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| Mirrors > Home > MPE Home > Th. List > nvlinv | Structured version Visualization version GIF version | ||
| Description: Minus a vector plus itself. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| nvrinv.1 | β’ π = (BaseSetβπ) |
| nvrinv.2 | β’ πΊ = ( +π£ βπ) |
| nvrinv.4 | β’ π = ( Β·π OLD βπ) |
| nvrinv.6 | β’ π = (0vecβπ) |
| Ref | Expression |
|---|---|
| nvlinv | β’ ((π β NrmCVec β§ π΄ β π) β ((-1ππ΄)πΊπ΄) = π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nvrinv.2 | . . . 4 β’ πΊ = ( +π£ βπ) | |
| 2 | 1 | nvgrp 30125 | . . 3 β’ (π β NrmCVec β πΊ β GrpOp) |
| 3 | nvrinv.1 | . . . . 5 β’ π = (BaseSetβπ) | |
| 4 | 3, 1 | bafval 30112 | . . . 4 β’ π = ran πΊ |
| 5 | eqid 2732 | . . . 4 β’ (GIdβπΊ) = (GIdβπΊ) | |
| 6 | eqid 2732 | . . . 4 β’ (invβπΊ) = (invβπΊ) | |
| 7 | 4, 5, 6 | grpolinv 30034 | . . 3 β’ ((πΊ β GrpOp β§ π΄ β π) β (((invβπΊ)βπ΄)πΊπ΄) = (GIdβπΊ)) |
| 8 | 2, 7 | sylan 580 | . 2 β’ ((π β NrmCVec β§ π΄ β π) β (((invβπΊ)βπ΄)πΊπ΄) = (GIdβπΊ)) |
| 9 | nvrinv.4 | . . . 4 β’ π = ( Β·π OLD βπ) | |
| 10 | 3, 1, 9, 6 | nvinv 30147 | . . 3 β’ ((π β NrmCVec β§ π΄ β π) β (-1ππ΄) = ((invβπΊ)βπ΄)) |
| 11 | 10 | oveq1d 7426 | . 2 β’ ((π β NrmCVec β§ π΄ β π) β ((-1ππ΄)πΊπ΄) = (((invβπΊ)βπ΄)πΊπ΄)) |
| 12 | nvrinv.6 | . . . 4 β’ π = (0vecβπ) | |
| 13 | 1, 12 | 0vfval 30114 | . . 3 β’ (π β NrmCVec β π = (GIdβπΊ)) |
| 14 | 13 | adantr 481 | . 2 β’ ((π β NrmCVec β§ π΄ β π) β π = (GIdβπΊ)) |
| 15 | 8, 11, 14 | 3eqtr4d 2782 | 1 β’ ((π β NrmCVec β§ π΄ β π) β ((-1ππ΄)πΊπ΄) = π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 βcfv 6543 (class class class)co 7411 1c1 11113 -cneg 11449 GrpOpcgr 29997 GIdcgi 29998 invcgn 29999 NrmCVeccnv 30092 +π£ cpv 30093 BaseSetcba 30094 Β·π OLD cns 30095 0veccn0v 30096 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-ltxr 11257 df-sub 11450 df-neg 11451 df-grpo 30001 df-gid 30002 df-ginv 30003 df-ablo 30053 df-vc 30067 df-nv 30100 df-va 30103 df-ba 30104 df-sm 30105 df-0v 30106 df-nmcv 30108 |
| This theorem is referenced by: nvabs 30180 imsmetlem 30198 lno0 30264 |
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