Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > nv0 | Structured version Visualization version GIF version | ||
| Description: Zero times a vector is the zero vector. (Contributed by NM, 27-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| nv0.1 | β’ π = (BaseSetβπ) |
| nv0.4 | β’ π = ( Β·π OLD βπ) |
| nv0.6 | β’ π = (0vecβπ) |
| Ref | Expression |
|---|---|
| nv0 | β’ ((π β NrmCVec β§ π΄ β π) β (0ππ΄) = π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2732 | . . . 4 β’ (1st βπ) = (1st βπ) | |
| 2 | 1 | nvvc 29906 | . . 3 β’ (π β NrmCVec β (1st βπ) β CVecOLD) |
| 3 | eqid 2732 | . . . . 5 β’ ( +π£ βπ) = ( +π£ βπ) | |
| 4 | 3 | vafval 29894 | . . . 4 β’ ( +π£ βπ) = (1st β(1st βπ)) |
| 5 | nv0.4 | . . . . 5 β’ π = ( Β·π OLD βπ) | |
| 6 | 5 | smfval 29896 | . . . 4 β’ π = (2nd β(1st βπ)) |
| 7 | nv0.1 | . . . . 5 β’ π = (BaseSetβπ) | |
| 8 | 7, 3 | bafval 29895 | . . . 4 β’ π = ran ( +π£ βπ) |
| 9 | eqid 2732 | . . . 4 β’ (GIdβ( +π£ βπ)) = (GIdβ( +π£ βπ)) | |
| 10 | 4, 6, 8, 9 | vc0 29865 | . . 3 β’ (((1st βπ) β CVecOLD β§ π΄ β π) β (0ππ΄) = (GIdβ( +π£ βπ))) |
| 11 | 2, 10 | sylan 580 | . 2 β’ ((π β NrmCVec β§ π΄ β π) β (0ππ΄) = (GIdβ( +π£ βπ))) |
| 12 | nv0.6 | . . . 4 β’ π = (0vecβπ) | |
| 13 | 3, 12 | 0vfval 29897 | . . 3 β’ (π β NrmCVec β π = (GIdβ( +π£ βπ))) |
| 14 | 13 | adantr 481 | . 2 β’ ((π β NrmCVec β§ π΄ β π) β π = (GIdβ( +π£ βπ))) |
| 15 | 11, 14 | eqtr4d 2775 | 1 β’ ((π β NrmCVec β§ π΄ β π) β (0ππ΄) = 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 1st c1st 7975 0cc0 11112 GIdcgi 29781 CVecOLDcvc 29849 NrmCVeccnv 29875 +π£ cpv 29876 BaseSetcba 29877 Β·π OLD cns 29878 0veccn0v 29879 |
| 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-1st 7977 df-2nd 7978 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11252 df-mnf 11253 df-ltxr 11255 df-grpo 29784 df-gid 29785 df-ginv 29786 df-ablo 29836 df-vc 29850 df-nv 29883 df-va 29886 df-ba 29887 df-sm 29888 df-0v 29889 df-nmcv 29891 |
| This theorem is referenced by: nvmul0or 29941 nvz0 29959 nvge0 29964 ipasslem1 30122 hlmul0 30200 |
| Copyright terms: Public domain | W3C validator |