| Mathbox for Norm Megill |
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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 32396 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 39004 | . . . 4 ⊢ (𝜑 → (𝑅 ⊆ 𝑄 ↔ 𝑅 = 𝑄)) |
| 6 | eqcom 2744 | . . . 4 ⊢ (𝑅 = 𝑄 ↔ 𝑄 = 𝑅) | |
| 7 | 5, 6 | bitrdi 287 | . . 3 ⊢ (𝜑 → (𝑅 ⊆ 𝑄 ↔ 𝑄 = 𝑅)) |
| 8 | 7 | necon3bbid 2978 | . 2 ⊢ (𝜑 → (¬ 𝑅 ⊆ 𝑄 ↔ 𝑄 ≠ 𝑅)) |
| 9 | lsatnem0.o | . . 3 ⊢ 0 = (0g‘𝑊) | |
| 10 | eqid 2737 | . . 3 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
| 11 | lveclmod 21105 | . . . . 5 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
| 12 | 2, 11 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) |
| 13 | 10, 1, 12, 4 | lsatlssel 38998 | . . 3 ⊢ (𝜑 → 𝑄 ∈ (LSubSp‘𝑊)) |
| 14 | 9, 10, 1, 2, 13, 3 | lsatnle 39045 | . 2 ⊢ (𝜑 → (¬ 𝑅 ⊆ 𝑄 ↔ (𝑄 ∩ 𝑅) = { 0 })) |
| 15 | 8, 14 | bitr3d 281 | 1 ⊢ (𝜑 → (𝑄 ≠ 𝑅 ↔ (𝑄 ∩ 𝑅) = { 0 })) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∩ cin 3950 ⊆ wss 3951 {csn 4626 ‘cfv 6561 0gc0g 17484 LModclmod 20858 LSubSpclss 20929 LVecclvec 21101 LSAtomsclsa 38975 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-0g 17486 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cntz 19335 df-oppg 19364 df-lsm 19654 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-drng 20731 df-lmod 20860 df-lss 20930 df-lsp 20970 df-lvec 21102 df-lsatoms 38977 df-lcv 39020 |
| This theorem is referenced by: lsatexch1 39047 lsatcv0eq 39048 lsatcvatlem 39050 |
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