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Mirrors > Home > MPE Home > Th. List > lssvancl1 | Structured version Visualization version GIF version |
Description: Non-closure: if one vector belongs to a subspace but another does not, their sum does not belong. Useful for obtaining a new vector not in a subspace. TODO: notice similarity to lspindp3 21028. Can it be used along with lspsnne1 21009, lspsnne2 21010 to shorten this proof? (Contributed by NM, 14-May-2015.) |
Ref | Expression |
---|---|
lssvancl.v | ⊢ 𝑉 = (Base‘𝑊) |
lssvancl.p | ⊢ + = (+g‘𝑊) |
lssvancl.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lssvancl.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lssvancl.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lssvancl.x | ⊢ (𝜑 → 𝑋 ∈ 𝑈) |
lssvancl.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
lssvancl.n | ⊢ (𝜑 → ¬ 𝑌 ∈ 𝑈) |
Ref | Expression |
---|---|
lssvancl1 | ⊢ (𝜑 → ¬ (𝑋 + 𝑌) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lssvancl.n | . 2 ⊢ (𝜑 → ¬ 𝑌 ∈ 𝑈) | |
2 | lssvancl.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
3 | lmodabl 20796 | . . . . . 6 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ Abel) |
5 | lssvancl.u | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
6 | lssvancl.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑈) | |
7 | lssvancl.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑊) | |
8 | lssvancl.s | . . . . . . 7 ⊢ 𝑆 = (LSubSp‘𝑊) | |
9 | 7, 8 | lssel 20825 | . . . . . 6 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑉) |
10 | 5, 6, 9 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
11 | lssvancl.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
12 | lssvancl.p | . . . . . 6 ⊢ + = (+g‘𝑊) | |
13 | eqid 2725 | . . . . . 6 ⊢ (-g‘𝑊) = (-g‘𝑊) | |
14 | 7, 12, 13 | ablpncan2 19774 | . . . . 5 ⊢ ((𝑊 ∈ Abel ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝑋 + 𝑌)(-g‘𝑊)𝑋) = 𝑌) |
15 | 4, 10, 11, 14 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → ((𝑋 + 𝑌)(-g‘𝑊)𝑋) = 𝑌) |
16 | 15 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑈) → ((𝑋 + 𝑌)(-g‘𝑊)𝑋) = 𝑌) |
17 | 2 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑈) → 𝑊 ∈ LMod) |
18 | 5 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑈) → 𝑈 ∈ 𝑆) |
19 | simpr 483 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑈) → (𝑋 + 𝑌) ∈ 𝑈) | |
20 | 6 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑈) → 𝑋 ∈ 𝑈) |
21 | 13, 8 | lssvsubcl 20832 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ ((𝑋 + 𝑌) ∈ 𝑈 ∧ 𝑋 ∈ 𝑈)) → ((𝑋 + 𝑌)(-g‘𝑊)𝑋) ∈ 𝑈) |
22 | 17, 18, 19, 20, 21 | syl22anc 837 | . . 3 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑈) → ((𝑋 + 𝑌)(-g‘𝑊)𝑋) ∈ 𝑈) |
23 | 16, 22 | eqeltrrd 2826 | . 2 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑈) → 𝑌 ∈ 𝑈) |
24 | 1, 23 | mtand 814 | 1 ⊢ (𝜑 → ¬ (𝑋 + 𝑌) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ‘cfv 6543 (class class class)co 7416 Basecbs 17179 +gcplusg 17232 -gcsg 18896 Abelcabl 19740 LModclmod 20747 LSubSpclss 20819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-minusg 18898 df-sbg 18899 df-cmn 19741 df-abl 19742 df-mgp 20079 df-ur 20126 df-ring 20179 df-lmod 20749 df-lss 20820 |
This theorem is referenced by: lssvancl2 20834 dvh3dim2 40977 dvh3dim3N 40978 hdmap11lem2 41371 |
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