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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 20998 | . . . . 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 3961 | . . . 4 β’ (π β π€ β π) |
6 | mapdindp1.y | . . . . 5 β’ (π β π β (π β { 0 })) | |
7 | 6 | eldifad 3961 | . . . 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 20988 | . . . 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 1368 | . . 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 21028 | . . . 4 β’ (π β ((πβ{π€}) β (πβ{π}) β§ Β¬ π β (πβ{π€, π}))) |
18 | 17 | simprd 494 | . . 3 β’ (π β Β¬ π β (πβ{π€, π})) |
19 | 12, 18 | ssneldd 3985 | . 2 β’ (π β Β¬ π β (πβ{(π€ + π)})) |
20 | 14 | eldifad 3961 | . . . . 5 β’ (π β π β π) |
21 | 8, 10 | lspsnid 20884 | . . . . 5 β’ ((π β LMod β§ π β π) β π β (πβ{π})) |
22 | 3, 20, 21 | syl2anc 582 | . . . 4 β’ (π β π β (πβ{π})) |
23 | eleq2 2818 | . . . 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 2951 | . 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 1533 β wcel 2098 β wne 2937 β cdif 3946 β wss 3949 {csn 4632 {cpr 4634 βcfv 6553 (class class class)co 7426 Basecbs 17187 +gcplusg 17240 0gc0g 17428 LModclmod 20750 LSpanclspn 20862 LVecclvec 20994 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-grp 18900 df-minusg 18901 df-sbg 18902 df-subg 19085 df-cntz 19275 df-lsm 19598 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-oppr 20280 df-dvdsr 20303 df-unit 20304 df-invr 20334 df-drng 20633 df-lmod 20752 df-lss 20823 df-lsp 20863 df-lvec 20995 |
This theorem is referenced by: mapdh6eN 41245 hdmap1l6e 41319 |
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