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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatcv0 | Structured version Visualization version GIF version |
Description: An atom covers the zero subspace. (atcv0 29756 analog.) (Contributed by NM, 7-Jan-2015.) |
Ref | Expression |
---|---|
lsatcv0.o | ⊢ 0 = (0g‘𝑊) |
lsatcv0.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lsatcv0.c | ⊢ 𝐶 = ( ⋖L ‘𝑊) |
lsatcv0.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lsatcv0.q | ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
Ref | Expression |
---|---|
lsatcv0 | ⊢ (𝜑 → { 0 }𝐶𝑄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsatcv0.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
2 | lveclmod 19465 | . . . . 5 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) |
4 | eqid 2825 | . . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
5 | lsatcv0.a | . . . . 5 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
6 | lsatcv0.q | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐴) | |
7 | 4, 5, 3, 6 | lsatlssel 35072 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ (LSubSp‘𝑊)) |
8 | lsatcv0.o | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
9 | 8, 4 | lss0ss 19305 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑄 ∈ (LSubSp‘𝑊)) → { 0 } ⊆ 𝑄) |
10 | 3, 7, 9 | syl2anc 581 | . . 3 ⊢ (𝜑 → { 0 } ⊆ 𝑄) |
11 | 8, 5, 3, 6 | lsatn0 35074 | . . . 4 ⊢ (𝜑 → 𝑄 ≠ { 0 }) |
12 | 11 | necomd 3054 | . . 3 ⊢ (𝜑 → { 0 } ≠ 𝑄) |
13 | df-pss 3814 | . . 3 ⊢ ({ 0 } ⊊ 𝑄 ↔ ({ 0 } ⊆ 𝑄 ∧ { 0 } ≠ 𝑄)) | |
14 | 10, 12, 13 | sylanbrc 580 | . 2 ⊢ (𝜑 → { 0 } ⊊ 𝑄) |
15 | eqid 2825 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
16 | eqid 2825 | . . . . . 6 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
17 | 15, 16, 8, 5 | islsat 35066 | . . . . 5 ⊢ (𝑊 ∈ LMod → (𝑄 ∈ 𝐴 ↔ ∃𝑥 ∈ ((Base‘𝑊) ∖ { 0 })𝑄 = ((LSpan‘𝑊)‘{𝑥}))) |
18 | 3, 17 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ↔ ∃𝑥 ∈ ((Base‘𝑊) ∖ { 0 })𝑄 = ((LSpan‘𝑊)‘{𝑥}))) |
19 | 6, 18 | mpbid 224 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ ((Base‘𝑊) ∖ { 0 })𝑄 = ((LSpan‘𝑊)‘{𝑥})) |
20 | 1 | adantr 474 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑊) ∖ { 0 })) → 𝑊 ∈ LVec) |
21 | eldifi 3959 | . . . . . . . 8 ⊢ (𝑥 ∈ ((Base‘𝑊) ∖ { 0 }) → 𝑥 ∈ (Base‘𝑊)) | |
22 | 21 | adantl 475 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑊) ∖ { 0 })) → 𝑥 ∈ (Base‘𝑊)) |
23 | 15, 8, 4, 16, 20, 22 | lspsncv0 19506 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑊) ∖ { 0 })) → ¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥}))) |
24 | 23 | ex 403 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ ((Base‘𝑊) ∖ { 0 }) → ¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥})))) |
25 | psseq2 3921 | . . . . . . . . 9 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑥}) → (𝑠 ⊊ 𝑄 ↔ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥}))) | |
26 | 25 | anbi2d 624 | . . . . . . . 8 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑥}) → (({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄) ↔ ({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥})))) |
27 | 26 | rexbidv 3262 | . . . . . . 7 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑥}) → (∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄) ↔ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥})))) |
28 | 27 | notbid 310 | . . . . . 6 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑥}) → (¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄) ↔ ¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥})))) |
29 | 28 | biimprcd 242 | . . . . 5 ⊢ (¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥})) → (𝑄 = ((LSpan‘𝑊)‘{𝑥}) → ¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄))) |
30 | 24, 29 | syl6 35 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ((Base‘𝑊) ∖ { 0 }) → (𝑄 = ((LSpan‘𝑊)‘{𝑥}) → ¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄)))) |
31 | 30 | rexlimdv 3239 | . . 3 ⊢ (𝜑 → (∃𝑥 ∈ ((Base‘𝑊) ∖ { 0 })𝑄 = ((LSpan‘𝑊)‘{𝑥}) → ¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄))) |
32 | 19, 31 | mpd 15 | . 2 ⊢ (𝜑 → ¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄)) |
33 | lsatcv0.c | . . 3 ⊢ 𝐶 = ( ⋖L ‘𝑊) | |
34 | 8, 4 | lsssn0 19304 | . . . 4 ⊢ (𝑊 ∈ LMod → { 0 } ∈ (LSubSp‘𝑊)) |
35 | 3, 34 | syl 17 | . . 3 ⊢ (𝜑 → { 0 } ∈ (LSubSp‘𝑊)) |
36 | 4, 33, 1, 35, 7 | lcvbr 35096 | . 2 ⊢ (𝜑 → ({ 0 }𝐶𝑄 ↔ ({ 0 } ⊊ 𝑄 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄)))) |
37 | 14, 32, 36 | mpbir2and 706 | 1 ⊢ (𝜑 → { 0 }𝐶𝑄) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ≠ wne 2999 ∃wrex 3118 ∖ cdif 3795 ⊆ wss 3798 ⊊ wpss 3799 {csn 4397 class class class wbr 4873 ‘cfv 6123 Basecbs 16222 0gc0g 16453 LModclmod 19219 LSubSpclss 19288 LSpanclspn 19330 LVecclvec 19461 LSAtomsclsa 35049 ⋖L clcv 35093 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-tpos 7617 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-0g 16455 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-grp 17779 df-minusg 17780 df-sbg 17781 df-cmn 18548 df-abl 18549 df-mgp 18844 df-ur 18856 df-ring 18903 df-oppr 18977 df-dvdsr 18995 df-unit 18996 df-invr 19026 df-drng 19105 df-lmod 19221 df-lss 19289 df-lsp 19331 df-lvec 19462 df-lsatoms 35051 df-lcv 35094 |
This theorem is referenced by: lsatcveq0 35107 lsat0cv 35108 lsatcv0eq 35122 mapdcnvatN 37741 mapdat 37742 |
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