Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
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 30725 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 37003 | . . . 4 ⊢ (𝜑 → (𝑅 ⊆ 𝑄 ↔ 𝑅 = 𝑄)) |
6 | eqcom 2745 | . . . 4 ⊢ (𝑅 = 𝑄 ↔ 𝑄 = 𝑅) | |
7 | 5, 6 | bitrdi 287 | . . 3 ⊢ (𝜑 → (𝑅 ⊆ 𝑄 ↔ 𝑄 = 𝑅)) |
8 | 7 | necon3bbid 2981 | . 2 ⊢ (𝜑 → (¬ 𝑅 ⊆ 𝑄 ↔ 𝑄 ≠ 𝑅)) |
9 | lsatnem0.o | . . 3 ⊢ 0 = (0g‘𝑊) | |
10 | eqid 2738 | . . 3 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
11 | lveclmod 20356 | . . . . 5 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
12 | 2, 11 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) |
13 | 10, 1, 12, 4 | lsatlssel 36997 | . . 3 ⊢ (𝜑 → 𝑄 ∈ (LSubSp‘𝑊)) |
14 | 9, 10, 1, 2, 13, 3 | lsatnle 37044 | . 2 ⊢ (𝜑 → (¬ 𝑅 ⊆ 𝑄 ↔ (𝑄 ∩ 𝑅) = { 0 })) |
15 | 8, 14 | bitr3d 280 | 1 ⊢ (𝜑 → (𝑄 ≠ 𝑅 ↔ (𝑄 ∩ 𝑅) = { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∩ cin 3886 ⊆ wss 3887 {csn 4562 ‘cfv 6427 0gc0g 17138 LModclmod 20111 LSubSpclss 20181 LVecclvec 20352 LSAtomsclsa 36974 |
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 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-cnex 10915 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936 |
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 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-iin 4928 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7704 df-1st 7821 df-2nd 7822 df-tpos 8030 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-1o 8285 df-er 8486 df-en 8722 df-dom 8723 df-sdom 8724 df-fin 8725 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-nn 11962 df-2 12024 df-3 12025 df-sets 16853 df-slot 16871 df-ndx 16883 df-base 16901 df-ress 16930 df-plusg 16963 df-mulr 16964 df-0g 17140 df-mre 17283 df-mrc 17284 df-acs 17286 df-mgm 18314 df-sgrp 18363 df-mnd 18374 df-submnd 18419 df-grp 18568 df-minusg 18569 df-sbg 18570 df-subg 18740 df-cntz 18911 df-oppg 18938 df-lsm 19229 df-cmn 19376 df-abl 19377 df-mgp 19709 df-ur 19726 df-ring 19773 df-oppr 19850 df-dvdsr 19871 df-unit 19872 df-invr 19902 df-drng 19981 df-lmod 20113 df-lss 20182 df-lsp 20222 df-lvec 20353 df-lsatoms 36976 df-lcv 37019 |
This theorem is referenced by: lsatexch1 37046 lsatcv0eq 37047 lsatcvatlem 37049 |
Copyright terms: Public domain | W3C validator |