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Mathbox for Norm Megill |
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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 4786 | . . . . . 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 20950 | . . . . . . . . . 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 20851 | . . . . . . . . 9 β’ (π β LMod β (πβ{ 0 }) = { 0 }) |
10 | 6, 9 | syl 17 | . . . . . . . 8 β’ (π β (πβ{ 0 }) = { 0 }) |
11 | 10 | eqeq2d 2735 | . . . . . . 7 β’ (π β ((πβ{π}) = (πβ{ 0 }) β (πβ{π}) = { 0 })) |
12 | 1 | eldifad 3953 | . . . . . . . 8 β’ (π β π β π) |
13 | mapdindp1.v | . . . . . . . . 9 β’ π = (Baseβπ) | |
14 | 13, 7, 8 | lspsneq0 20855 | . . . . . . . 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 2977 | . . . . 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 4631 | . . . . 5 β’ ((π + π) = 0 β {(π + π)} = { 0 }) | |
21 | 20 | fveq2d 6886 | . . . 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 3007 | . 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 41094 | . . 3 β’ ((π β§ (π + π) β 0 ) β (πβ{(π + π)}) = (πβ{π})) |
41 | 25, 40 | neeqtrrd 3007 | . 2 β’ ((π β§ (π + π) β 0 ) β (πβ{π}) β (πβ{(π + π)})) |
42 | 23, 41 | pm2.61dane 3021 | 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 1533 β wcel 2098 β wne 2932 β cdif 3938 {csn 4621 {cpr 4623 βcfv 6534 (class class class)co 7402 Basecbs 17149 +gcplusg 17202 0gc0g 17390 LModclmod 20702 LSpanclspn 20814 LVecclvec 20946 |
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 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-0g 17392 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18710 df-grp 18862 df-minusg 18863 df-sbg 18864 df-subg 19046 df-cntz 19229 df-lsm 19552 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-drng 20585 df-lmod 20704 df-lss 20775 df-lsp 20815 df-lvec 20947 |
This theorem is referenced by: mapdh6dN 41114 mapdh6hN 41118 hdmap1l6d 41188 hdmap1l6h 41192 |
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