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Mirrors > Home > MPE Home > Th. List > lspsntrim | Structured version Visualization version GIF version |
Description: Triangle-type inequality for span of a singleton of vector difference. (Contributed by NM, 25-Apr-2014.) (Revised by Mario Carneiro, 21-Jun-2014.) |
Ref | Expression |
---|---|
lspsntrim.v | β’ π = (Baseβπ) |
lspsntrim.s | β’ β = (-gβπ) |
lspsntrim.p | β’ β = (LSSumβπ) |
lspsntrim.n | β’ π = (LSpanβπ) |
Ref | Expression |
---|---|
lspsntrim | β’ ((π β LMod β§ π β π β§ π β π) β (πβ{(π β π)}) β ((πβ{π}) β (πβ{π}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspsntrim.v | . . . . 5 β’ π = (Baseβπ) | |
2 | eqid 2733 | . . . . 5 β’ (invgβπ) = (invgβπ) | |
3 | 1, 2 | lmodvnegcl 20378 | . . . 4 β’ ((π β LMod β§ π β π) β ((invgβπ)βπ) β π) |
4 | 3 | 3adant2 1132 | . . 3 β’ ((π β LMod β§ π β π β§ π β π) β ((invgβπ)βπ) β π) |
5 | eqid 2733 | . . . 4 β’ (+gβπ) = (+gβπ) | |
6 | lspsntrim.n | . . . 4 β’ π = (LSpanβπ) | |
7 | lspsntrim.p | . . . 4 β’ β = (LSSumβπ) | |
8 | 1, 5, 6, 7 | lspsntri 20573 | . . 3 β’ ((π β LMod β§ π β π β§ ((invgβπ)βπ) β π) β (πβ{(π(+gβπ)((invgβπ)βπ))}) β ((πβ{π}) β (πβ{((invgβπ)βπ)}))) |
9 | 4, 8 | syld3an3 1410 | . 2 β’ ((π β LMod β§ π β π β§ π β π) β (πβ{(π(+gβπ)((invgβπ)βπ))}) β ((πβ{π}) β (πβ{((invgβπ)βπ)}))) |
10 | lspsntrim.s | . . . . . 6 β’ β = (-gβπ) | |
11 | 1, 5, 2, 10 | grpsubval 18801 | . . . . 5 β’ ((π β π β§ π β π) β (π β π) = (π(+gβπ)((invgβπ)βπ))) |
12 | 11 | sneqd 4599 | . . . 4 β’ ((π β π β§ π β π) β {(π β π)} = {(π(+gβπ)((invgβπ)βπ))}) |
13 | 12 | fveq2d 6847 | . . 3 β’ ((π β π β§ π β π) β (πβ{(π β π)}) = (πβ{(π(+gβπ)((invgβπ)βπ))})) |
14 | 13 | 3adant1 1131 | . 2 β’ ((π β LMod β§ π β π β§ π β π) β (πβ{(π β π)}) = (πβ{(π(+gβπ)((invgβπ)βπ))})) |
15 | 1, 2, 6 | lspsnneg 20482 | . . . . 5 β’ ((π β LMod β§ π β π) β (πβ{((invgβπ)βπ)}) = (πβ{π})) |
16 | 15 | 3adant2 1132 | . . . 4 β’ ((π β LMod β§ π β π β§ π β π) β (πβ{((invgβπ)βπ)}) = (πβ{π})) |
17 | 16 | eqcomd 2739 | . . 3 β’ ((π β LMod β§ π β π β§ π β π) β (πβ{π}) = (πβ{((invgβπ)βπ)})) |
18 | 17 | oveq2d 7374 | . 2 β’ ((π β LMod β§ π β π β§ π β π) β ((πβ{π}) β (πβ{π})) = ((πβ{π}) β (πβ{((invgβπ)βπ)}))) |
19 | 9, 14, 18 | 3sstr4d 3992 | 1 β’ ((π β LMod β§ π β π β§ π β π) β (πβ{(π β π)}) β ((πβ{π}) β (πβ{π}))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wss 3911 {csn 4587 βcfv 6497 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 invgcminusg 18754 -gcsg 18755 LSSumclsm 19421 LModclmod 20336 LSpanclspn 20447 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
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 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-submnd 18607 df-grp 18756 df-minusg 18757 df-sbg 18758 df-subg 18930 df-cntz 19102 df-lsm 19423 df-cmn 19569 df-abl 19570 df-mgp 19902 df-ur 19919 df-ring 19971 df-lmod 20338 df-lss 20408 df-lsp 20448 |
This theorem is referenced by: mapdpglem1 40181 baerlem3lem2 40219 baerlem5alem2 40220 baerlem5blem2 40221 |
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