![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatcv0 | Structured version Visualization version GIF version |
Description: An atom covers the zero subspace. (atcv0 32224 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 21003 | . . . . 5 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) |
4 | eqid 2725 | . . . . 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 38599 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ (LSubSp‘𝑊)) |
8 | lsatcv0.o | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
9 | 8, 4 | lss0ss 20845 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑄 ∈ (LSubSp‘𝑊)) → { 0 } ⊆ 𝑄) |
10 | 3, 7, 9 | syl2anc 582 | . . 3 ⊢ (𝜑 → { 0 } ⊆ 𝑄) |
11 | 8, 5, 3, 6 | lsatn0 38601 | . . . 4 ⊢ (𝜑 → 𝑄 ≠ { 0 }) |
12 | 11 | necomd 2985 | . . 3 ⊢ (𝜑 → { 0 } ≠ 𝑄) |
13 | df-pss 3964 | . . 3 ⊢ ({ 0 } ⊊ 𝑄 ↔ ({ 0 } ⊆ 𝑄 ∧ { 0 } ≠ 𝑄)) | |
14 | 10, 12, 13 | sylanbrc 581 | . 2 ⊢ (𝜑 → { 0 } ⊊ 𝑄) |
15 | eqid 2725 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
16 | eqid 2725 | . . . . . 6 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
17 | 15, 16, 8, 5 | islsat 38593 | . . . . 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 479 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑊) ∖ { 0 })) → 𝑊 ∈ LVec) |
21 | eldifi 4123 | . . . . . . . 8 ⊢ (𝑥 ∈ ((Base‘𝑊) ∖ { 0 }) → 𝑥 ∈ (Base‘𝑊)) | |
22 | 21 | adantl 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑊) ∖ { 0 })) → 𝑥 ∈ (Base‘𝑊)) |
23 | 15, 8, 4, 16, 20, 22 | lspsncv0 21046 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑊) ∖ { 0 })) → ¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥}))) |
24 | 23 | ex 411 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ ((Base‘𝑊) ∖ { 0 }) → ¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥})))) |
25 | psseq2 4084 | . . . . . . . . 9 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑥}) → (𝑠 ⊊ 𝑄 ↔ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥}))) | |
26 | 25 | anbi2d 628 | . . . . . . . 8 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑥}) → (({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄) ↔ ({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥})))) |
27 | 26 | rexbidv 3168 | . . . . . . 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 3142 | . . 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 20844 | . . . 4 ⊢ (𝑊 ∈ LMod → { 0 } ∈ (LSubSp‘𝑊)) |
35 | 3, 34 | syl 17 | . . 3 ⊢ (𝜑 → { 0 } ∈ (LSubSp‘𝑊)) |
36 | 4, 33, 1, 35, 7 | lcvbr 38623 | . 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 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∃wrex 3059 ∖ cdif 3941 ⊆ wss 3944 ⊊ wpss 3945 {csn 4630 class class class wbr 5149 ‘cfv 6549 Basecbs 17183 0gc0g 17424 LModclmod 20755 LSubSpclss 20827 LSpanclspn 20867 LVecclvec 20999 LSAtomsclsa 38576 ⋖L clcv 38620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-0g 17426 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18901 df-minusg 18902 df-sbg 18903 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-oppr 20285 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-drng 20638 df-lmod 20757 df-lss 20828 df-lsp 20868 df-lvec 21000 df-lsatoms 38578 df-lcv 38621 |
This theorem is referenced by: lsatcveq0 38634 lsat0cv 38635 lsatcv0eq 38649 mapdcnvatN 41269 mapdat 41270 |
Copyright terms: Public domain | W3C validator |