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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcv1 | Structured version Visualization version GIF version |
Description: Covering property of a subspace plus an atom. (chcv1 30436 analog.) (Contributed by NM, 10-Jan-2015.) |
Ref | Expression |
---|---|
lcv1.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lcv1.p | ⊢ ⊕ = (LSSum‘𝑊) |
lcv1.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lcv1.c | ⊢ 𝐶 = ( ⋖L ‘𝑊) |
lcv1.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lcv1.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lcv1.q | ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
Ref | Expression |
---|---|
lcv1 | ⊢ (𝜑 → (¬ 𝑄 ⊆ 𝑈 ↔ 𝑈𝐶(𝑈 ⊕ 𝑄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcv1.q | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐴) | |
2 | lcv1.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
3 | eqid 2737 | . . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | eqid 2737 | . . . . . . 7 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
5 | eqid 2737 | . . . . . . 7 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
6 | lcv1.a | . . . . . . 7 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
7 | 3, 4, 5, 6 | islsat 36742 | . . . . . 6 ⊢ (𝑊 ∈ LVec → (𝑄 ∈ 𝐴 ↔ ∃𝑥 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)})𝑄 = ((LSpan‘𝑊)‘{𝑥}))) |
8 | 2, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ↔ ∃𝑥 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)})𝑄 = ((LSpan‘𝑊)‘{𝑥}))) |
9 | 1, 8 | mpbid 235 | . . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)})𝑄 = ((LSpan‘𝑊)‘{𝑥})) |
10 | 9 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → ∃𝑥 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)})𝑄 = ((LSpan‘𝑊)‘{𝑥})) |
11 | lcv1.s | . . . . . 6 ⊢ 𝑆 = (LSubSp‘𝑊) | |
12 | lcv1.p | . . . . . 6 ⊢ ⊕ = (LSSum‘𝑊) | |
13 | lcv1.c | . . . . . 6 ⊢ 𝐶 = ( ⋖L ‘𝑊) | |
14 | 2 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑊 ∈ LVec) |
15 | 14 | 3ad2ant1 1135 | . . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)}) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑥})) → 𝑊 ∈ LVec) |
16 | lcv1.u | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
17 | 16 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ∈ 𝑆) |
18 | 17 | 3ad2ant1 1135 | . . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)}) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑥})) → 𝑈 ∈ 𝑆) |
19 | eldifi 4041 | . . . . . . 7 ⊢ (𝑥 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)}) → 𝑥 ∈ (Base‘𝑊)) | |
20 | 19 | 3ad2ant2 1136 | . . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)}) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑥})) → 𝑥 ∈ (Base‘𝑊)) |
21 | simp1r 1200 | . . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)}) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑥})) → ¬ 𝑄 ⊆ 𝑈) | |
22 | simp3 1140 | . . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)}) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑥})) → 𝑄 = ((LSpan‘𝑊)‘{𝑥})) | |
23 | 22 | sseq1d 3932 | . . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)}) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑥})) → (𝑄 ⊆ 𝑈 ↔ ((LSpan‘𝑊)‘{𝑥}) ⊆ 𝑈)) |
24 | 21, 23 | mtbid 327 | . . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)}) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑥})) → ¬ ((LSpan‘𝑊)‘{𝑥}) ⊆ 𝑈) |
25 | 3, 11, 4, 12, 13, 15, 18, 20, 24 | lsmcv2 36780 | . . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)}) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑥})) → 𝑈𝐶(𝑈 ⊕ ((LSpan‘𝑊)‘{𝑥}))) |
26 | 22 | oveq2d 7229 | . . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)}) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑥})) → (𝑈 ⊕ 𝑄) = (𝑈 ⊕ ((LSpan‘𝑊)‘{𝑥}))) |
27 | 25, 26 | breqtrrd 5081 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)}) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑥})) → 𝑈𝐶(𝑈 ⊕ 𝑄)) |
28 | 27 | rexlimdv3a 3205 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → (∃𝑥 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)})𝑄 = ((LSpan‘𝑊)‘{𝑥}) → 𝑈𝐶(𝑈 ⊕ 𝑄))) |
29 | 10, 28 | mpd 15 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈𝐶(𝑈 ⊕ 𝑄)) |
30 | 2 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑈𝐶(𝑈 ⊕ 𝑄)) → 𝑊 ∈ LVec) |
31 | 16 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑈𝐶(𝑈 ⊕ 𝑄)) → 𝑈 ∈ 𝑆) |
32 | lveclmod 20143 | . . . . . . 7 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
33 | 2, 32 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod) |
34 | 11, 6, 33, 1 | lsatlssel 36748 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ 𝑆) |
35 | 11, 12 | lsmcl 20120 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑄 ∈ 𝑆) → (𝑈 ⊕ 𝑄) ∈ 𝑆) |
36 | 33, 16, 34, 35 | syl3anc 1373 | . . . . 5 ⊢ (𝜑 → (𝑈 ⊕ 𝑄) ∈ 𝑆) |
37 | 36 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑈𝐶(𝑈 ⊕ 𝑄)) → (𝑈 ⊕ 𝑄) ∈ 𝑆) |
38 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ 𝑈𝐶(𝑈 ⊕ 𝑄)) → 𝑈𝐶(𝑈 ⊕ 𝑄)) | |
39 | 11, 13, 30, 31, 37, 38 | lcvpss 36775 | . . 3 ⊢ ((𝜑 ∧ 𝑈𝐶(𝑈 ⊕ 𝑄)) → 𝑈 ⊊ (𝑈 ⊕ 𝑄)) |
40 | 11 | lsssssubg 19995 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) |
41 | 33, 40 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊)) |
42 | 41, 16 | sseldd 3902 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
43 | 41, 34 | sseldd 3902 | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑊)) |
44 | 12, 42, 43 | lssnle 19064 | . . . 4 ⊢ (𝜑 → (¬ 𝑄 ⊆ 𝑈 ↔ 𝑈 ⊊ (𝑈 ⊕ 𝑄))) |
45 | 44 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑈𝐶(𝑈 ⊕ 𝑄)) → (¬ 𝑄 ⊆ 𝑈 ↔ 𝑈 ⊊ (𝑈 ⊕ 𝑄))) |
46 | 39, 45 | mpbird 260 | . 2 ⊢ ((𝜑 ∧ 𝑈𝐶(𝑈 ⊕ 𝑄)) → ¬ 𝑄 ⊆ 𝑈) |
47 | 29, 46 | impbida 801 | 1 ⊢ (𝜑 → (¬ 𝑄 ⊆ 𝑈 ↔ 𝑈𝐶(𝑈 ⊕ 𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∃wrex 3062 ∖ cdif 3863 ⊆ wss 3866 ⊊ wpss 3867 {csn 4541 class class class wbr 5053 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 0gc0g 16944 SubGrpcsubg 18537 LSSumclsm 19023 LModclmod 19899 LSubSpclss 19968 LSpanclspn 20008 LVecclvec 20139 LSAtomsclsa 36725 ⋖L clcv 36769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-tpos 7968 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-0g 16946 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-grp 18368 df-minusg 18369 df-sbg 18370 df-subg 18540 df-cntz 18711 df-lsm 19025 df-cmn 19172 df-abl 19173 df-mgp 19505 df-ur 19517 df-ring 19564 df-oppr 19641 df-dvdsr 19659 df-unit 19660 df-invr 19690 df-drng 19769 df-lmod 19901 df-lss 19969 df-lsp 20009 df-lvec 20140 df-lsatoms 36727 df-lcv 36770 |
This theorem is referenced by: lcv2 36793 lsatnle 36795 lsatcvat3 36803 |
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