Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| 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 2725 | . . . . 5 β’ (invgβπ) = (invgβπ) | |
| 3 | 1, 2 | lmodvnegcl 20790 | . . . 4 β’ ((π β LMod β§ π β π) β ((invgβπ)βπ) β π) |
| 4 | 3 | 3adant2 1128 | . . 3 β’ ((π β LMod β§ π β π β§ π β π) β ((invgβπ)βπ) β π) |
| 5 | eqid 2725 | . . . 4 β’ (+gβπ) = (+gβπ) | |
| 6 | lspsntrim.n | . . . 4 β’ π = (LSpanβπ) | |
| 7 | lspsntrim.p | . . . 4 β’ β = (LSSumβπ) | |
| 8 | 1, 5, 6, 7 | lspsntri 20986 | . . 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 1406 | . 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 18946 | . . . . 5 β’ ((π β π β§ π β π) β (π β π) = (π(+gβπ)((invgβπ)βπ))) |
| 12 | 11 | sneqd 4636 | . . . 4 β’ ((π β π β§ π β π) β {(π β π)} = {(π(+gβπ)((invgβπ)βπ))}) |
| 13 | 12 | fveq2d 6896 | . . 3 β’ ((π β π β§ π β π) β (πβ{(π β π)}) = (πβ{(π(+gβπ)((invgβπ)βπ))})) |
| 14 | 13 | 3adant1 1127 | . 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 20894 | . . . . 5 β’ ((π β LMod β§ π β π) β (πβ{((invgβπ)βπ)}) = (πβ{π})) |
| 16 | 15 | 3adant2 1128 | . . . 4 β’ ((π β LMod β§ π β π β§ π β π) β (πβ{((invgβπ)βπ)}) = (πβ{π})) |
| 17 | 16 | eqcomd 2731 | . . 3 β’ ((π β LMod β§ π β π β§ π β π) β (πβ{π}) = (πβ{((invgβπ)βπ)})) |
| 18 | 17 | oveq2d 7432 | . 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 4020 | 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 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β wss 3939 {csn 4624 βcfv 6543 (class class class)co 7416 Basecbs 17179 +gcplusg 17232 invgcminusg 18895 -gcsg 18896 LSSumclsm 19593 LModclmod 20747 LSpanclspn 20859 |
| 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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
| 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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-grp 18897 df-minusg 18898 df-sbg 18899 df-subg 19082 df-cntz 19272 df-lsm 19595 df-cmn 19741 df-abl 19742 df-mgp 20079 df-ur 20126 df-ring 20179 df-lmod 20749 df-lss 20820 df-lsp 20860 |
| This theorem is referenced by: mapdpglem1 41201 baerlem3lem2 41239 baerlem5alem2 41240 baerlem5blem2 41241 |
| Copyright terms: Public domain | W3C validator |