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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 30125 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 19871 | . . . . 5 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) |
4 | eqid 2798 | . . . . 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 36293 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ (LSubSp‘𝑊)) |
8 | lsatcv0.o | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
9 | 8, 4 | lss0ss 19713 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑄 ∈ (LSubSp‘𝑊)) → { 0 } ⊆ 𝑄) |
10 | 3, 7, 9 | syl2anc 587 | . . 3 ⊢ (𝜑 → { 0 } ⊆ 𝑄) |
11 | 8, 5, 3, 6 | lsatn0 36295 | . . . 4 ⊢ (𝜑 → 𝑄 ≠ { 0 }) |
12 | 11 | necomd 3042 | . . 3 ⊢ (𝜑 → { 0 } ≠ 𝑄) |
13 | df-pss 3900 | . . 3 ⊢ ({ 0 } ⊊ 𝑄 ↔ ({ 0 } ⊆ 𝑄 ∧ { 0 } ≠ 𝑄)) | |
14 | 10, 12, 13 | sylanbrc 586 | . 2 ⊢ (𝜑 → { 0 } ⊊ 𝑄) |
15 | eqid 2798 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
16 | eqid 2798 | . . . . . 6 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
17 | 15, 16, 8, 5 | islsat 36287 | . . . . 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 235 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ ((Base‘𝑊) ∖ { 0 })𝑄 = ((LSpan‘𝑊)‘{𝑥})) |
20 | 1 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑊) ∖ { 0 })) → 𝑊 ∈ LVec) |
21 | eldifi 4054 | . . . . . . . 8 ⊢ (𝑥 ∈ ((Base‘𝑊) ∖ { 0 }) → 𝑥 ∈ (Base‘𝑊)) | |
22 | 21 | adantl 485 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑊) ∖ { 0 })) → 𝑥 ∈ (Base‘𝑊)) |
23 | 15, 8, 4, 16, 20, 22 | lspsncv0 19911 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑊) ∖ { 0 })) → ¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥}))) |
24 | 23 | ex 416 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ ((Base‘𝑊) ∖ { 0 }) → ¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥})))) |
25 | psseq2 4016 | . . . . . . . . 9 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑥}) → (𝑠 ⊊ 𝑄 ↔ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥}))) | |
26 | 25 | anbi2d 631 | . . . . . . . 8 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑥}) → (({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄) ↔ ({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥})))) |
27 | 26 | rexbidv 3256 | . . . . . . 7 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑥}) → (∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄) ↔ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥})))) |
28 | 27 | notbid 321 | . . . . . 6 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑥}) → (¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄) ↔ ¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥})))) |
29 | 28 | biimprcd 253 | . . . . 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 3242 | . . 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 19712 | . . . 4 ⊢ (𝑊 ∈ LMod → { 0 } ∈ (LSubSp‘𝑊)) |
35 | 3, 34 | syl 17 | . . 3 ⊢ (𝜑 → { 0 } ∈ (LSubSp‘𝑊)) |
36 | 4, 33, 1, 35, 7 | lcvbr 36317 | . 2 ⊢ (𝜑 → ({ 0 }𝐶𝑄 ↔ ({ 0 } ⊊ 𝑄 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄)))) |
37 | 14, 32, 36 | mpbir2and 712 | 1 ⊢ (𝜑 → { 0 }𝐶𝑄) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∃wrex 3107 ∖ cdif 3878 ⊆ wss 3881 ⊊ wpss 3882 {csn 4525 class class class wbr 5030 ‘cfv 6324 Basecbs 16475 0gc0g 16705 LModclmod 19627 LSubSpclss 19696 LSpanclspn 19736 LVecclvec 19867 LSAtomsclsa 36270 ⋖L clcv 36314 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-sbg 18100 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-drng 19497 df-lmod 19629 df-lss 19697 df-lsp 19737 df-lvec 19868 df-lsatoms 36272 df-lcv 36315 |
This theorem is referenced by: lsatcveq0 36328 lsat0cv 36329 lsatcv0eq 36343 mapdcnvatN 38962 mapdat 38963 |
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