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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 43271 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 17135 | . . 3 β’ πΈ = Slot (πΈβndx) |
4 | 1, 2 | ndxarg 17134 | . . . 4 β’ (πΈβndx) = π |
5 | mnringnmulrdOLD.4 | . . . 4 β’ π β (.rβndx) | |
6 | 4, 5 | eqnetri 3010 | . . 3 β’ (πΈβndx) β (.rβndx) |
7 | 3, 6 | setsnid 17147 | . 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 2731 | . . . 4 β’ (.rβπ ) = (.rβπ ) | |
10 | eqid 2731 | . . . 4 β’ (0gβπ ) = (0gβπ ) | |
11 | mnringnmulrdOLD.5 | . . . 4 β’ π΄ = (Baseβπ) | |
12 | eqid 2731 | . . . 4 β’ (+gβπ) = (+gβπ) | |
13 | mnringnmulrdOLD.6 | . . . 4 β’ π = (π freeLMod π΄) | |
14 | eqid 2731 | . . . 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 43270 | . . 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 6895 | . 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 2790 | 1 β’ (π β (πΈβπ) = (πΈβπΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1540 β wcel 2105 β wne 2939 ifcif 4528 β¨cop 4634 β¦ cmpt 5231 βcfv 6543 (class class class)co 7412 β cmpo 7414 βcn 12217 sSet csts 17101 Slot cslot 17119 ndxcnx 17131 Basecbs 17149 +gcplusg 17202 .rcmulr 17203 0gc0g 17390 Ξ£g cgsu 17391 freeLMod cfrlm 21521 MndRing cmnring 43268 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-1cn 11172 ax-addcl 11174 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-nn 12218 df-sets 17102 df-slot 17120 df-ndx 17132 df-mnring 43269 |
This theorem is referenced by: mnringbasedOLD 43274 mnringaddgdOLD 43280 mnringscadOLD 43285 mnringvscadOLD 43287 |
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