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Mirrors > Home > MPE Home > Th. List > lspindp2 | Structured version Visualization version GIF version |
Description: Alternate way to say 3 vectors are mutually independent (rotate right). (Contributed by NM, 12-Apr-2015.) |
Ref | Expression |
---|---|
lspindp1.v | ⊢ 𝑉 = (Base‘𝑊) |
lspindp1.o | ⊢ 0 = (0g‘𝑊) |
lspindp1.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lspindp1.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lspindp2.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lspindp2.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
lspindp2.z | ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
lspindp2.q | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
lspindp2.e | ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌})) |
Ref | Expression |
---|---|
lspindp2 | ⊢ (𝜑 → ((𝑁‘{𝑍}) ≠ (𝑁‘{𝑋}) ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑋}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspindp1.v | . 2 ⊢ 𝑉 = (Base‘𝑊) | |
2 | lspindp1.o | . 2 ⊢ 0 = (0g‘𝑊) | |
3 | lspindp1.n | . 2 ⊢ 𝑁 = (LSpan‘𝑊) | |
4 | lspindp1.w | . 2 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
5 | lspindp2.y | . 2 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
6 | lspindp2.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
7 | lspindp2.z | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
8 | lspindp2.q | . . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
9 | 8 | necomd 2992 | . 2 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑋})) |
10 | lspindp2.e | . . 3 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌})) | |
11 | prcom 4739 | . . . . 5 ⊢ {𝑋, 𝑌} = {𝑌, 𝑋} | |
12 | 11 | fveq2i 6903 | . . . 4 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘{𝑌, 𝑋}) |
13 | 12 | eleq2i 2820 | . . 3 ⊢ (𝑍 ∈ (𝑁‘{𝑋, 𝑌}) ↔ 𝑍 ∈ (𝑁‘{𝑌, 𝑋})) |
14 | 10, 13 | sylnib 327 | . 2 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑌, 𝑋})) |
15 | 1, 2, 3, 4, 5, 6, 7, 9, 14 | lspindp1 21026 | 1 ⊢ (𝜑 → ((𝑁‘{𝑍}) ≠ (𝑁‘{𝑋}) ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑋}))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2936 ∖ cdif 3944 {csn 4630 {cpr 4632 ‘cfv 6551 Basecbs 17185 0gc0g 17426 LSpanclspn 20860 LVecclvec 20992 |
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 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-tpos 8236 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-2 12311 df-3 12312 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-ress 17215 df-plusg 17251 df-mulr 17252 df-0g 17428 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-submnd 18746 df-grp 18898 df-minusg 18899 df-sbg 18900 df-subg 19083 df-cntz 19273 df-lsm 19596 df-cmn 19742 df-abl 19743 df-mgp 20080 df-rng 20098 df-ur 20127 df-ring 20180 df-oppr 20278 df-dvdsr 20301 df-unit 20302 df-invr 20332 df-drng 20631 df-lmod 20750 df-lss 20821 df-lsp 20861 df-lvec 20993 |
This theorem is referenced by: mapdheq4lem 41208 mapdheq4 41209 mapdh6lem1N 41210 mapdh6lem2N 41211 mapdh6aN 41212 hdmap1l6lem1 41284 hdmap1l6lem2 41285 hdmap1l6a 41286 |
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