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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdindp2 | Structured version Visualization version GIF version |
Description: Vector independence lemma. (Contributed by NM, 1-May-2015.) |
Ref | Expression |
---|---|
mapdindp1.v | ⊢ 𝑉 = (Base‘𝑊) |
mapdindp1.p | ⊢ + = (+g‘𝑊) |
mapdindp1.o | ⊢ 0 = (0g‘𝑊) |
mapdindp1.n | ⊢ 𝑁 = (LSpan‘𝑊) |
mapdindp1.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
mapdindp1.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
mapdindp1.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
mapdindp1.z | ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) |
mapdindp1.W | ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) |
mapdindp1.e | ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) |
mapdindp1.ne | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
mapdindp1.f | ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) |
Ref | Expression |
---|---|
mapdindp2 | ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, (𝑌 + 𝑍)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq2 4501 | . . . . . 6 ⊢ ((𝑌 + 𝑍) = 0 → {𝑋, (𝑌 + 𝑍)} = {𝑋, 0 }) | |
2 | 1 | fveq2d 6450 | . . . . 5 ⊢ ((𝑌 + 𝑍) = 0 → (𝑁‘{𝑋, (𝑌 + 𝑍)}) = (𝑁‘{𝑋, 0 })) |
3 | mapdindp1.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
4 | mapdindp1.o | . . . . . 6 ⊢ 0 = (0g‘𝑊) | |
5 | mapdindp1.n | . . . . . 6 ⊢ 𝑁 = (LSpan‘𝑊) | |
6 | mapdindp1.w | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
7 | lveclmod 19501 | . . . . . . 7 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod) |
9 | mapdindp1.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
10 | 9 | eldifad 3804 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
11 | 3, 4, 5, 8, 10 | lsppr0 19487 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋, 0 }) = (𝑁‘{𝑋})) |
12 | 2, 11 | sylan9eqr 2836 | . . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝑁‘{𝑋, (𝑌 + 𝑍)}) = (𝑁‘{𝑋})) |
13 | mapdindp1.y | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
14 | 13 | eldifad 3804 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
15 | prssi 4583 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ⊆ 𝑉) | |
16 | 10, 14, 15 | syl2anc 579 | . . . . . 6 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ 𝑉) |
17 | snsspr1 4576 | . . . . . . 7 ⊢ {𝑋} ⊆ {𝑋, 𝑌} | |
18 | 17 | a1i 11 | . . . . . 6 ⊢ (𝜑 → {𝑋} ⊆ {𝑋, 𝑌}) |
19 | 3, 5 | lspss 19379 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ {𝑋, 𝑌} ⊆ 𝑉 ∧ {𝑋} ⊆ {𝑋, 𝑌}) → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑋, 𝑌})) |
20 | 8, 16, 18, 19 | syl3anc 1439 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑋, 𝑌})) |
21 | 20 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑋, 𝑌})) |
22 | 12, 21 | eqsstrd 3858 | . . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝑁‘{𝑋, (𝑌 + 𝑍)}) ⊆ (𝑁‘{𝑋, 𝑌})) |
23 | mapdindp1.f | . . . 4 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) | |
24 | 23 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) |
25 | 22, 24 | ssneldd 3824 | . 2 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → ¬ 𝑤 ∈ (𝑁‘{𝑋, (𝑌 + 𝑍)})) |
26 | 23 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) |
27 | mapdindp1.p | . . . . . 6 ⊢ + = (+g‘𝑊) | |
28 | 6 | adantr 474 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑊 ∈ LVec) |
29 | 9 | adantr 474 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
30 | 13 | adantr 474 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑌 ∈ (𝑉 ∖ { 0 })) |
31 | mapdindp1.z | . . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) | |
32 | 31 | adantr 474 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑍 ∈ (𝑉 ∖ { 0 })) |
33 | mapdindp1.W | . . . . . . 7 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) | |
34 | 33 | adantr 474 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑤 ∈ (𝑉 ∖ { 0 })) |
35 | mapdindp1.e | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) | |
36 | 35 | adantr 474 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) |
37 | mapdindp1.ne | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
38 | 37 | adantr 474 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
39 | simpr 479 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑌 + 𝑍) ≠ 0 ) | |
40 | 3, 27, 4, 5, 28, 29, 30, 32, 34, 36, 38, 26, 39 | mapdindp0 37873 | . . . . 5 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{(𝑌 + 𝑍)}) = (𝑁‘{𝑌})) |
41 | 40 | oveq2d 6938 | . . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑌 + 𝑍)})) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌}))) |
42 | eqid 2778 | . . . . . 6 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊) | |
43 | 31 | eldifad 3804 | . . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
44 | 3, 27 | lmodvacl 19269 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑌 + 𝑍) ∈ 𝑉) |
45 | 8, 14, 43, 44 | syl3anc 1439 | . . . . . 6 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ 𝑉) |
46 | 3, 5, 42, 8, 10, 45 | lsmpr 19484 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋, (𝑌 + 𝑍)}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑌 + 𝑍)}))) |
47 | 46 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑋, (𝑌 + 𝑍)}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑌 + 𝑍)}))) |
48 | 3, 5, 42, 8, 10, 14 | lsmpr 19484 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌}))) |
49 | 48 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌}))) |
50 | 41, 47, 49 | 3eqtr4d 2824 | . . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑋, (𝑌 + 𝑍)}) = (𝑁‘{𝑋, 𝑌})) |
51 | 26, 50 | neleqtrrd 2881 | . 2 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → ¬ 𝑤 ∈ (𝑁‘{𝑋, (𝑌 + 𝑍)})) |
52 | 25, 51 | pm2.61dane 3057 | 1 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, (𝑌 + 𝑍)})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∖ cdif 3789 ⊆ wss 3792 {csn 4398 {cpr 4400 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 +gcplusg 16338 0gc0g 16486 LSSumclsm 18433 LModclmod 19255 LSpanclspn 19366 LVecclvec 19497 |
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-tpos 7634 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-3 11439 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-0g 16488 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-grp 17812 df-minusg 17813 df-sbg 17814 df-subg 17975 df-cntz 18133 df-lsm 18435 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 df-drng 19141 df-lmod 19257 df-lss 19325 df-lsp 19367 df-lvec 19498 |
This theorem is referenced by: mapdh6dN 37893 hdmap1l6d 37967 |
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