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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 31284 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 20567 | . . . . 5 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) |
4 | eqid 2736 | . . . . 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 37459 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ (LSubSp‘𝑊)) |
8 | lsatcv0.o | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
9 | 8, 4 | lss0ss 20409 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑄 ∈ (LSubSp‘𝑊)) → { 0 } ⊆ 𝑄) |
10 | 3, 7, 9 | syl2anc 584 | . . 3 ⊢ (𝜑 → { 0 } ⊆ 𝑄) |
11 | 8, 5, 3, 6 | lsatn0 37461 | . . . 4 ⊢ (𝜑 → 𝑄 ≠ { 0 }) |
12 | 11 | necomd 2999 | . . 3 ⊢ (𝜑 → { 0 } ≠ 𝑄) |
13 | df-pss 3929 | . . 3 ⊢ ({ 0 } ⊊ 𝑄 ↔ ({ 0 } ⊆ 𝑄 ∧ { 0 } ≠ 𝑄)) | |
14 | 10, 12, 13 | sylanbrc 583 | . 2 ⊢ (𝜑 → { 0 } ⊊ 𝑄) |
15 | eqid 2736 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
16 | eqid 2736 | . . . . . 6 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
17 | 15, 16, 8, 5 | islsat 37453 | . . . . 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 231 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ ((Base‘𝑊) ∖ { 0 })𝑄 = ((LSpan‘𝑊)‘{𝑥})) |
20 | 1 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑊) ∖ { 0 })) → 𝑊 ∈ LVec) |
21 | eldifi 4086 | . . . . . . . 8 ⊢ (𝑥 ∈ ((Base‘𝑊) ∖ { 0 }) → 𝑥 ∈ (Base‘𝑊)) | |
22 | 21 | adantl 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑊) ∖ { 0 })) → 𝑥 ∈ (Base‘𝑊)) |
23 | 15, 8, 4, 16, 20, 22 | lspsncv0 20607 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑊) ∖ { 0 })) → ¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥}))) |
24 | 23 | ex 413 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ ((Base‘𝑊) ∖ { 0 }) → ¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥})))) |
25 | psseq2 4048 | . . . . . . . . 9 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑥}) → (𝑠 ⊊ 𝑄 ↔ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥}))) | |
26 | 25 | anbi2d 629 | . . . . . . . 8 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑥}) → (({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄) ↔ ({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥})))) |
27 | 26 | rexbidv 3175 | . . . . . . 7 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑥}) → (∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄) ↔ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥})))) |
28 | 27 | notbid 317 | . . . . . 6 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑥}) → (¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄) ↔ ¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥})))) |
29 | 28 | biimprcd 249 | . . . . 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 3150 | . . 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 20408 | . . . 4 ⊢ (𝑊 ∈ LMod → { 0 } ∈ (LSubSp‘𝑊)) |
35 | 3, 34 | syl 17 | . . 3 ⊢ (𝜑 → { 0 } ∈ (LSubSp‘𝑊)) |
36 | 4, 33, 1, 35, 7 | lcvbr 37483 | . 2 ⊢ (𝜑 → ({ 0 }𝐶𝑄 ↔ ({ 0 } ⊊ 𝑄 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄)))) |
37 | 14, 32, 36 | mpbir2and 711 | 1 ⊢ (𝜑 → { 0 }𝐶𝑄) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3073 ∖ cdif 3907 ⊆ wss 3910 ⊊ wpss 3911 {csn 4586 class class class wbr 5105 ‘cfv 6496 Basecbs 17083 0gc0g 17321 LModclmod 20322 LSubSpclss 20392 LSpanclspn 20432 LVecclvec 20563 LSAtomsclsa 37436 ⋖L clcv 37480 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-tpos 8157 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-0g 17323 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-grp 18751 df-minusg 18752 df-sbg 18753 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-drng 20187 df-lmod 20324 df-lss 20393 df-lsp 20433 df-lvec 20564 df-lsatoms 37438 df-lcv 37481 |
This theorem is referenced by: lsatcveq0 37494 lsat0cv 37495 lsatcv0eq 37509 mapdcnvatN 40129 mapdat 40130 |
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