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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdindp3 | Structured version Visualization version GIF version |
Description: Vector independence lemma. (Contributed by NM, 29-Apr-2015.) |
Ref | Expression |
---|---|
mapdindp1.v | β’ π = (Baseβπ) |
mapdindp1.p | β’ + = (+gβπ) |
mapdindp1.o | β’ 0 = (0gβπ) |
mapdindp1.n | β’ π = (LSpanβπ) |
mapdindp1.w | β’ (π β π β LVec) |
mapdindp1.x | β’ (π β π β (π β { 0 })) |
mapdindp1.y | β’ (π β π β (π β { 0 })) |
mapdindp1.z | β’ (π β π β (π β { 0 })) |
mapdindp1.W | β’ (π β π€ β (π β { 0 })) |
mapdindp1.e | β’ (π β (πβ{π}) = (πβ{π})) |
mapdindp1.ne | β’ (π β (πβ{π}) β (πβ{π})) |
mapdindp1.f | β’ (π β Β¬ π€ β (πβ{π, π})) |
Ref | Expression |
---|---|
mapdindp3 | β’ (π β (πβ{π}) β (πβ{(π€ + π)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdindp1.w | . . . . 5 β’ (π β π β LVec) | |
2 | lveclmod 20611 | . . . . 5 β’ (π β LVec β π β LMod) | |
3 | 1, 2 | syl 17 | . . . 4 β’ (π β π β LMod) |
4 | mapdindp1.W | . . . . 5 β’ (π β π€ β (π β { 0 })) | |
5 | 4 | eldifad 3926 | . . . 4 β’ (π β π€ β π) |
6 | mapdindp1.y | . . . . 5 β’ (π β π β (π β { 0 })) | |
7 | 6 | eldifad 3926 | . . . 4 β’ (π β π β π) |
8 | mapdindp1.v | . . . . 5 β’ π = (Baseβπ) | |
9 | mapdindp1.p | . . . . 5 β’ + = (+gβπ) | |
10 | mapdindp1.n | . . . . 5 β’ π = (LSpanβπ) | |
11 | 8, 9, 10 | lspvadd 20601 | . . . 4 β’ ((π β LMod β§ π€ β π β§ π β π) β (πβ{(π€ + π)}) β (πβ{π€, π})) |
12 | 3, 5, 7, 11 | syl3anc 1372 | . . 3 β’ (π β (πβ{(π€ + π)}) β (πβ{π€, π})) |
13 | mapdindp1.o | . . . . 5 β’ 0 = (0gβπ) | |
14 | mapdindp1.x | . . . . 5 β’ (π β π β (π β { 0 })) | |
15 | mapdindp1.ne | . . . . 5 β’ (π β (πβ{π}) β (πβ{π})) | |
16 | mapdindp1.f | . . . . 5 β’ (π β Β¬ π€ β (πβ{π, π})) | |
17 | 8, 13, 10, 1, 14, 7, 5, 15, 16 | lspindp1 20639 | . . . 4 β’ (π β ((πβ{π€}) β (πβ{π}) β§ Β¬ π β (πβ{π€, π}))) |
18 | 17 | simprd 497 | . . 3 β’ (π β Β¬ π β (πβ{π€, π})) |
19 | 12, 18 | ssneldd 3951 | . 2 β’ (π β Β¬ π β (πβ{(π€ + π)})) |
20 | 14 | eldifad 3926 | . . . . 5 β’ (π β π β π) |
21 | 8, 10 | lspsnid 20498 | . . . . 5 β’ ((π β LMod β§ π β π) β π β (πβ{π})) |
22 | 3, 20, 21 | syl2anc 585 | . . . 4 β’ (π β π β (πβ{π})) |
23 | eleq2 2823 | . . . 4 β’ ((πβ{π}) = (πβ{(π€ + π)}) β (π β (πβ{π}) β π β (πβ{(π€ + π)}))) | |
24 | 22, 23 | syl5ibcom 244 | . . 3 β’ (π β ((πβ{π}) = (πβ{(π€ + π)}) β π β (πβ{(π€ + π)}))) |
25 | 24 | necon3bd 2954 | . 2 β’ (π β (Β¬ π β (πβ{(π€ + π)}) β (πβ{π}) β (πβ{(π€ + π)}))) |
26 | 19, 25 | mpd 15 | 1 β’ (π β (πβ{π}) β (πβ{(π€ + π)})) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 = wceq 1542 β wcel 2107 β wne 2940 β cdif 3911 β wss 3914 {csn 4590 {cpr 4592 βcfv 6500 (class class class)co 7361 Basecbs 17091 +gcplusg 17141 0gc0g 17329 LModclmod 20365 LSpanclspn 20476 LVecclvec 20607 |
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 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-tpos 8161 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-0g 17331 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-submnd 18610 df-grp 18759 df-minusg 18760 df-sbg 18761 df-subg 18933 df-cntz 19105 df-lsm 19426 df-cmn 19572 df-abl 19573 df-mgp 19905 df-ur 19922 df-ring 19974 df-oppr 20057 df-dvdsr 20078 df-unit 20079 df-invr 20109 df-drng 20221 df-lmod 20367 df-lss 20437 df-lsp 20477 df-lvec 20608 |
This theorem is referenced by: mapdh6eN 40253 hdmap1l6e 40327 |
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