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| Mirrors > Home > MPE Home > Th. List > nvcom | Structured version Visualization version GIF version | ||
| Description: The vector addition (group) operation is commutative. (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 |
|---|---|
| nvcom | β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (π΄πΊπ΅) = (π΅πΊπ΄)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nvgcl.2 | . . 3 β’ πΊ = ( +π£ βπ) | |
| 2 | 1 | nvablo 30439 | . 2 β’ (π β NrmCVec β πΊ β AbelOp) |
| 3 | nvgcl.1 | . . . 4 β’ π = (BaseSetβπ) | |
| 4 | 3, 1 | bafval 30427 | . . 3 β’ π = ran πΊ |
| 5 | 4 | ablocom 30371 | . 2 β’ ((πΊ β AbelOp β§ π΄ β π β§ π΅ β π) β (π΄πΊπ΅) = (π΅πΊπ΄)) |
| 6 | 2, 5 | syl3an1 1161 | 1 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (π΄πΊπ΅) = (π΅πΊπ΄)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ w3a 1085 = wceq 1534 β wcel 2099 βcfv 6548 (class class class)co 7420 AbelOpcablo 30367 NrmCVeccnv 30407 +π£ cpv 30408 BaseSetcba 30409 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-1st 7993 df-2nd 7994 df-ablo 30368 df-vc 30382 df-nv 30415 df-va 30418 df-ba 30419 df-sm 30420 df-0v 30421 df-nmcv 30423 |
| This theorem is referenced by: nvmval2 30466 nvpncan 30477 nvdif 30489 nvpi 30490 nvabs 30495 dipcj 30537 hlcom 30723 |
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