Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > nvass | Structured version Visualization version GIF version | ||
| Description: The vector addition (group) operation is associative. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| nvgcl.1 | β’ π = (BaseSetβπ) |
| nvgcl.2 | β’ πΊ = ( +π£ βπ) |
| Ref | Expression |
|---|---|
| nvass | β’ ((π β NrmCVec β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((π΄πΊπ΅)πΊπΆ) = (π΄πΊ(π΅πΊπΆ))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nvgcl.2 | . . 3 β’ πΊ = ( +π£ βπ) | |
| 2 | 1 | nvgrp 29908 | . 2 β’ (π β NrmCVec β πΊ β GrpOp) |
| 3 | nvgcl.1 | . . . 4 β’ π = (BaseSetβπ) | |
| 4 | 3, 1 | bafval 29895 | . . 3 β’ π = ran πΊ |
| 5 | 4 | grpoass 29794 | . 2 β’ ((πΊ β GrpOp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((π΄πΊπ΅)πΊπΆ) = (π΄πΊ(π΅πΊπΆ))) |
| 6 | 2, 5 | sylan 580 | 1 β’ ((π β NrmCVec β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((π΄πΊπ΅)πΊπΆ) = (π΄πΊ(π΅πΊπΆ))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 βcfv 6543 (class class class)co 7411 GrpOpcgr 29780 NrmCVeccnv 29875 +π£ cpv 29876 BaseSetcba 29877 |
| 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-pr 5427 ax-un 7727 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 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-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-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-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-ov 7414 df-oprab 7415 df-1st 7977 df-2nd 7978 df-grpo 29784 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: nvabs 29963 imsmetlem 29981 hlass 30192 |
| Copyright terms: Public domain | W3C validator |