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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatcvat2 | Structured version Visualization version GIF version |
Description: A subspace covered by the sum of two distinct atoms is an atom. (atcvat2i 30091 analog.) (Contributed by NM, 10-Jan-2015.) |
Ref | Expression |
---|---|
lsatcvat2.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lsatcvat2.p | ⊢ ⊕ = (LSSum‘𝑊) |
lsatcvat2.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lsatcvat2.c | ⊢ 𝐶 = ( ⋖L ‘𝑊) |
lsatcvat2.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lsatcvat2.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lsatcvat2.q | ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
lsatcvat2.r | ⊢ (𝜑 → 𝑅 ∈ 𝐴) |
lsatcvat2.n | ⊢ (𝜑 → 𝑄 ≠ 𝑅) |
lsatcvat2.l | ⊢ (𝜑 → 𝑈𝐶(𝑄 ⊕ 𝑅)) |
Ref | Expression |
---|---|
lsatcvat2 | ⊢ (𝜑 → 𝑈 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . 2 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
2 | lsatcvat2.s | . 2 ⊢ 𝑆 = (LSubSp‘𝑊) | |
3 | lsatcvat2.p | . 2 ⊢ ⊕ = (LSSum‘𝑊) | |
4 | lsatcvat2.a | . 2 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
5 | lsatcvat2.w | . 2 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
6 | lsatcvat2.u | . 2 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
7 | lsatcvat2.q | . 2 ⊢ (𝜑 → 𝑄 ∈ 𝐴) | |
8 | lsatcvat2.r | . 2 ⊢ (𝜑 → 𝑅 ∈ 𝐴) | |
9 | lsatcvat2.n | . . 3 ⊢ (𝜑 → 𝑄 ≠ 𝑅) | |
10 | lsatcvat2.c | . . . . 5 ⊢ 𝐶 = ( ⋖L ‘𝑊) | |
11 | lsatcvat2.l | . . . . 5 ⊢ (𝜑 → 𝑈𝐶(𝑄 ⊕ 𝑅)) | |
12 | 1, 3, 2, 4, 10, 5, 6, 7, 8, 11 | lsatcv1 36064 | . . . 4 ⊢ (𝜑 → (𝑈 = {(0g‘𝑊)} ↔ 𝑄 = 𝑅)) |
13 | 12 | necon3bid 3057 | . . 3 ⊢ (𝜑 → (𝑈 ≠ {(0g‘𝑊)} ↔ 𝑄 ≠ 𝑅)) |
14 | 9, 13 | mpbird 258 | . 2 ⊢ (𝜑 → 𝑈 ≠ {(0g‘𝑊)}) |
15 | lveclmod 19807 | . . . . 5 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
16 | 5, 15 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) |
17 | 2, 4, 16, 7 | lsatlssel 36013 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝑆) |
18 | 2, 4, 16, 8 | lsatlssel 36013 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑆) |
19 | 2, 3 | lsmcl 19784 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑄 ∈ 𝑆 ∧ 𝑅 ∈ 𝑆) → (𝑄 ⊕ 𝑅) ∈ 𝑆) |
20 | 16, 17, 18, 19 | syl3anc 1363 | . . 3 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) ∈ 𝑆) |
21 | 2, 10, 5, 6, 20, 11 | lcvpss 36040 | . 2 ⊢ (𝜑 → 𝑈 ⊊ (𝑄 ⊕ 𝑅)) |
22 | 1, 2, 3, 4, 5, 6, 7, 8, 14, 21 | lsatcvat 36066 | 1 ⊢ (𝜑 → 𝑈 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 {csn 4557 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 0gc0g 16701 LSSumclsm 18688 LModclmod 19563 LSubSpclss 19632 LVecclvec 19803 LSAtomsclsa 35990 ⋖L clcv 36034 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-0g 16703 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-subg 18214 df-cntz 18385 df-oppg 18412 df-lsm 18690 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-drng 19433 df-lmod 19565 df-lss 19633 df-lsp 19673 df-lvec 19804 df-lsatoms 35992 df-lcv 36035 |
This theorem is referenced by: lsatcvat3 36068 |
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