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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 28585 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 18927 | . . . . 5 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) |
4 | eqid 2610 | . . . . 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 33302 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ (LSubSp‘𝑊)) |
8 | lsatcv0.o | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
9 | 8, 4 | lss0ss 18770 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑄 ∈ (LSubSp‘𝑊)) → { 0 } ⊆ 𝑄) |
10 | 3, 7, 9 | syl2anc 691 | . . 3 ⊢ (𝜑 → { 0 } ⊆ 𝑄) |
11 | 8, 5, 3, 6 | lsatn0 33304 | . . . 4 ⊢ (𝜑 → 𝑄 ≠ { 0 }) |
12 | 11 | necomd 2837 | . . 3 ⊢ (𝜑 → { 0 } ≠ 𝑄) |
13 | df-pss 3556 | . . 3 ⊢ ({ 0 } ⊊ 𝑄 ↔ ({ 0 } ⊆ 𝑄 ∧ { 0 } ≠ 𝑄)) | |
14 | 10, 12, 13 | sylanbrc 695 | . 2 ⊢ (𝜑 → { 0 } ⊊ 𝑄) |
15 | eqid 2610 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
16 | eqid 2610 | . . . . . 6 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
17 | 15, 16, 8, 5 | islsat 33296 | . . . . 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 221 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ ((Base‘𝑊) ∖ { 0 })𝑄 = ((LSpan‘𝑊)‘{𝑥})) |
20 | 1 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑊) ∖ { 0 })) → 𝑊 ∈ LVec) |
21 | eldifi 3694 | . . . . . . . 8 ⊢ (𝑥 ∈ ((Base‘𝑊) ∖ { 0 }) → 𝑥 ∈ (Base‘𝑊)) | |
22 | 21 | adantl 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑊) ∖ { 0 })) → 𝑥 ∈ (Base‘𝑊)) |
23 | eldifsni 4261 | . . . . . . . 8 ⊢ (𝑥 ∈ ((Base‘𝑊) ∖ { 0 }) → 𝑥 ≠ 0 ) | |
24 | 23 | adantl 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑊) ∖ { 0 })) → 𝑥 ≠ 0 ) |
25 | 15, 8, 4, 16, 20, 22, 24 | lspsncv0 18967 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑊) ∖ { 0 })) → ¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥}))) |
26 | 25 | ex 449 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ ((Base‘𝑊) ∖ { 0 }) → ¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥})))) |
27 | psseq2 3657 | . . . . . . . . 9 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑥}) → (𝑠 ⊊ 𝑄 ↔ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥}))) | |
28 | 27 | anbi2d 736 | . . . . . . . 8 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑥}) → (({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄) ↔ ({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥})))) |
29 | 28 | rexbidv 3034 | . . . . . . 7 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑥}) → (∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄) ↔ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥})))) |
30 | 29 | notbid 307 | . . . . . 6 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑥}) → (¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄) ↔ ¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥})))) |
31 | 30 | biimprcd 239 | . . . . 5 ⊢ (¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ ((LSpan‘𝑊)‘{𝑥})) → (𝑄 = ((LSpan‘𝑊)‘{𝑥}) → ¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄))) |
32 | 26, 31 | syl6 34 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ((Base‘𝑊) ∖ { 0 }) → (𝑄 = ((LSpan‘𝑊)‘{𝑥}) → ¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄)))) |
33 | 32 | rexlimdv 3012 | . . 3 ⊢ (𝜑 → (∃𝑥 ∈ ((Base‘𝑊) ∖ { 0 })𝑄 = ((LSpan‘𝑊)‘{𝑥}) → ¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄))) |
34 | 19, 33 | mpd 15 | . 2 ⊢ (𝜑 → ¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄)) |
35 | lsatcv0.c | . . 3 ⊢ 𝐶 = ( ⋖L ‘𝑊) | |
36 | 8, 4 | lsssn0 18769 | . . . 4 ⊢ (𝑊 ∈ LMod → { 0 } ∈ (LSubSp‘𝑊)) |
37 | 3, 36 | syl 17 | . . 3 ⊢ (𝜑 → { 0 } ∈ (LSubSp‘𝑊)) |
38 | 4, 35, 1, 37, 7 | lcvbr 33326 | . 2 ⊢ (𝜑 → ({ 0 }𝐶𝑄 ↔ ({ 0 } ⊊ 𝑄 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑊)({ 0 } ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑄)))) |
39 | 14, 34, 38 | mpbir2and 959 | 1 ⊢ (𝜑 → { 0 }𝐶𝑄) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 ∖ cdif 3537 ⊆ wss 3540 ⊊ wpss 3541 {csn 4125 class class class wbr 4583 ‘cfv 5804 Basecbs 15695 0gc0g 15923 LModclmod 18686 LSubSpclss 18753 LSpanclspn 18792 LVecclvec 18923 LSAtomsclsa 33279 ⋖L clcv 33323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-sbg 17250 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-drng 18572 df-lmod 18688 df-lss 18754 df-lsp 18793 df-lvec 18924 df-lsatoms 33281 df-lcv 33324 |
This theorem is referenced by: lsatcveq0 33337 lsat0cv 33338 lsatcv0eq 33352 mapdcnvatN 35973 mapdat 35974 |
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