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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 21798 | . 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 494 | 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 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6553 (class class class)co 7426 Basecbs 17187 .rcmulr 17241 Scalarcsca 17243 Β·π cvsca 17244 AssAlgcasa 21791 |
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 2699 ax-nul 5310 |
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-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rab 3431 df-v 3475 df-sbc 3779 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-br 5153 df-iota 6505 df-fv 6561 df-ov 7429 df-assa 21794 |
This theorem is referenced by: assa2ass 21804 issubassa3 21806 sraassab 21808 asclmul2 21827 assamulgscmlem2 21840 mplmon2mul 22020 matinv 22599 cpmadugsumlemC 22797 |
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