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Mirrors > Home > MPE Home > Th. List > rlmval2 | Structured version Visualization version GIF version |
Description: Value of the ring module extended. (Contributed by AV, 2-Dec-2018.) (Revised by Thierry Arnoux, 16-Jun-2019.) |
Ref | Expression |
---|---|
rlmval2 | β’ (π β π β (ringLModβπ) = (((π sSet β¨(Scalarβndx), πβ©) sSet β¨( Β·π βndx), (.rβπ)β©) sSet β¨(Β·πβndx), (.rβπ)β©)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rlmval 21091 | . . 3 β’ (ringLModβπ) = ((subringAlg βπ)β(Baseβπ)) | |
2 | 1 | a1i 11 | . 2 β’ (π β π β (ringLModβπ) = ((subringAlg βπ)β(Baseβπ))) |
3 | ssid 4004 | . . 3 β’ (Baseβπ) β (Baseβπ) | |
4 | sraval 21067 | . . 3 β’ ((π β π β§ (Baseβπ) β (Baseβπ)) β ((subringAlg βπ)β(Baseβπ)) = (((π sSet β¨(Scalarβndx), (π βΎs (Baseβπ))β©) sSet β¨( Β·π βndx), (.rβπ)β©) sSet β¨(Β·πβndx), (.rβπ)β©)) | |
5 | 3, 4 | mpan2 689 | . 2 β’ (π β π β ((subringAlg βπ)β(Baseβπ)) = (((π sSet β¨(Scalarβndx), (π βΎs (Baseβπ))β©) sSet β¨( Β·π βndx), (.rβπ)β©) sSet β¨(Β·πβndx), (.rβπ)β©)) |
6 | eqid 2728 | . . . . . . 7 β’ (Baseβπ) = (Baseβπ) | |
7 | 6 | ressid 17232 | . . . . . 6 β’ (π β π β (π βΎs (Baseβπ)) = π) |
8 | 7 | opeq2d 4885 | . . . . 5 β’ (π β π β β¨(Scalarβndx), (π βΎs (Baseβπ))β© = β¨(Scalarβndx), πβ©) |
9 | 8 | oveq2d 7442 | . . . 4 β’ (π β π β (π sSet β¨(Scalarβndx), (π βΎs (Baseβπ))β©) = (π sSet β¨(Scalarβndx), πβ©)) |
10 | 9 | oveq1d 7441 | . . 3 β’ (π β π β ((π sSet β¨(Scalarβndx), (π βΎs (Baseβπ))β©) sSet β¨( Β·π βndx), (.rβπ)β©) = ((π sSet β¨(Scalarβndx), πβ©) sSet β¨( Β·π βndx), (.rβπ)β©)) |
11 | 10 | oveq1d 7441 | . 2 β’ (π β π β (((π sSet β¨(Scalarβndx), (π βΎs (Baseβπ))β©) sSet β¨( Β·π βndx), (.rβπ)β©) sSet β¨(Β·πβndx), (.rβπ)β©) = (((π sSet β¨(Scalarβndx), πβ©) sSet β¨( Β·π βndx), (.rβπ)β©) sSet β¨(Β·πβndx), (.rβπ)β©)) |
12 | 2, 5, 11 | 3eqtrd 2772 | 1 β’ (π β π β (ringLModβπ) = (((π sSet β¨(Scalarβndx), πβ©) sSet β¨( Β·π βndx), (.rβπ)β©) sSet β¨(Β·πβndx), (.rβπ)β©)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wss 3949 β¨cop 4638 βcfv 6553 (class class class)co 7426 sSet csts 17139 ndxcnx 17169 Basecbs 17187 βΎs cress 17216 .rcmulr 17241 Scalarcsca 17243 Β·π cvsca 17244 Β·πcip 17245 subringAlg csra 21063 ringLModcrglmod 21064 |
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-pow 5369 ax-pr 5433 |
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-pw 4608 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-mpo 7431 df-ress 17217 df-sra 21065 df-rgmod 21066 |
This theorem is referenced by: (None) |
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