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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mnringnmulrdOLD | Structured version Visualization version GIF version |
Description: Obsolete version of mnringnmulrd 42581 as of 1-Nov-2024. Components of a monoid ring other than its ring product match its underlying free module. (Contributed by Rohan Ridenour, 14-May-2024.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
mnringnmulrdOLD.1 | β’ πΉ = (π MndRing π) |
mnringnmulrdOLD.2 | β’ πΈ = Slot π |
mnringnmulrdOLD.3 | β’ π β β |
mnringnmulrdOLD.4 | β’ π β (.rβndx) |
mnringnmulrdOLD.5 | β’ π΄ = (Baseβπ) |
mnringnmulrdOLD.6 | β’ π = (π freeLMod π΄) |
mnringnmulrdOLD.7 | β’ (π β π β π) |
mnringnmulrdOLD.8 | β’ (π β π β π) |
Ref | Expression |
---|---|
mnringnmulrdOLD | β’ (π β (πΈβπ) = (πΈβπΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnringnmulrdOLD.2 | . . . 4 β’ πΈ = Slot π | |
2 | mnringnmulrdOLD.3 | . . . 4 β’ π β β | |
3 | 1, 2 | ndxid 17077 | . . 3 β’ πΈ = Slot (πΈβndx) |
4 | 1, 2 | ndxarg 17076 | . . . 4 β’ (πΈβndx) = π |
5 | mnringnmulrdOLD.4 | . . . 4 β’ π β (.rβndx) | |
6 | 4, 5 | eqnetri 3011 | . . 3 β’ (πΈβndx) β (.rβndx) |
7 | 3, 6 | setsnid 17089 | . 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©)) |
8 | mnringnmulrdOLD.1 | . . . 4 β’ πΉ = (π MndRing π) | |
9 | eqid 2733 | . . . 4 β’ (.rβπ ) = (.rβπ ) | |
10 | eqid 2733 | . . . 4 β’ (0gβπ ) = (0gβπ ) | |
11 | mnringnmulrdOLD.5 | . . . 4 β’ π΄ = (Baseβπ) | |
12 | eqid 2733 | . . . 4 β’ (+gβπ) = (+gβπ) | |
13 | mnringnmulrdOLD.6 | . . . 4 β’ π = (π freeLMod π΄) | |
14 | eqid 2733 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
15 | mnringnmulrdOLD.7 | . . . 4 β’ (π β π β π) | |
16 | mnringnmulrdOLD.8 | . . . 4 β’ (π β π β π) | |
17 | 8, 9, 10, 11, 12, 13, 14, 15, 16 | mnringvald 42580 | . . 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©)) |
18 | 17 | fveq2d 6850 | . 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©))) |
19 | 7, 18 | 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 2940 ifcif 4490 β¨cop 4596 β¦ cmpt 5192 βcfv 6500 (class class class)co 7361 β cmpo 7363 βcn 12161 sSet csts 17043 Slot cslot 17061 ndxcnx 17073 Basecbs 17091 +gcplusg 17141 .rcmulr 17142 0gc0g 17329 Ξ£g cgsu 17330 freeLMod cfrlm 21175 MndRing cmnring 42578 |
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 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-1cn 11117 ax-addcl 11119 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-nn 12162 df-sets 17044 df-slot 17062 df-ndx 17074 df-mnring 42579 |
This theorem is referenced by: mnringbasedOLD 42584 mnringaddgdOLD 42590 mnringscadOLD 42595 mnringvscadOLD 42597 |
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