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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdindp4 | Structured version Visualization version GIF version |
Description: Vector independence lemma. (Contributed by NM, 29-Apr-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 |
---|---|
mapdindp4 | ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, (𝑤 + 𝑌)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdindp1.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
2 | mapdindp1.o | . . 3 ⊢ 0 = (0g‘𝑊) | |
3 | mapdindp1.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
4 | mapdindp1.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
5 | mapdindp1.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) | |
6 | lveclmod 21123 | . . . . 5 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
7 | 4, 6 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) |
8 | mapdindp1.W | . . . . 5 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) | |
9 | 8 | eldifad 3975 | . . . 4 ⊢ (𝜑 → 𝑤 ∈ 𝑉) |
10 | mapdindp1.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
11 | 10 | eldifad 3975 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
12 | mapdindp1.p | . . . . 5 ⊢ + = (+g‘𝑊) | |
13 | 1, 12 | lmodvacl 20890 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑤 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑤 + 𝑌) ∈ 𝑉) |
14 | 7, 9, 11, 13 | syl3anc 1370 | . . 3 ⊢ (𝜑 → (𝑤 + 𝑌) ∈ 𝑉) |
15 | mapdindp1.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
16 | 15 | eldifad 3975 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
17 | mapdindp1.e | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) | |
18 | mapdindp1.f | . . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) | |
19 | 1, 3, 4, 9, 16, 11, 18 | lspindpi 21152 | . . . . . . . 8 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}))) |
20 | 19 | simprd 495 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌})) |
21 | 20 | necomd 2994 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤})) |
22 | 1, 12, 2, 3, 4, 11, 8, 21 | lspindp3 21156 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{(𝑌 + 𝑤)})) |
23 | 1, 12 | lmodcom 20923 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑤 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑤 + 𝑌) = (𝑌 + 𝑤)) |
24 | 7, 9, 11, 23 | syl3anc 1370 | . . . . . . 7 ⊢ (𝜑 → (𝑤 + 𝑌) = (𝑌 + 𝑤)) |
25 | 24 | sneqd 4643 | . . . . . 6 ⊢ (𝜑 → {(𝑤 + 𝑌)} = {(𝑌 + 𝑤)}) |
26 | 25 | fveq2d 6911 | . . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑤 + 𝑌)}) = (𝑁‘{(𝑌 + 𝑤)})) |
27 | 22, 26 | neeqtrrd 3013 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{(𝑤 + 𝑌)})) |
28 | 17, 27 | eqnetrrd 3007 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{(𝑤 + 𝑌)})) |
29 | mapdindp1.ne | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
30 | 1, 2, 3, 4, 15, 11, 9, 29, 18 | lspindp1 21153 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}) ∧ ¬ 𝑋 ∈ (𝑁‘{𝑤, 𝑌}))) |
31 | 30 | simprd 495 | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑤, 𝑌})) |
32 | eqid 2735 | . . . . . 6 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊) | |
33 | 5 | eldifad 3975 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
34 | 1, 3, 32, 7, 33, 14 | lsmpr 21106 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑍, (𝑤 + 𝑌)}) = ((𝑁‘{𝑍})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)}))) |
35 | 1, 12 | lmodcom 20923 | . . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) → (𝑌 + 𝑤) = (𝑤 + 𝑌)) |
36 | 7, 11, 9, 35 | syl3anc 1370 | . . . . . . . . 9 ⊢ (𝜑 → (𝑌 + 𝑤) = (𝑤 + 𝑌)) |
37 | 36 | preq2d 4745 | . . . . . . . 8 ⊢ (𝜑 → {𝑌, (𝑌 + 𝑤)} = {𝑌, (𝑤 + 𝑌)}) |
38 | 37 | fveq2d 6911 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌, (𝑌 + 𝑤)}) = (𝑁‘{𝑌, (𝑤 + 𝑌)})) |
39 | 1, 12, 3, 7, 11, 9 | lspprabs 21112 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌, (𝑌 + 𝑤)}) = (𝑁‘{𝑌, 𝑤})) |
40 | 1, 3, 32, 7, 11, 14 | lsmpr 21106 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌, (𝑤 + 𝑌)}) = ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)}))) |
41 | 38, 39, 40 | 3eqtr3rd 2784 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)})) = (𝑁‘{𝑌, 𝑤})) |
42 | 17 | oveq1d 7446 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)})) = ((𝑁‘{𝑍})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)}))) |
43 | prcom 4737 | . . . . . . . 8 ⊢ {𝑌, 𝑤} = {𝑤, 𝑌} | |
44 | 43 | fveq2i 6910 | . . . . . . 7 ⊢ (𝑁‘{𝑌, 𝑤}) = (𝑁‘{𝑤, 𝑌}) |
45 | 44 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑤}) = (𝑁‘{𝑤, 𝑌})) |
46 | 41, 42, 45 | 3eqtr3d 2783 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑍})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)})) = (𝑁‘{𝑤, 𝑌})) |
47 | 34, 46 | eqtrd 2775 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑍, (𝑤 + 𝑌)}) = (𝑁‘{𝑤, 𝑌})) |
48 | 31, 47 | neleqtrrd 2862 | . . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑍, (𝑤 + 𝑌)})) |
49 | 1, 2, 3, 4, 5, 14, 16, 28, 48 | lspindp1 21153 | . 2 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{(𝑤 + 𝑌)}) ∧ ¬ 𝑍 ∈ (𝑁‘{𝑋, (𝑤 + 𝑌)}))) |
50 | 49 | simprd 495 | 1 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, (𝑤 + 𝑌)})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∖ cdif 3960 {csn 4631 {cpr 4633 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 0gc0g 17486 LSSumclsm 19667 LModclmod 20875 LSpanclspn 20987 LVecclvec 21119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cntz 19348 df-lsm 19669 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lvec 21120 |
This theorem is referenced by: mapdh6eN 41723 hdmap1l6e 41797 |
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