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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 20987. Can it be used along with lspsnne1 20968, lspsnne2 20969 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 20755 | . . . . . 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 20784 | . . . . . 6 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑉) |
10 | 5, 6, 9 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
11 | lssvancl.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
12 | lssvancl.p | . . . . . 6 ⊢ + = (+g‘𝑊) | |
13 | eqid 2726 | . . . . . 6 ⊢ (-g‘𝑊) = (-g‘𝑊) | |
14 | 7, 12, 13 | ablpncan2 19735 | . . . . 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 480 | . . 3 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑈) → ((𝑋 + 𝑌)(-g‘𝑊)𝑋) = 𝑌) |
17 | 2 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑈) → 𝑊 ∈ LMod) |
18 | 5 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑈) → 𝑈 ∈ 𝑆) |
19 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑈) → (𝑋 + 𝑌) ∈ 𝑈) | |
20 | 6 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑈) → 𝑋 ∈ 𝑈) |
21 | 13, 8 | lssvsubcl 20791 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ ((𝑋 + 𝑌) ∈ 𝑈 ∧ 𝑋 ∈ 𝑈)) → ((𝑋 + 𝑌)(-g‘𝑊)𝑋) ∈ 𝑈) |
22 | 17, 18, 19, 20, 21 | syl22anc 836 | . . 3 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑈) → ((𝑋 + 𝑌)(-g‘𝑊)𝑋) ∈ 𝑈) |
23 | 16, 22 | eqeltrrd 2828 | . 2 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑈) → 𝑌 ∈ 𝑈) |
24 | 1, 23 | mtand 813 | 1 ⊢ (𝜑 → ¬ (𝑋 + 𝑌) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ‘cfv 6537 (class class class)co 7405 Basecbs 17153 +gcplusg 17206 -gcsg 18865 Abelcabl 19701 LModclmod 20706 LSubSpclss 20778 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-plusg 17219 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-sbg 18868 df-cmn 19702 df-abl 19703 df-mgp 20040 df-ur 20087 df-ring 20140 df-lmod 20708 df-lss 20779 |
This theorem is referenced by: lssvancl2 20793 dvh3dim2 40832 dvh3dim3N 40833 hdmap11lem2 41226 |
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