 df-mulg 18711 df-cntz 18933 df-cmn 19398 df-psmet 20599 df-xmet 20600 df-met 20601 df-bl 20602 df-mopn 20603 df-fbas 20604 df-fg 20605 df-cnfld 20608 df-top 22053 df-topon 22070 df-topsp 22092 df-bases 22106 df-cld 22180 df-ntr 22181 df-cls 22182 df-nei 22259 df-cn 22388 df-cnp 22389 df-lm 22390 df-haus 22476 df-tx 22723 df-hmeo 22916 df-fil 23007 df-fm 23099 df-flim 23100 df-flf 23101 df-xms 23483 df-ms 23484 df-tms 23485 df-cfil 24429 df-cau 24430 df-cmet 24431 df-grpo 28863 df-gid 28864 df-ginv 28865 df-gdiv 28866 df-ablo 28915 df-vc 28929 df-nv 28962 df-va 28965 df-ba 28966 df-sm 28967 df-0v 28968 df-vs 28969 df-nmcv 28970 df-ims 28971 df-dip 29071 df-ssp 29092 df-ph 29183 df-cbn 29233 df-hnorm 29338 df-hba 29339 df-hvsub 29341 df-hlim 29342 df-hcau 29343 df-sh 29577 df-ch 29591 df-oc 29622 df-ch0 29623 df-shs 29678 df-chj 29680 df-pjh 29765 df-cm 29953 df-md 30650 df-dmd 30651 |
This theorem is referenced by: (None) |
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