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Mirrors > Home > MPE Home > Th. List > lspabs2 | Structured version Visualization version GIF version |
Description: Absorption law for span of vector sum. (Contributed by NM, 30-Apr-2015.) |
Ref | Expression |
---|---|
lspabs2.v | β’ π = (Baseβπ) |
lspabs2.p | β’ + = (+gβπ) |
lspabs2.o | β’ 0 = (0gβπ) |
lspabs2.n | β’ π = (LSpanβπ) |
lspabs2.w | β’ (π β π β LVec) |
lspabs2.x | β’ (π β π β π) |
lspabs2.y | β’ (π β π β (π β { 0 })) |
lspabs2.e | β’ (π β (πβ{π}) = (πβ{(π + π)})) |
Ref | Expression |
---|---|
lspabs2 | β’ (π β (πβ{π}) = (πβ{π})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspabs2.w | . . . . . . 7 β’ (π β π β LVec) | |
2 | lveclmod 20583 | . . . . . . 7 β’ (π β LVec β π β LMod) | |
3 | 1, 2 | syl 17 | . . . . . 6 β’ (π β π β LMod) |
4 | lspabs2.x | . . . . . 6 β’ (π β π β π) | |
5 | lspabs2.v | . . . . . . 7 β’ π = (Baseβπ) | |
6 | lspabs2.n | . . . . . . 7 β’ π = (LSpanβπ) | |
7 | 5, 6 | lspsnsubg 20457 | . . . . . 6 β’ ((π β LMod β§ π β π) β (πβ{π}) β (SubGrpβπ)) |
8 | 3, 4, 7 | syl2anc 585 | . . . . 5 β’ (π β (πβ{π}) β (SubGrpβπ)) |
9 | lspabs2.y | . . . . . . 7 β’ (π β π β (π β { 0 })) | |
10 | 9 | eldifad 3927 | . . . . . 6 β’ (π β π β π) |
11 | 5, 6 | lspsnsubg 20457 | . . . . . 6 β’ ((π β LMod β§ π β π) β (πβ{π}) β (SubGrpβπ)) |
12 | 3, 10, 11 | syl2anc 585 | . . . . 5 β’ (π β (πβ{π}) β (SubGrpβπ)) |
13 | eqid 2737 | . . . . . 6 β’ (LSSumβπ) = (LSSumβπ) | |
14 | 13 | lsmub2 19447 | . . . . 5 β’ (((πβ{π}) β (SubGrpβπ) β§ (πβ{π}) β (SubGrpβπ)) β (πβ{π}) β ((πβ{π})(LSSumβπ)(πβ{π}))) |
15 | 8, 12, 14 | syl2anc 585 | . . . 4 β’ (π β (πβ{π}) β ((πβ{π})(LSSumβπ)(πβ{π}))) |
16 | lspabs2.e | . . . . . 6 β’ (π β (πβ{π}) = (πβ{(π + π)})) | |
17 | 16 | oveq2d 7378 | . . . . 5 β’ (π β ((πβ{π})(LSSumβπ)(πβ{π})) = ((πβ{π})(LSSumβπ)(πβ{(π + π)}))) |
18 | 13 | lsmidm 19452 | . . . . . 6 β’ ((πβ{π}) β (SubGrpβπ) β ((πβ{π})(LSSumβπ)(πβ{π})) = (πβ{π})) |
19 | 8, 18 | syl 17 | . . . . 5 β’ (π β ((πβ{π})(LSSumβπ)(πβ{π})) = (πβ{π})) |
20 | lspabs2.p | . . . . . . 7 β’ + = (+gβπ) | |
21 | 5, 20, 6, 3, 4, 10 | lspprabs 20572 | . . . . . 6 β’ (π β (πβ{π, (π + π)}) = (πβ{π, π})) |
22 | 5, 20 | lmodvacl 20352 | . . . . . . . 8 β’ ((π β LMod β§ π β π β§ π β π) β (π + π) β π) |
23 | 3, 4, 10, 22 | syl3anc 1372 | . . . . . . 7 β’ (π β (π + π) β π) |
24 | 5, 6, 13, 3, 4, 23 | lsmpr 20566 | . . . . . 6 β’ (π β (πβ{π, (π + π)}) = ((πβ{π})(LSSumβπ)(πβ{(π + π)}))) |
25 | 5, 6, 13, 3, 4, 10 | lsmpr 20566 | . . . . . 6 β’ (π β (πβ{π, π}) = ((πβ{π})(LSSumβπ)(πβ{π}))) |
26 | 21, 24, 25 | 3eqtr3d 2785 | . . . . 5 β’ (π β ((πβ{π})(LSSumβπ)(πβ{(π + π)})) = ((πβ{π})(LSSumβπ)(πβ{π}))) |
27 | 17, 19, 26 | 3eqtr3rd 2786 | . . . 4 β’ (π β ((πβ{π})(LSSumβπ)(πβ{π})) = (πβ{π})) |
28 | 15, 27 | sseqtrd 3989 | . . 3 β’ (π β (πβ{π}) β (πβ{π})) |
29 | lspabs2.o | . . . 4 β’ 0 = (0gβπ) | |
30 | 5, 29, 6, 1, 9, 4 | lspsncmp 20593 | . . 3 β’ (π β ((πβ{π}) β (πβ{π}) β (πβ{π}) = (πβ{π}))) |
31 | 28, 30 | mpbid 231 | . 2 β’ (π β (πβ{π}) = (πβ{π})) |
32 | 31 | eqcomd 2743 | 1 β’ (π β (πβ{π}) = (πβ{π})) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β cdif 3912 β wss 3915 {csn 4591 {cpr 4593 βcfv 6501 (class class class)co 7362 Basecbs 17090 +gcplusg 17140 0gc0g 17328 SubGrpcsubg 18929 LSSumclsm 19423 LModclmod 20338 LSpanclspn 20448 LVecclvec 20579 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-tpos 8162 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-3 12224 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-0g 17330 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-submnd 18609 df-grp 18758 df-minusg 18759 df-sbg 18760 df-subg 18932 df-cntz 19104 df-lsm 19425 df-cmn 19571 df-abl 19572 df-mgp 19904 df-ur 19921 df-ring 19973 df-oppr 20056 df-dvdsr 20077 df-unit 20078 df-invr 20108 df-drng 20201 df-lmod 20340 df-lss 20409 df-lsp 20449 df-lvec 20580 |
This theorem is referenced by: lspindp3 20613 |
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