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Mathbox for Norm Megill |
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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 32383 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 2734 | . . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | eqid 2734 | . . . . . . 7 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
5 | eqid 2734 | . . . . . . 7 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
6 | lcv1.a | . . . . . . 7 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
7 | 3, 4, 5, 6 | islsat 38972 | . . . . . 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 232 | . . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)})𝑄 = ((LSpan‘𝑊)‘{𝑥})) |
10 | 9 | adantr 480 | . . 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 480 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑊 ∈ LVec) |
15 | 14 | 3ad2ant1 1132 | . . . . . 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 480 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ∈ 𝑆) |
18 | 17 | 3ad2ant1 1132 | . . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)}) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑥})) → 𝑈 ∈ 𝑆) |
19 | eldifi 4140 | . . . . . . 7 ⊢ (𝑥 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)}) → 𝑥 ∈ (Base‘𝑊)) | |
20 | 19 | 3ad2ant2 1133 | . . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)}) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑥})) → 𝑥 ∈ (Base‘𝑊)) |
21 | simp1r 1197 | . . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)}) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑥})) → ¬ 𝑄 ⊆ 𝑈) | |
22 | simp3 1137 | . . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)}) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑥})) → 𝑄 = ((LSpan‘𝑊)‘{𝑥})) | |
23 | 22 | sseq1d 4026 | . . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)}) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑥})) → (𝑄 ⊆ 𝑈 ↔ ((LSpan‘𝑊)‘{𝑥}) ⊆ 𝑈)) |
24 | 21, 23 | mtbid 324 | . . . . . 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 39010 | . . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)}) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑥})) → 𝑈𝐶(𝑈 ⊕ ((LSpan‘𝑊)‘{𝑥}))) |
26 | 22 | oveq2d 7446 | . . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)}) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑥})) → (𝑈 ⊕ 𝑄) = (𝑈 ⊕ ((LSpan‘𝑊)‘{𝑥}))) |
27 | 25, 26 | breqtrrd 5175 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) ∧ 𝑥 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)}) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑥})) → 𝑈𝐶(𝑈 ⊕ 𝑄)) |
28 | 27 | rexlimdv3a 3156 | . . 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 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑈𝐶(𝑈 ⊕ 𝑄)) → 𝑊 ∈ LVec) |
31 | 16 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑈𝐶(𝑈 ⊕ 𝑄)) → 𝑈 ∈ 𝑆) |
32 | lveclmod 21122 | . . . . . . 7 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
33 | 2, 32 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod) |
34 | 11, 6, 33, 1 | lsatlssel 38978 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ 𝑆) |
35 | 11, 12 | lsmcl 21099 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑄 ∈ 𝑆) → (𝑈 ⊕ 𝑄) ∈ 𝑆) |
36 | 33, 16, 34, 35 | syl3anc 1370 | . . . . 5 ⊢ (𝜑 → (𝑈 ⊕ 𝑄) ∈ 𝑆) |
37 | 36 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑈𝐶(𝑈 ⊕ 𝑄)) → (𝑈 ⊕ 𝑄) ∈ 𝑆) |
38 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑈𝐶(𝑈 ⊕ 𝑄)) → 𝑈𝐶(𝑈 ⊕ 𝑄)) | |
39 | 11, 13, 30, 31, 37, 38 | lcvpss 39005 | . . 3 ⊢ ((𝜑 ∧ 𝑈𝐶(𝑈 ⊕ 𝑄)) → 𝑈 ⊊ (𝑈 ⊕ 𝑄)) |
40 | 11 | lsssssubg 20973 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) |
41 | 33, 40 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊)) |
42 | 41, 16 | sseldd 3995 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
43 | 41, 34 | sseldd 3995 | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑊)) |
44 | 12, 42, 43 | lssnle 19706 | . . . 4 ⊢ (𝜑 → (¬ 𝑄 ⊆ 𝑈 ↔ 𝑈 ⊊ (𝑈 ⊕ 𝑄))) |
45 | 44 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑈𝐶(𝑈 ⊕ 𝑄)) → (¬ 𝑄 ⊆ 𝑈 ↔ 𝑈 ⊊ (𝑈 ⊕ 𝑄))) |
46 | 39, 45 | mpbird 257 | . 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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ∃wrex 3067 ∖ cdif 3959 ⊆ wss 3962 ⊊ wpss 3963 {csn 4630 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 0gc0g 17485 SubGrpcsubg 19150 LSSumclsm 19666 LModclmod 20874 LSubSpclss 20946 LSpanclspn 20986 LVecclvec 21118 LSAtomsclsa 38955 ⋖L clcv 38999 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-tpos 8249 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-grp 18966 df-minusg 18967 df-sbg 18968 df-subg 19153 df-cntz 19347 df-lsm 19668 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-invr 20404 df-drng 20747 df-lmod 20876 df-lss 20947 df-lsp 20987 df-lvec 21119 df-lsatoms 38957 df-lcv 39000 |
This theorem is referenced by: lcv2 39023 lsatnle 39025 lsatcvat3 39033 |
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