Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 19911. Can it be used along with lspsnne1 19892, lspsnne2 19893 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 19684 | . . . . . 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 19712 | . . . . . 6 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑉) |
10 | 5, 6, 9 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
11 | lssvancl.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
12 | lssvancl.p | . . . . . 6 ⊢ + = (+g‘𝑊) | |
13 | eqid 2824 | . . . . . 6 ⊢ (-g‘𝑊) = (-g‘𝑊) | |
14 | 7, 12, 13 | ablpncan2 18939 | . . . . 5 ⊢ ((𝑊 ∈ Abel ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝑋 + 𝑌)(-g‘𝑊)𝑋) = 𝑌) |
15 | 4, 10, 11, 14 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → ((𝑋 + 𝑌)(-g‘𝑊)𝑋) = 𝑌) |
16 | 15 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑈) → ((𝑋 + 𝑌)(-g‘𝑊)𝑋) = 𝑌) |
17 | 2 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑈) → 𝑊 ∈ LMod) |
18 | 5 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑈) → 𝑈 ∈ 𝑆) |
19 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑈) → (𝑋 + 𝑌) ∈ 𝑈) | |
20 | 6 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) ∈ 𝑈) → 𝑋 ∈ 𝑈) |
21 | 13, 8 | lssvsubcl 19718 | . . . 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 2917 | . 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 398 = wceq 1536 ∈ wcel 2113 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 +gcplusg 16568 -gcsg 18108 Abelcabl 18910 LModclmod 19637 LSubSpclss 19706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-plusg 16581 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-grp 18109 df-minusg 18110 df-sbg 18111 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-lmod 19639 df-lss 19707 |
This theorem is referenced by: lssvancl2 19720 dvh3dim2 38588 dvh3dim3N 38589 hdmap11lem2 38982 |
Copyright terms: Public domain | W3C validator |