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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 20613. Can it be used along with lspsnne1 20594, lspsnne2 20595 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 20384 | . . . . . 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 20413 | . . . . . 6 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑉) |
10 | 5, 6, 9 | syl2anc 585 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
11 | lssvancl.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
12 | lssvancl.p | . . . . . 6 ⊢ + = (+g‘𝑊) | |
13 | eqid 2733 | . . . . . 6 ⊢ (-g‘𝑊) = (-g‘𝑊) | |
14 | 7, 12, 13 | ablpncan2 19599 | . . . . 5 ⊢ ((𝑊 ∈ Abel ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝑋 + 𝑌)(-g‘𝑊)𝑋) = 𝑌) |
15 | 4, 10, 11, 14 | syl3anc 1372 | . . . 4 ⊢ (𝜑 → ((𝑋 + 𝑌)(-g‘𝑊)𝑋) = 𝑌) |
16 | 15 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑈) → ((𝑋 + 𝑌)(-g‘𝑊)𝑋) = 𝑌) |
17 | 2 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑈) → 𝑊 ∈ LMod) |
18 | 5 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑈) → 𝑈 ∈ 𝑆) |
19 | simpr 486 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑈) → (𝑋 + 𝑌) ∈ 𝑈) | |
20 | 6 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑈) → 𝑋 ∈ 𝑈) |
21 | 13, 8 | lssvsubcl 20419 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ ((𝑋 + 𝑌) ∈ 𝑈 ∧ 𝑋 ∈ 𝑈)) → ((𝑋 + 𝑌)(-g‘𝑊)𝑋) ∈ 𝑈) |
22 | 17, 18, 19, 20, 21 | syl22anc 838 | . . 3 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑈) → ((𝑋 + 𝑌)(-g‘𝑊)𝑋) ∈ 𝑈) |
23 | 16, 22 | eqeltrrd 2835 | . 2 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑈) → 𝑌 ∈ 𝑈) |
24 | 1, 23 | mtand 815 | 1 ⊢ (𝜑 → ¬ (𝑋 + 𝑌) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ‘cfv 6497 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 -gcsg 18755 Abelcabl 19568 LModclmod 20336 LSubSpclss 20407 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-minusg 18757 df-sbg 18758 df-cmn 19569 df-abl 19570 df-mgp 19902 df-ur 19919 df-ring 19971 df-lmod 20338 df-lss 20408 |
This theorem is referenced by: lssvancl2 20421 dvh3dim2 39957 dvh3dim3N 39958 hdmap11lem2 40351 |
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