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Mirrors > Home > MPE Home > Th. List > lmodindp1 | Structured version Visualization version GIF version |
Description: Two independent (non-colinear) vectors have nonzero sum. (Contributed by NM, 22-Apr-2015.) |
Ref | Expression |
---|---|
lmodindp1.v | ⊢ 𝑉 = (Base‘𝑊) |
lmodindp1.p | ⊢ + = (+g‘𝑊) |
lmodindp1.o | ⊢ 0 = (0g‘𝑊) |
lmodindp1.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lmodindp1.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lmodindp1.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lmodindp1.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
lmodindp1.q | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
Ref | Expression |
---|---|
lmodindp1 | ⊢ (𝜑 → (𝑋 + 𝑌) ≠ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodindp1.q | . 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
2 | lmodindp1.w | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
3 | lmodindp1.x | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
4 | lmodindp1.v | . . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑊) | |
5 | eqid 2778 | . . . . . . . . 9 ⊢ (invg‘𝑊) = (invg‘𝑊) | |
6 | lmodindp1.n | . . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑊) | |
7 | 4, 5, 6 | lspsnneg 19401 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{((invg‘𝑊)‘𝑋)}) = (𝑁‘{𝑋})) |
8 | 2, 3, 7 | syl2anc 579 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{((invg‘𝑊)‘𝑋)}) = (𝑁‘{𝑋})) |
9 | 8 | eqcomd 2784 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{((invg‘𝑊)‘𝑋)})) |
10 | 9 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) = 0 ) → (𝑁‘{𝑋}) = (𝑁‘{((invg‘𝑊)‘𝑋)})) |
11 | lmodgrp 19262 | . . . . . . . . . 10 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
12 | 2, 11 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ Grp) |
13 | lmodindp1.y | . . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
14 | lmodindp1.p | . . . . . . . . . 10 ⊢ + = (+g‘𝑊) | |
15 | lmodindp1.o | . . . . . . . . . 10 ⊢ 0 = (0g‘𝑊) | |
16 | 4, 14, 15, 5 | grpinvid1 17857 | . . . . . . . . 9 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (((invg‘𝑊)‘𝑋) = 𝑌 ↔ (𝑋 + 𝑌) = 0 )) |
17 | 12, 3, 13, 16 | syl3anc 1439 | . . . . . . . 8 ⊢ (𝜑 → (((invg‘𝑊)‘𝑋) = 𝑌 ↔ (𝑋 + 𝑌) = 0 )) |
18 | 17 | biimpar 471 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) = 0 ) → ((invg‘𝑊)‘𝑋) = 𝑌) |
19 | 18 | sneqd 4410 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) = 0 ) → {((invg‘𝑊)‘𝑋)} = {𝑌}) |
20 | 19 | fveq2d 6450 | . . . . 5 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) = 0 ) → (𝑁‘{((invg‘𝑊)‘𝑋)}) = (𝑁‘{𝑌})) |
21 | 10, 20 | eqtrd 2814 | . . . 4 ⊢ ((𝜑 ∧ (𝑋 + 𝑌) = 0 ) → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) |
22 | 21 | ex 403 | . . 3 ⊢ (𝜑 → ((𝑋 + 𝑌) = 0 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))) |
23 | 22 | necon3d 2990 | . 2 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) → (𝑋 + 𝑌) ≠ 0 )) |
24 | 1, 23 | mpd 15 | 1 ⊢ (𝜑 → (𝑋 + 𝑌) ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 {csn 4398 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 +gcplusg 16338 0gc0g 16486 Grpcgrp 17809 invgcminusg 17810 LModclmod 19255 LSpanclspn 19366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-plusg 16351 df-0g 16488 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-grp 17812 df-minusg 17813 df-sbg 17814 df-mgp 18877 df-ur 18889 df-ring 18936 df-lmod 19257 df-lss 19325 df-lsp 19367 |
This theorem is referenced by: lcfrlem17 37713 mapdh6aN 37889 mapdh6eN 37894 hdmap1l6a 37963 hdmap1l6e 37968 hdmaprnlem3eN 38012 |
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