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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcvp | Structured version Visualization version GIF version |
Description: Covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 32308 analog.) (Contributed by NM, 10-Jan-2015.) |
Ref | Expression |
---|---|
lcvp.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lcvp.p | ⊢ ⊕ = (LSSum‘𝑊) |
lcvp.o | ⊢ 0 = (0g‘𝑊) |
lcvp.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lcvp.c | ⊢ 𝐶 = ( ⋖L ‘𝑊) |
lcvp.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lcvp.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lcvp.q | ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
Ref | Expression |
---|---|
lcvp | ⊢ (𝜑 → ((𝑈 ∩ 𝑄) = { 0 } ↔ 𝑈𝐶(𝑈 ⊕ 𝑄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcvp.o | . . 3 ⊢ 0 = (0g‘𝑊) | |
2 | lcvp.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
3 | lcvp.a | . . 3 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
4 | lcvp.c | . . 3 ⊢ 𝐶 = ( ⋖L ‘𝑊) | |
5 | lcvp.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
6 | lveclmod 21084 | . . . . 5 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) |
8 | lcvp.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
9 | lcvp.q | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐴) | |
10 | 2, 3, 7, 9 | lsatlssel 38695 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝑆) |
11 | 2 | lssincl 20942 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑄 ∈ 𝑆) → (𝑈 ∩ 𝑄) ∈ 𝑆) |
12 | 7, 8, 10, 11 | syl3anc 1368 | . . 3 ⊢ (𝜑 → (𝑈 ∩ 𝑄) ∈ 𝑆) |
13 | 1, 2, 3, 4, 5, 12, 9 | lsatcveq0 38730 | . 2 ⊢ (𝜑 → ((𝑈 ∩ 𝑄)𝐶𝑄 ↔ (𝑈 ∩ 𝑄) = { 0 })) |
14 | lcvp.p | . . 3 ⊢ ⊕ = (LSSum‘𝑊) | |
15 | 2, 14, 4, 7, 8, 10 | lcvexch 38737 | . 2 ⊢ (𝜑 → ((𝑈 ∩ 𝑄)𝐶𝑄 ↔ 𝑈𝐶(𝑈 ⊕ 𝑄))) |
16 | 13, 15 | bitr3d 280 | 1 ⊢ (𝜑 → ((𝑈 ∩ 𝑄) = { 0 } ↔ 𝑈𝐶(𝑈 ⊕ 𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 ∩ cin 3946 {csn 4633 class class class wbr 5153 ‘cfv 6554 (class class class)co 7424 0gc0g 17454 LSSumclsm 19632 LModclmod 20836 LSubSpclss 20908 LVecclvec 21080 LSAtomsclsa 38672 ⋖L clcv 38716 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-tpos 8241 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-0g 17456 df-mre 17599 df-mrc 17600 df-acs 17602 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-submnd 18774 df-grp 18931 df-minusg 18932 df-sbg 18933 df-subg 19117 df-cntz 19311 df-oppg 19340 df-lsm 19634 df-cmn 19780 df-abl 19781 df-mgp 20118 df-rng 20136 df-ur 20165 df-ring 20218 df-oppr 20316 df-dvdsr 20339 df-unit 20340 df-invr 20370 df-drng 20709 df-lmod 20838 df-lss 20909 df-lsp 20949 df-lvec 21081 df-lsatoms 38674 df-lcv 38717 |
This theorem is referenced by: lsatexch 38741 lsatnle 38742 lsatcv0eq 38745 lsatcvatlem 38747 |
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