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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 6891 | . . . 4 β’ (π = π β (subringAlg βπ) = (subringAlg βπ)) | |
2 | fveq2 6891 | . . . 4 β’ (π = π β (Baseβπ) = (Baseβπ)) | |
3 | 1, 2 | fveq12d 6898 | . . 3 β’ (π = π β ((subringAlg βπ)β(Baseβπ)) = ((subringAlg βπ)β(Baseβπ))) |
4 | df-rgmod 21048 | . . 3 β’ ringLMod = (π β V β¦ ((subringAlg βπ)β(Baseβπ))) | |
5 | fvex 6904 | . . 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 6935 | . . . 4 β’ (β β(Baseβπ)) = β | |
8 | 7 | eqcomi 2736 | . . 3 β’ β = (β β(Baseβπ)) |
9 | fvprc 6883 | . . 3 β’ (Β¬ π β V β (ringLModβπ) = β ) | |
10 | fvprc 6883 | . . . 4 β’ (Β¬ π β V β (subringAlg βπ) = β ) | |
11 | 10 | fveq1d 6893 | . . 3 β’ (Β¬ π β V β ((subringAlg βπ)β(Baseβπ)) = (β β(Baseβπ))) |
12 | 8, 9, 11 | 3eqtr4a 2793 | . 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 1534 β wcel 2099 Vcvv 3469 β c0 4318 βcfv 6542 Basecbs 17171 subringAlg csra 21045 ringLModcrglmod 21046 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 df-rgmod 21048 |
This theorem is referenced by: rlmval2 21074 rlmbas 21075 rlmplusg 21076 rlm0 21077 rlmmulr 21079 rlmsca 21080 rlmsca2 21081 rlmvsca 21082 rlmtopn 21083 rlmds 21084 rlmlmod 21085 frlmip 21699 rlmassa 21791 rlmnlm 24592 rlmbn 25276 rrxprds 25304 rlmdim 33239 rgmoddimOLD 33240 extdgid 33284 |
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