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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatnem0 | Structured version Visualization version GIF version |
Description: The meet of distinct atoms is the zero subspace. (atnemeq0 29841 analog.) (Contributed by NM, 10-Jan-2015.) |
Ref | Expression |
---|---|
lsatnem0.o | ⊢ 0 = (0g‘𝑊) |
lsatnem0.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lsatnem0.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lsatnem0.q | ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
lsatnem0.r | ⊢ (𝜑 → 𝑅 ∈ 𝐴) |
Ref | Expression |
---|---|
lsatnem0 | ⊢ (𝜑 → (𝑄 ≠ 𝑅 ↔ (𝑄 ∩ 𝑅) = { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsatnem0.a | . . . . 5 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
2 | lsatnem0.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
3 | lsatnem0.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝐴) | |
4 | lsatnem0.q | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐴) | |
5 | 1, 2, 3, 4 | lsatcmp 35691 | . . . 4 ⊢ (𝜑 → (𝑅 ⊆ 𝑄 ↔ 𝑅 = 𝑄)) |
6 | eqcom 2804 | . . . 4 ⊢ (𝑅 = 𝑄 ↔ 𝑄 = 𝑅) | |
7 | 5, 6 | syl6bb 288 | . . 3 ⊢ (𝜑 → (𝑅 ⊆ 𝑄 ↔ 𝑄 = 𝑅)) |
8 | 7 | necon3bbid 3023 | . 2 ⊢ (𝜑 → (¬ 𝑅 ⊆ 𝑄 ↔ 𝑄 ≠ 𝑅)) |
9 | lsatnem0.o | . . 3 ⊢ 0 = (0g‘𝑊) | |
10 | eqid 2797 | . . 3 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
11 | lveclmod 19572 | . . . . 5 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
12 | 2, 11 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) |
13 | 10, 1, 12, 4 | lsatlssel 35685 | . . 3 ⊢ (𝜑 → 𝑄 ∈ (LSubSp‘𝑊)) |
14 | 9, 10, 1, 2, 13, 3 | lsatnle 35732 | . 2 ⊢ (𝜑 → (¬ 𝑅 ⊆ 𝑄 ↔ (𝑄 ∩ 𝑅) = { 0 })) |
15 | 8, 14 | bitr3d 282 | 1 ⊢ (𝜑 → (𝑄 ≠ 𝑅 ↔ (𝑄 ∩ 𝑅) = { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 = wceq 1525 ∈ wcel 2083 ≠ wne 2986 ∩ cin 3864 ⊆ wss 3865 {csn 4478 ‘cfv 6232 0gc0g 16546 LModclmod 19328 LSubSpclss 19397 LVecclvec 19568 LSAtomsclsa 35662 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-iin 4834 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-tpos 7750 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-oadd 7964 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-nn 11493 df-2 11554 df-3 11555 df-ndx 16319 df-slot 16320 df-base 16322 df-sets 16323 df-ress 16324 df-plusg 16411 df-mulr 16412 df-0g 16548 df-mre 16690 df-mrc 16691 df-acs 16693 df-mgm 17685 df-sgrp 17727 df-mnd 17738 df-submnd 17779 df-grp 17868 df-minusg 17869 df-sbg 17870 df-subg 18034 df-cntz 18192 df-oppg 18219 df-lsm 18495 df-cmn 18639 df-abl 18640 df-mgp 18934 df-ur 18946 df-ring 18993 df-oppr 19067 df-dvdsr 19085 df-unit 19086 df-invr 19116 df-drng 19198 df-lmod 19330 df-lss 19398 df-lsp 19438 df-lvec 19569 df-lsatoms 35664 df-lcv 35707 |
This theorem is referenced by: lsatexch1 35734 lsatcv0eq 35735 lsatcvatlem 35737 |
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