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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mnringnmulrd | Structured version Visualization version GIF version |
Description: Components of a monoid ring other than its ring product match its underlying free module. (Contributed by Rohan Ridenour, 14-May-2024.) (Revised by AV, 1-Nov-2024.) |
Ref | Expression |
---|---|
mnringnmulrd.1 | β’ πΉ = (π MndRing π) |
mnringnmulrd.2 | β’ πΈ = Slot (πΈβndx) |
mnringnmulrd.4 | β’ (πΈβndx) β (.rβndx) |
mnringnmulrd.5 | β’ π΄ = (Baseβπ) |
mnringnmulrd.6 | β’ π = (π freeLMod π΄) |
mnringnmulrd.7 | β’ (π β π β π) |
mnringnmulrd.8 | β’ (π β π β π) |
Ref | Expression |
---|---|
mnringnmulrd | β’ (π β (πΈβπ) = (πΈβπΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnringnmulrd.2 | . . 3 β’ πΈ = Slot (πΈβndx) | |
2 | mnringnmulrd.4 | . . 3 β’ (πΈβndx) β (.rβndx) | |
3 | 1, 2 | setsnid 17142 | . 2 β’ (πΈβπ) = (πΈβ(π sSet β¨(.rβndx), (π₯ β (Baseβπ), π¦ β (Baseβπ) β¦ (π Ξ£g (π β π΄, π β π΄ β¦ (π β π΄ β¦ if(π = (π(+gβπ)π), ((π₯βπ)(.rβπ )(π¦βπ)), (0gβπ ))))))β©)) |
4 | mnringnmulrd.1 | . . . 4 β’ πΉ = (π MndRing π) | |
5 | eqid 2733 | . . . 4 β’ (.rβπ ) = (.rβπ ) | |
6 | eqid 2733 | . . . 4 β’ (0gβπ ) = (0gβπ ) | |
7 | mnringnmulrd.5 | . . . 4 β’ π΄ = (Baseβπ) | |
8 | eqid 2733 | . . . 4 β’ (+gβπ) = (+gβπ) | |
9 | mnringnmulrd.6 | . . . 4 β’ π = (π freeLMod π΄) | |
10 | eqid 2733 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
11 | mnringnmulrd.7 | . . . 4 β’ (π β π β π) | |
12 | mnringnmulrd.8 | . . . 4 β’ (π β π β π) | |
13 | 4, 5, 6, 7, 8, 9, 10, 11, 12 | mnringvald 42967 | . . 3 β’ (π β πΉ = (π sSet β¨(.rβndx), (π₯ β (Baseβπ), π¦ β (Baseβπ) β¦ (π Ξ£g (π β π΄, π β π΄ β¦ (π β π΄ β¦ if(π = (π(+gβπ)π), ((π₯βπ)(.rβπ )(π¦βπ)), (0gβπ ))))))β©)) |
14 | 13 | fveq2d 6896 | . 2 β’ (π β (πΈβπΉ) = (πΈβ(π sSet β¨(.rβndx), (π₯ β (Baseβπ), π¦ β (Baseβπ) β¦ (π Ξ£g (π β π΄, π β π΄ β¦ (π β π΄ β¦ if(π = (π(+gβπ)π), ((π₯βπ)(.rβπ )(π¦βπ)), (0gβπ ))))))β©))) |
15 | 3, 14 | eqtr4id 2792 | 1 β’ (π β (πΈβπ) = (πΈβπΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β wne 2941 ifcif 4529 β¨cop 4635 β¦ cmpt 5232 βcfv 6544 (class class class)co 7409 β cmpo 7411 sSet csts 17096 Slot cslot 17114 ndxcnx 17126 Basecbs 17144 +gcplusg 17197 .rcmulr 17198 0gc0g 17385 Ξ£g cgsu 17386 freeLMod cfrlm 21301 MndRing cmnring 42965 |
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 ax-un 7725 |
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-sbc 3779 df-csb 3895 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-res 5689 df-iota 6496 df-fun 6546 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-sets 17097 df-slot 17115 df-mnring 42966 |
This theorem is referenced by: mnringbased 42970 mnringaddgd 42976 mnringscad 42981 mnringvscad 42983 |
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