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Mirrors > Home > MPE Home > Th. List > rlmval | Structured version Visualization version GIF version |
Description: Value of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.) |
Ref | Expression |
---|---|
rlmval | β’ (ringLModβπ) = ((subringAlg βπ)β(Baseβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6892 | . . . 4 β’ (π = π β (subringAlg βπ) = (subringAlg βπ)) | |
2 | fveq2 6892 | . . . 4 β’ (π = π β (Baseβπ) = (Baseβπ)) | |
3 | 1, 2 | fveq12d 6899 | . . 3 β’ (π = π β ((subringAlg βπ)β(Baseβπ)) = ((subringAlg βπ)β(Baseβπ))) |
4 | df-rgmod 20786 | . . 3 β’ ringLMod = (π β V β¦ ((subringAlg βπ)β(Baseβπ))) | |
5 | fvex 6905 | . . 3 β’ ((subringAlg βπ)β(Baseβπ)) β V | |
6 | 3, 4, 5 | fvmpt 6999 | . 2 β’ (π β V β (ringLModβπ) = ((subringAlg βπ)β(Baseβπ))) |
7 | 0fv 6936 | . . . 4 β’ (β β(Baseβπ)) = β | |
8 | 7 | eqcomi 2742 | . . 3 β’ β = (β β(Baseβπ)) |
9 | fvprc 6884 | . . 3 β’ (Β¬ π β V β (ringLModβπ) = β ) | |
10 | fvprc 6884 | . . . 4 β’ (Β¬ π β V β (subringAlg βπ) = β ) | |
11 | 10 | fveq1d 6894 | . . 3 β’ (Β¬ π β V β ((subringAlg βπ)β(Baseβπ)) = (β β(Baseβπ))) |
12 | 8, 9, 11 | 3eqtr4a 2799 | . 2 β’ (Β¬ π β V β (ringLModβπ) = ((subringAlg βπ)β(Baseβπ))) |
13 | 6, 12 | pm2.61i 182 | 1 β’ (ringLModβπ) = ((subringAlg βπ)β(Baseβπ)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 = wceq 1542 β wcel 2107 Vcvv 3475 β c0 4323 βcfv 6544 Basecbs 17144 subringAlg csra 20781 ringLModcrglmod 20782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-rgmod 20786 |
This theorem is referenced by: rlmval2 20816 rlmbas 20817 rlmplusg 20818 rlm0 20819 rlmmulr 20821 rlmsca 20822 rlmsca2 20823 rlmvsca 20824 rlmtopn 20825 rlmds 20826 rlmlmod 20827 frlmip 21333 rlmassa 21425 rlmnlm 24205 rlmbn 24878 rrxprds 24906 rlmdim 32694 rgmoddimOLD 32695 extdgid 32739 |
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