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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 30364 | . 2 β’ (π β NrmCVec β πΊ β AbelOp) |
| 3 | nvgcl.1 | . . . 4 β’ π = (BaseSetβπ) | |
| 4 | 3, 1 | bafval 30352 | . . 3 β’ π = ran πΊ |
| 5 | 4 | ablocom 30296 | . 2 β’ ((πΊ β AbelOp β§ π΄ β π β§ π΅ β π) β (π΄πΊπ΅) = (π΅πΊπ΄)) |
| 6 | 2, 5 | syl3an1 1160 | 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 1084 = wceq 1533 β wcel 2098 βcfv 6534 (class class class)co 7402 AbelOpcablo 30292 NrmCVeccnv 30332 +π£ cpv 30333 BaseSetcba 30334 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pr 5418 ax-un 7719 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-1st 7969 df-2nd 7970 df-ablo 30293 df-vc 30307 df-nv 30340 df-va 30343 df-ba 30344 df-sm 30345 df-0v 30346 df-nmcv 30348 |
| This theorem is referenced by: nvmval2 30391 nvpncan 30402 nvdif 30414 nvpi 30415 nvabs 30420 dipcj 30462 hlcom 30648 |
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