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Mirrors > Home > MPE Home > Th. List > assaassr | Structured version Visualization version GIF version |
Description: Right-associative property of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.) |
Ref | Expression |
---|---|
isassa.v | β’ π = (Baseβπ) |
isassa.f | β’ πΉ = (Scalarβπ) |
isassa.b | β’ π΅ = (BaseβπΉ) |
isassa.s | β’ Β· = ( Β·π βπ) |
isassa.t | β’ Γ = (.rβπ) |
Ref | Expression |
---|---|
assaassr | β’ ((π β AssAlg β§ (π΄ β π΅ β§ π β π β§ π β π)) β (π Γ (π΄ Β· π)) = (π΄ Β· (π Γ π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isassa.v | . . 3 β’ π = (Baseβπ) | |
2 | isassa.f | . . 3 β’ πΉ = (Scalarβπ) | |
3 | isassa.b | . . 3 β’ π΅ = (BaseβπΉ) | |
4 | isassa.s | . . 3 β’ Β· = ( Β·π βπ) | |
5 | isassa.t | . . 3 β’ Γ = (.rβπ) | |
6 | 1, 2, 3, 4, 5 | assalem 21752 | . 2 β’ ((π β AssAlg β§ (π΄ β π΅ β§ π β π β§ π β π)) β (((π΄ Β· π) Γ π) = (π΄ Β· (π Γ π)) β§ (π Γ (π΄ Β· π)) = (π΄ Β· (π Γ π)))) |
7 | 6 | simprd 495 | 1 β’ ((π β AssAlg β§ (π΄ β π΅ β§ π β π β§ π β π)) β (π Γ (π΄ Β· π)) = (π΄ Β· (π Γ π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6537 (class class class)co 7405 Basecbs 17153 .rcmulr 17207 Scalarcsca 17209 Β·π cvsca 17210 AssAlgcasa 21745 |
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-ext 2697 ax-nul 5299 |
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-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-iota 6489 df-fv 6545 df-ov 7408 df-assa 21748 |
This theorem is referenced by: assa2ass 21758 issubassa3 21760 sraassab 21762 asclmul2 21781 assamulgscmlem2 21794 mplmon2mul 21972 matinv 22534 cpmadugsumlemC 22732 |
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