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| Mirrors > Home > MPE Home > Th. List > nvaddsub | Structured version Visualization version GIF version | ||
| Description: Commutative/associative law for vector addition and subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| nvpncan2.1 | β’ π = (BaseSetβπ) |
| nvpncan2.2 | β’ πΊ = ( +π£ βπ) |
| nvpncan2.3 | β’ π = ( βπ£ βπ) |
| Ref | Expression |
|---|---|
| nvaddsub | β’ ((π β NrmCVec β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((π΄πΊπ΅)ππΆ) = ((π΄ππΆ)πΊπ΅)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nvpncan2.2 | . . 3 β’ πΊ = ( +π£ βπ) | |
| 2 | 1 | nvablo 30304 | . 2 β’ (π β NrmCVec β πΊ β AbelOp) |
| 3 | nvpncan2.1 | . . . 4 β’ π = (BaseSetβπ) | |
| 4 | 3, 1 | bafval 30292 | . . 3 β’ π = ran πΊ |
| 5 | nvpncan2.3 | . . . 4 β’ π = ( βπ£ βπ) | |
| 6 | 1, 5 | vsfval 30321 | . . 3 β’ π = ( /π βπΊ) |
| 7 | 4, 6 | ablomuldiv 30240 | . 2 β’ ((πΊ β AbelOp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((π΄πΊπ΅)ππΆ) = ((π΄ππΆ)πΊπ΅)) |
| 8 | 2, 7 | sylan 579 | 1 β’ ((π β NrmCVec β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((π΄πΊπ΅)ππΆ) = ((π΄ππΆ)πΊπ΅)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6533 (class class class)co 7401 AbelOpcablo 30232 NrmCVeccnv 30272 +π£ cpv 30273 BaseSetcba 30274 βπ£ cnsb 30277 |
| 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 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
| 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 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-grpo 30181 df-gid 30182 df-ginv 30183 df-gdiv 30184 df-ablo 30233 df-vc 30247 df-nv 30280 df-va 30283 df-ba 30284 df-sm 30285 df-0v 30286 df-vs 30287 df-nmcv 30288 |
| This theorem is referenced by: nvnpcan 30344 |
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