Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > nv0lid | Structured version Visualization version GIF version | ||
| Description: The zero vector is a left identity element. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| nv0id.1 | β’ π = (BaseSetβπ) |
| nv0id.2 | β’ πΊ = ( +π£ βπ) |
| nv0id.6 | β’ π = (0vecβπ) |
| Ref | Expression |
|---|---|
| nv0lid | β’ ((π β NrmCVec β§ π΄ β π) β (ππΊπ΄) = π΄) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nv0id.2 | . . . . 5 β’ πΊ = ( +π£ βπ) | |
| 2 | nv0id.6 | . . . . 5 β’ π = (0vecβπ) | |
| 3 | 1, 2 | 0vfval 30124 | . . . 4 β’ (π β NrmCVec β π = (GIdβπΊ)) |
| 4 | 3 | oveq1d 7428 | . . 3 β’ (π β NrmCVec β (ππΊπ΄) = ((GIdβπΊ)πΊπ΄)) |
| 5 | 4 | adantr 479 | . 2 β’ ((π β NrmCVec β§ π΄ β π) β (ππΊπ΄) = ((GIdβπΊ)πΊπ΄)) |
| 6 | 1 | nvgrp 30135 | . . 3 β’ (π β NrmCVec β πΊ β GrpOp) |
| 7 | nv0id.1 | . . . . 5 β’ π = (BaseSetβπ) | |
| 8 | 7, 1 | bafval 30122 | . . . 4 β’ π = ran πΊ |
| 9 | eqid 2730 | . . . 4 β’ (GIdβπΊ) = (GIdβπΊ) | |
| 10 | 8, 9 | grpolid 30034 | . . 3 β’ ((πΊ β GrpOp β§ π΄ β π) β ((GIdβπΊ)πΊπ΄) = π΄) |
| 11 | 6, 10 | sylan 578 | . 2 β’ ((π β NrmCVec β§ π΄ β π) β ((GIdβπΊ)πΊπ΄) = π΄) |
| 12 | 5, 11 | eqtrd 2770 | 1 β’ ((π β NrmCVec β§ π΄ β π) β (ππΊπ΄) = π΄) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 394 = wceq 1539 β wcel 2104 βcfv 6544 (class class class)co 7413 GrpOpcgr 30007 GIdcgi 30008 NrmCVeccnv 30102 +π£ cpv 30103 BaseSetcba 30104 0veccn0v 30106 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7729 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-1st 7979 df-2nd 7980 df-grpo 30011 df-gid 30012 df-ablo 30063 df-vc 30077 df-nv 30110 df-va 30113 df-ba 30114 df-sm 30115 df-0v 30116 df-nmcv 30118 |
| This theorem is referenced by: nvpncan2 30171 nvmeq0 30176 imsmetlem 30208 ipdirilem 30347 |
| Copyright terms: Public domain | W3C validator |