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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 29857 | . 2 β’ (π β NrmCVec β πΊ β AbelOp) |
| 3 | nvgcl.1 | . . . 4 β’ π = (BaseSetβπ) | |
| 4 | 3, 1 | bafval 29845 | . . 3 β’ π = ran πΊ |
| 5 | 4 | ablocom 29789 | . 2 β’ ((πΊ β AbelOp β§ π΄ β π β§ π΅ β π) β (π΄πΊπ΅) = (π΅πΊπ΄)) |
| 6 | 2, 5 | syl3an1 1164 | 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 1088 = wceq 1542 β wcel 2107 βcfv 6541 (class class class)co 7406 AbelOpcablo 29785 NrmCVeccnv 29825 +π£ cpv 29826 BaseSetcba 29827 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7722 |
| This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 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 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7409 df-oprab 7410 df-1st 7972 df-2nd 7973 df-ablo 29786 df-vc 29800 df-nv 29833 df-va 29836 df-ba 29837 df-sm 29838 df-0v 29839 df-nmcv 29841 |
| This theorem is referenced by: nvmval2 29884 nvpncan 29895 nvdif 29907 nvpi 29908 nvabs 29913 dipcj 29955 hlcom 30141 |
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