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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 19106 | . . . . 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 3586 | . . . 4 ⊢ (𝜑 → 𝑤 ∈ 𝑉) |
10 | mapdindp1.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
11 | 10 | eldifad 3586 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
12 | mapdindp1.p | . . . . 5 ⊢ + = (+g‘𝑊) | |
13 | 1, 12 | lmodvacl 18877 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑤 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑤 + 𝑌) ∈ 𝑉) |
14 | 7, 9, 11, 13 | syl3anc 1326 | . . 3 ⊢ (𝜑 → (𝑤 + 𝑌) ∈ 𝑉) |
15 | mapdindp1.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
16 | 15 | eldifad 3586 | . . 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 19132 | . . . . . . . 8 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}))) |
20 | 19 | simprd 479 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌})) |
21 | 20 | necomd 2849 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤})) |
22 | 1, 12, 2, 3, 4, 11, 8, 21 | lspindp3 19136 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{(𝑌 + 𝑤)})) |
23 | 1, 12 | lmodcom 18909 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑤 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑤 + 𝑌) = (𝑌 + 𝑤)) |
24 | 7, 9, 11, 23 | syl3anc 1326 | . . . . . . 7 ⊢ (𝜑 → (𝑤 + 𝑌) = (𝑌 + 𝑤)) |
25 | 24 | sneqd 4189 | . . . . . 6 ⊢ (𝜑 → {(𝑤 + 𝑌)} = {(𝑌 + 𝑤)}) |
26 | 25 | fveq2d 6195 | . . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑤 + 𝑌)}) = (𝑁‘{(𝑌 + 𝑤)})) |
27 | 22, 26 | neeqtrrd 2868 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{(𝑤 + 𝑌)})) |
28 | 17, 27 | eqnetrrd 2862 | . . 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 19133 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}) ∧ ¬ 𝑋 ∈ (𝑁‘{𝑤, 𝑌}))) |
31 | 30 | simprd 479 | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑤, 𝑌})) |
32 | eqid 2622 | . . . . . 6 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊) | |
33 | 5 | eldifad 3586 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
34 | 1, 3, 32, 7, 33, 14 | lsmpr 19089 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑍, (𝑤 + 𝑌)}) = ((𝑁‘{𝑍})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)}))) |
35 | 1, 12 | lmodcom 18909 | . . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) → (𝑌 + 𝑤) = (𝑤 + 𝑌)) |
36 | 7, 11, 9, 35 | syl3anc 1326 | . . . . . . . . 9 ⊢ (𝜑 → (𝑌 + 𝑤) = (𝑤 + 𝑌)) |
37 | 36 | preq2d 4275 | . . . . . . . 8 ⊢ (𝜑 → {𝑌, (𝑌 + 𝑤)} = {𝑌, (𝑤 + 𝑌)}) |
38 | 37 | fveq2d 6195 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌, (𝑌 + 𝑤)}) = (𝑁‘{𝑌, (𝑤 + 𝑌)})) |
39 | 1, 12, 3, 7, 11, 9 | lspprabs 19095 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌, (𝑌 + 𝑤)}) = (𝑁‘{𝑌, 𝑤})) |
40 | 1, 3, 32, 7, 11, 14 | lsmpr 19089 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌, (𝑤 + 𝑌)}) = ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)}))) |
41 | 38, 39, 40 | 3eqtr3rd 2665 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)})) = (𝑁‘{𝑌, 𝑤})) |
42 | 17 | oveq1d 6665 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑌})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)})) = ((𝑁‘{𝑍})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)}))) |
43 | prcom 4267 | . . . . . . . 8 ⊢ {𝑌, 𝑤} = {𝑤, 𝑌} | |
44 | 43 | fveq2i 6194 | . . . . . . 7 ⊢ (𝑁‘{𝑌, 𝑤}) = (𝑁‘{𝑤, 𝑌}) |
45 | 44 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑤}) = (𝑁‘{𝑤, 𝑌})) |
46 | 41, 42, 45 | 3eqtr3d 2664 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑍})(LSSum‘𝑊)(𝑁‘{(𝑤 + 𝑌)})) = (𝑁‘{𝑤, 𝑌})) |
47 | 34, 46 | eqtrd 2656 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑍, (𝑤 + 𝑌)}) = (𝑁‘{𝑤, 𝑌})) |
48 | 31, 47 | neleqtrrd 2723 | . . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑍, (𝑤 + 𝑌)})) |
49 | 1, 2, 3, 4, 5, 14, 16, 28, 48 | lspindp1 19133 | . 2 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{(𝑤 + 𝑌)}) ∧ ¬ 𝑍 ∈ (𝑁‘{𝑋, (𝑤 + 𝑌)}))) |
50 | 49 | simprd 479 | 1 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, (𝑤 + 𝑌)})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∖ cdif 3571 {csn 4177 {cpr 4179 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 0gc0g 16100 LSSumclsm 18049 LModclmod 18863 LSpanclspn 18971 LVecclvec 19102 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-cntz 17750 df-lsm 18051 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-drng 18749 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lvec 19103 |
This theorem is referenced by: mapdh6eN 37029 hdmap1l6e 37104 |
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