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| Mirrors > Home > MPE Home > Th. List > nvadd32 | Structured version Visualization version GIF version | ||
| Description: Commutative/associative law for vector addition. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| nvgcl.1 | β’ π = (BaseSetβπ) |
| nvgcl.2 | β’ πΊ = ( +π£ βπ) |
| Ref | Expression |
|---|---|
| nvadd32 | β’ ((π β NrmCVec β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((π΄πΊπ΅)πΊπΆ) = ((π΄πΊπΆ)πΊπ΅)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nvgcl.2 | . . 3 β’ πΊ = ( +π£ βπ) | |
| 2 | 1 | nvablo 30446 | . 2 β’ (π β NrmCVec β πΊ β AbelOp) |
| 3 | nvgcl.1 | . . . 4 β’ π = (BaseSetβπ) | |
| 4 | 3, 1 | bafval 30434 | . . 3 β’ π = ran πΊ |
| 5 | 4 | ablo32 30379 | . 2 β’ ((πΊ β AbelOp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((π΄πΊπ΅)πΊπΆ) = ((π΄πΊπΆ)πΊπ΅)) |
| 6 | 2, 5 | sylan 578 | 1 β’ ((π β NrmCVec β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((π΄πΊπ΅)πΊπΆ) = ((π΄πΊπΆ)πΊπ΅)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6553 (class class class)co 7426 AbelOpcablo 30374 NrmCVeccnv 30414 +π£ cpv 30415 BaseSetcba 30416 |
| 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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-1st 7999 df-2nd 8000 df-grpo 30323 df-ablo 30375 df-vc 30389 df-nv 30422 df-va 30425 df-ba 30426 df-sm 30427 df-0v 30428 df-nmcv 30430 |
| This theorem is referenced by: nvpncan2 30483 |
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