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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 | 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 |
---|---|
mapdindp1 | β’ (π β (πβ{π}) β (πβ{(π + π)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdindp1.x | . . . . . 6 β’ (π β π β (π β { 0 })) | |
2 | eldifsni 4794 | . . . . . 6 β’ (π β (π β { 0 }) β π β 0 ) | |
3 | 1, 2 | syl 17 | . . . . 5 β’ (π β π β 0 ) |
4 | mapdindp1.w | . . . . . . . . . 10 β’ (π β π β LVec) | |
5 | lveclmod 20991 | . . . . . . . . . 10 β’ (π β LVec β π β LMod) | |
6 | 4, 5 | syl 17 | . . . . . . . . 9 β’ (π β π β LMod) |
7 | mapdindp1.o | . . . . . . . . . 10 β’ 0 = (0gβπ) | |
8 | mapdindp1.n | . . . . . . . . . 10 β’ π = (LSpanβπ) | |
9 | 7, 8 | lspsn0 20892 | . . . . . . . . 9 β’ (π β LMod β (πβ{ 0 }) = { 0 }) |
10 | 6, 9 | syl 17 | . . . . . . . 8 β’ (π β (πβ{ 0 }) = { 0 }) |
11 | 10 | eqeq2d 2739 | . . . . . . 7 β’ (π β ((πβ{π}) = (πβ{ 0 }) β (πβ{π}) = { 0 })) |
12 | 1 | eldifad 3959 | . . . . . . . 8 β’ (π β π β π) |
13 | mapdindp1.v | . . . . . . . . 9 β’ π = (Baseβπ) | |
14 | 13, 7, 8 | lspsneq0 20896 | . . . . . . . 8 β’ ((π β LMod β§ π β π) β ((πβ{π}) = { 0 } β π = 0 )) |
15 | 6, 12, 14 | syl2anc 583 | . . . . . . 7 β’ (π β ((πβ{π}) = { 0 } β π = 0 )) |
16 | 11, 15 | bitrd 279 | . . . . . 6 β’ (π β ((πβ{π}) = (πβ{ 0 }) β π = 0 )) |
17 | 16 | necon3bid 2982 | . . . . 5 β’ (π β ((πβ{π}) β (πβ{ 0 }) β π β 0 )) |
18 | 3, 17 | mpbird 257 | . . . 4 β’ (π β (πβ{π}) β (πβ{ 0 })) |
19 | 18 | adantr 480 | . . 3 β’ ((π β§ (π + π) = 0 ) β (πβ{π}) β (πβ{ 0 })) |
20 | sneq 4639 | . . . . 5 β’ ((π + π) = 0 β {(π + π)} = { 0 }) | |
21 | 20 | fveq2d 6901 | . . . 4 β’ ((π + π) = 0 β (πβ{(π + π)}) = (πβ{ 0 })) |
22 | 21 | adantl 481 | . . 3 β’ ((π β§ (π + π) = 0 ) β (πβ{(π + π)}) = (πβ{ 0 })) |
23 | 19, 22 | neeqtrrd 3012 | . 2 β’ ((π β§ (π + π) = 0 ) β (πβ{π}) β (πβ{(π + π)})) |
24 | mapdindp1.ne | . . . 4 β’ (π β (πβ{π}) β (πβ{π})) | |
25 | 24 | adantr 480 | . . 3 β’ ((π β§ (π + π) β 0 ) β (πβ{π}) β (πβ{π})) |
26 | mapdindp1.p | . . . 4 β’ + = (+gβπ) | |
27 | 4 | adantr 480 | . . . 4 β’ ((π β§ (π + π) β 0 ) β π β LVec) |
28 | 1 | adantr 480 | . . . 4 β’ ((π β§ (π + π) β 0 ) β π β (π β { 0 })) |
29 | mapdindp1.y | . . . . 5 β’ (π β π β (π β { 0 })) | |
30 | 29 | adantr 480 | . . . 4 β’ ((π β§ (π + π) β 0 ) β π β (π β { 0 })) |
31 | mapdindp1.z | . . . . 5 β’ (π β π β (π β { 0 })) | |
32 | 31 | adantr 480 | . . . 4 β’ ((π β§ (π + π) β 0 ) β π β (π β { 0 })) |
33 | mapdindp1.W | . . . . 5 β’ (π β π€ β (π β { 0 })) | |
34 | 33 | adantr 480 | . . . 4 β’ ((π β§ (π + π) β 0 ) β π€ β (π β { 0 })) |
35 | mapdindp1.e | . . . . 5 β’ (π β (πβ{π}) = (πβ{π})) | |
36 | 35 | adantr 480 | . . . 4 β’ ((π β§ (π + π) β 0 ) β (πβ{π}) = (πβ{π})) |
37 | mapdindp1.f | . . . . 5 β’ (π β Β¬ π€ β (πβ{π, π})) | |
38 | 37 | adantr 480 | . . . 4 β’ ((π β§ (π + π) β 0 ) β Β¬ π€ β (πβ{π, π})) |
39 | simpr 484 | . . . 4 β’ ((π β§ (π + π) β 0 ) β (π + π) β 0 ) | |
40 | 13, 26, 7, 8, 27, 28, 30, 32, 34, 36, 25, 38, 39 | mapdindp0 41192 | . . 3 β’ ((π β§ (π + π) β 0 ) β (πβ{(π + π)}) = (πβ{π})) |
41 | 25, 40 | neeqtrrd 3012 | . 2 β’ ((π β§ (π + π) β 0 ) β (πβ{π}) β (πβ{(π + π)})) |
42 | 23, 41 | pm2.61dane 3026 | 1 β’ (π β (πβ{π}) β (πβ{(π + π)})) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 395 = wceq 1534 β wcel 2099 β wne 2937 β cdif 3944 {csn 4629 {cpr 4631 βcfv 6548 (class class class)co 7420 Basecbs 17180 +gcplusg 17233 0gc0g 17421 LModclmod 20743 LSpanclspn 20855 LVecclvec 20987 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 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 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-0g 17423 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18741 df-grp 18893 df-minusg 18894 df-sbg 18895 df-subg 19078 df-cntz 19268 df-lsm 19591 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-oppr 20273 df-dvdsr 20296 df-unit 20297 df-invr 20327 df-drng 20626 df-lmod 20745 df-lss 20816 df-lsp 20856 df-lvec 20988 |
This theorem is referenced by: mapdh6dN 41212 mapdh6hN 41216 hdmap1l6d 41286 hdmap1l6h 41290 |
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