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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihmeetlem14N | Structured version Visualization version GIF version |
Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dihmeetlem14.b | ⊢ 𝐵 = (Base‘𝐾) |
dihmeetlem14.l | ⊢ ≤ = (le‘𝐾) |
dihmeetlem14.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihmeetlem14.j | ⊢ ∨ = (join‘𝐾) |
dihmeetlem14.m | ⊢ ∧ = (meet‘𝐾) |
dihmeetlem14.a | ⊢ 𝐴 = (Atoms‘𝐾) |
dihmeetlem14.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dihmeetlem14.s | ⊢ ⊕ = (LSSum‘𝑈) |
dihmeetlem14.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dihmeetlem14N | ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑟 ≤ 𝑌 ∧ (𝑌 ∧ 𝑝) ≤ 𝑊)) → ((𝐼‘(𝑌 ∧ 𝑝)) ⊕ ((𝐼‘𝑟) ∩ (𝐼‘𝑝))) = ((𝐼‘𝑌) ∩ (𝐼‘𝑝))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dihmeetlem14.b | . 2 ⊢ 𝐵 = (Base‘𝐾) | |
2 | dihmeetlem14.l | . 2 ⊢ ≤ = (le‘𝐾) | |
3 | dihmeetlem14.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | dihmeetlem14.j | . 2 ⊢ ∨ = (join‘𝐾) | |
5 | dihmeetlem14.m | . 2 ⊢ ∧ = (meet‘𝐾) | |
6 | dihmeetlem14.a | . 2 ⊢ 𝐴 = (Atoms‘𝐾) | |
7 | dihmeetlem14.u | . 2 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
8 | dihmeetlem14.s | . 2 ⊢ ⊕ = (LSSum‘𝑈) | |
9 | dihmeetlem14.i | . 2 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | dihmeetlem12N 41030 | 1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑟 ≤ 𝑌 ∧ (𝑌 ∧ 𝑝) ≤ 𝑊)) → ((𝐼‘(𝑌 ∧ 𝑝)) ⊕ ((𝐼‘𝑟) ∩ (𝐼‘𝑝))) = ((𝐼‘𝑌) ∩ (𝐼‘𝑝))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ∩ cin 3945 class class class wbr 5145 ‘cfv 6546 (class class class)co 7416 Basecbs 17208 lecple 17268 joincjn 18331 meetcmee 18332 LSSumclsm 19628 Atomscatm 38974 HLchlt 39061 LHypclh 39696 DVecHcdvh 40790 DIsoHcdih 40940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-riotaBAD 38664 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-iin 4996 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-tpos 8233 df-undef 8280 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8726 df-map 8849 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-n0 12519 df-z 12605 df-uz 12869 df-fz 13533 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-sca 17277 df-vsca 17278 df-0g 17451 df-mre 17594 df-mrc 17595 df-acs 17597 df-proset 18315 df-poset 18333 df-plt 18350 df-lub 18366 df-glb 18367 df-join 18368 df-meet 18369 df-p0 18445 df-p1 18446 df-lat 18452 df-clat 18519 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-submnd 18769 df-grp 18926 df-minusg 18927 df-sbg 18928 df-subg 19113 df-cntz 19307 df-lsm 19630 df-cmn 19776 df-abl 19777 df-mgp 20114 df-rng 20132 df-ur 20161 df-ring 20214 df-oppr 20312 df-dvdsr 20335 df-unit 20336 df-invr 20366 df-dvr 20379 df-drng 20705 df-lmod 20834 df-lss 20905 df-lsp 20945 df-lvec 21077 df-oposet 38887 df-ol 38889 df-oml 38890 df-covers 38977 df-ats 38978 df-atl 39009 df-cvlat 39033 df-hlat 39062 df-llines 39210 df-lplanes 39211 df-lvols 39212 df-lines 39213 df-psubsp 39215 df-pmap 39216 df-padd 39508 df-lhyp 39700 df-laut 39701 df-ldil 39816 df-ltrn 39817 df-trl 39871 df-tendo 40467 df-edring 40469 df-disoa 40741 df-dvech 40791 df-dib 40851 df-dic 40885 df-dih 40941 |
This theorem is referenced by: dihmeetlem16N 41034 |
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