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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatcveq0 | Structured version Visualization version GIF version |
Description: A subspace covered by an atom must be the zero subspace. (atcveq0 29207 analog.) (Contributed by NM, 7-Jan-2015.) |
Ref | Expression |
---|---|
lsatcveq0.o | ⊢ 0 = (0g‘𝑊) |
lsatcveq0.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lsatcveq0.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lsatcveq0.c | ⊢ 𝐶 = ( ⋖L ‘𝑊) |
lsatcveq0.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lsatcveq0.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lsatcveq0.q | ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
Ref | Expression |
---|---|
lsatcveq0 | ⊢ (𝜑 → (𝑈𝐶𝑄 ↔ 𝑈 = { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsatcveq0.s | . . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊) | |
2 | lsatcveq0.c | . . . . 5 ⊢ 𝐶 = ( ⋖L ‘𝑊) | |
3 | lsatcveq0.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
4 | 3 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑈𝐶𝑄) → 𝑊 ∈ LVec) |
5 | lsatcveq0.u | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
6 | 5 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑈𝐶𝑄) → 𝑈 ∈ 𝑆) |
7 | lsatcveq0.a | . . . . . . 7 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
8 | lveclmod 19106 | . . . . . . . 8 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
9 | 3, 8 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LMod) |
10 | lsatcveq0.q | . . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ 𝐴) | |
11 | 1, 7, 9, 10 | lsatlssel 34284 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ 𝑆) |
12 | 11 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑈𝐶𝑄) → 𝑄 ∈ 𝑆) |
13 | simpr 477 | . . . . 5 ⊢ ((𝜑 ∧ 𝑈𝐶𝑄) → 𝑈𝐶𝑄) | |
14 | 1, 2, 4, 6, 12, 13 | lcvpss 34311 | . . . 4 ⊢ ((𝜑 ∧ 𝑈𝐶𝑄) → 𝑈 ⊊ 𝑄) |
15 | 14 | ex 450 | . . 3 ⊢ (𝜑 → (𝑈𝐶𝑄 → 𝑈 ⊊ 𝑄)) |
16 | lsatcveq0.o | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
17 | 16, 7, 2, 3, 10 | lsatcv0 34318 | . . . 4 ⊢ (𝜑 → { 0 }𝐶𝑄) |
18 | 3 | 3ad2ant1 1082 | . . . . . 6 ⊢ ((𝜑 ∧ { 0 }𝐶𝑄 ∧ 𝑈 ⊊ 𝑄) → 𝑊 ∈ LVec) |
19 | 16, 1 | lsssn0 18948 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → { 0 } ∈ 𝑆) |
20 | 9, 19 | syl 17 | . . . . . . 7 ⊢ (𝜑 → { 0 } ∈ 𝑆) |
21 | 20 | 3ad2ant1 1082 | . . . . . 6 ⊢ ((𝜑 ∧ { 0 }𝐶𝑄 ∧ 𝑈 ⊊ 𝑄) → { 0 } ∈ 𝑆) |
22 | 11 | 3ad2ant1 1082 | . . . . . 6 ⊢ ((𝜑 ∧ { 0 }𝐶𝑄 ∧ 𝑈 ⊊ 𝑄) → 𝑄 ∈ 𝑆) |
23 | 5 | 3ad2ant1 1082 | . . . . . 6 ⊢ ((𝜑 ∧ { 0 }𝐶𝑄 ∧ 𝑈 ⊊ 𝑄) → 𝑈 ∈ 𝑆) |
24 | simp2 1062 | . . . . . 6 ⊢ ((𝜑 ∧ { 0 }𝐶𝑄 ∧ 𝑈 ⊊ 𝑄) → { 0 }𝐶𝑄) | |
25 | 16, 1 | lss0ss 18949 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → { 0 } ⊆ 𝑈) |
26 | 9, 5, 25 | syl2anc 693 | . . . . . . 7 ⊢ (𝜑 → { 0 } ⊆ 𝑈) |
27 | 26 | 3ad2ant1 1082 | . . . . . 6 ⊢ ((𝜑 ∧ { 0 }𝐶𝑄 ∧ 𝑈 ⊊ 𝑄) → { 0 } ⊆ 𝑈) |
28 | simp3 1063 | . . . . . 6 ⊢ ((𝜑 ∧ { 0 }𝐶𝑄 ∧ 𝑈 ⊊ 𝑄) → 𝑈 ⊊ 𝑄) | |
29 | 1, 2, 18, 21, 22, 23, 24, 27, 28 | lcvnbtwn3 34315 | . . . . 5 ⊢ ((𝜑 ∧ { 0 }𝐶𝑄 ∧ 𝑈 ⊊ 𝑄) → 𝑈 = { 0 }) |
30 | 29 | 3exp 1264 | . . . 4 ⊢ (𝜑 → ({ 0 }𝐶𝑄 → (𝑈 ⊊ 𝑄 → 𝑈 = { 0 }))) |
31 | 17, 30 | mpd 15 | . . 3 ⊢ (𝜑 → (𝑈 ⊊ 𝑄 → 𝑈 = { 0 })) |
32 | 15, 31 | syld 47 | . 2 ⊢ (𝜑 → (𝑈𝐶𝑄 → 𝑈 = { 0 })) |
33 | breq1 4656 | . . 3 ⊢ (𝑈 = { 0 } → (𝑈𝐶𝑄 ↔ { 0 }𝐶𝑄)) | |
34 | 17, 33 | syl5ibrcom 237 | . 2 ⊢ (𝜑 → (𝑈 = { 0 } → 𝑈𝐶𝑄)) |
35 | 32, 34 | impbid 202 | 1 ⊢ (𝜑 → (𝑈𝐶𝑄 ↔ 𝑈 = { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 ⊊ wpss 3575 {csn 4177 class class class wbr 4653 ‘cfv 5888 0gc0g 16100 LModclmod 18863 LSubSpclss 18932 LVecclvec 19102 LSAtomsclsa 34261 ⋖L clcv 34305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-drng 18749 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lvec 19103 df-lsatoms 34263 df-lcv 34306 |
This theorem is referenced by: lcvp 34327 lsatcv1 34335 |
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