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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdindp1 | 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 |
---|---|
mapdindp1 | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑌 + 𝑍)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdindp1.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
2 | eldifsni 4749 | . . . . . 6 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → 𝑋 ≠ 0 ) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑋 ≠ 0 ) |
4 | mapdindp1.w | . . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
5 | lveclmod 20520 | . . . . . . . . . 10 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
6 | 4, 5 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ LMod) |
7 | mapdindp1.o | . . . . . . . . . 10 ⊢ 0 = (0g‘𝑊) | |
8 | mapdindp1.n | . . . . . . . . . 10 ⊢ 𝑁 = (LSpan‘𝑊) | |
9 | 7, 8 | lspsn0 20422 | . . . . . . . . 9 ⊢ (𝑊 ∈ LMod → (𝑁‘{ 0 }) = { 0 }) |
10 | 6, 9 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝑁‘{ 0 }) = { 0 }) |
11 | 10 | eqeq2d 2749 | . . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑋}) = (𝑁‘{ 0 }) ↔ (𝑁‘{𝑋}) = { 0 })) |
12 | 1 | eldifad 3921 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
13 | mapdindp1.v | . . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑊) | |
14 | 13, 7, 8 | lspsneq0 20426 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑁‘{𝑋}) = { 0 } ↔ 𝑋 = 0 )) |
15 | 6, 12, 14 | syl2anc 585 | . . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑋}) = { 0 } ↔ 𝑋 = 0 )) |
16 | 11, 15 | bitrd 279 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) = (𝑁‘{ 0 }) ↔ 𝑋 = 0 )) |
17 | 16 | necon3bid 2987 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{ 0 }) ↔ 𝑋 ≠ 0 )) |
18 | 3, 17 | mpbird 257 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{ 0 })) |
19 | 18 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{ 0 })) |
20 | sneq 4595 | . . . . 5 ⊢ ((𝑌 + 𝑍) = 0 → {(𝑌 + 𝑍)} = { 0 }) | |
21 | 20 | fveq2d 6844 | . . . 4 ⊢ ((𝑌 + 𝑍) = 0 → (𝑁‘{(𝑌 + 𝑍)}) = (𝑁‘{ 0 })) |
22 | 21 | adantl 483 | . . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝑁‘{(𝑌 + 𝑍)}) = (𝑁‘{ 0 })) |
23 | 19, 22 | neeqtrrd 3017 | . 2 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) = 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑌 + 𝑍)})) |
24 | mapdindp1.ne | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
25 | 24 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
26 | mapdindp1.p | . . . 4 ⊢ + = (+g‘𝑊) | |
27 | 4 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑊 ∈ LVec) |
28 | 1 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
29 | mapdindp1.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
30 | 29 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑌 ∈ (𝑉 ∖ { 0 })) |
31 | mapdindp1.z | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) | |
32 | 31 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑍 ∈ (𝑉 ∖ { 0 })) |
33 | mapdindp1.W | . . . . 5 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) | |
34 | 33 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → 𝑤 ∈ (𝑉 ∖ { 0 })) |
35 | mapdindp1.e | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) | |
36 | 35 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) |
37 | mapdindp1.f | . . . . 5 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) | |
38 | 37 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) |
39 | simpr 486 | . . . 4 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑌 + 𝑍) ≠ 0 ) | |
40 | 13, 26, 7, 8, 27, 28, 30, 32, 34, 36, 25, 38, 39 | mapdindp0 40114 | . . 3 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{(𝑌 + 𝑍)}) = (𝑁‘{𝑌})) |
41 | 25, 40 | neeqtrrd 3017 | . 2 ⊢ ((𝜑 ∧ (𝑌 + 𝑍) ≠ 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑌 + 𝑍)})) |
42 | 23, 41 | pm2.61dane 3031 | 1 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑌 + 𝑍)})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2942 ∖ cdif 3906 {csn 4585 {cpr 4587 ‘cfv 6494 (class class class)co 7352 Basecbs 17043 +gcplusg 17093 0gc0g 17281 LModclmod 20275 LSpanclspn 20385 LVecclvec 20516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-cnex 11066 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7796 df-1st 7914 df-2nd 7915 df-tpos 8150 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-er 8607 df-en 8843 df-dom 8844 df-sdom 8845 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-nn 12113 df-2 12175 df-3 12176 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-ress 17073 df-plusg 17106 df-mulr 17107 df-0g 17283 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-submnd 18562 df-grp 18711 df-minusg 18712 df-sbg 18713 df-subg 18884 df-cntz 19056 df-lsm 19377 df-cmn 19523 df-abl 19524 df-mgp 19856 df-ur 19873 df-ring 19920 df-oppr 20002 df-dvdsr 20023 df-unit 20024 df-invr 20054 df-drng 20140 df-lmod 20277 df-lss 20346 df-lsp 20386 df-lvec 20517 |
This theorem is referenced by: mapdh6dN 40134 mapdh6hN 40138 hdmap1l6d 40208 hdmap1l6h 40212 |
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