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Mirrors > Home > MPE Home > Th. List > Mathboxes > mnringvscadOLD | Structured version Visualization version GIF version |
Description: Obsolete version of mnringvscad 41879 as of 1-Nov-2024. The scalar product of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
mnringvscad.1 | β’ πΉ = (π MndRing π) |
mnringvscad.2 | β’ π΅ = (Baseβπ) |
mnringvscad.3 | β’ π = (π freeLMod π΅) |
mnringvscad.4 | β’ (π β π β π) |
mnringvscad.5 | β’ (π β π β π) |
Ref | Expression |
---|---|
mnringvscadOLD | β’ (π β ( Β·π βπ) = ( Β·π βπΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnringvscad.1 | . 2 β’ πΉ = (π MndRing π) | |
2 | df-vsca 17020 | . 2 β’ Β·π = Slot 6 | |
3 | 6nn 12104 | . 2 β’ 6 β β | |
4 | 3re 12095 | . . . 4 β’ 3 β β | |
5 | 3lt6 12198 | . . . 4 β’ 3 < 6 | |
6 | 4, 5 | gtneii 11129 | . . 3 β’ 6 β 3 |
7 | mulrndx 17044 | . . 3 β’ (.rβndx) = 3 | |
8 | 6, 7 | neeqtrri 3015 | . 2 β’ 6 β (.rβndx) |
9 | mnringvscad.2 | . 2 β’ π΅ = (Baseβπ) | |
10 | mnringvscad.3 | . 2 β’ π = (π freeLMod π΅) | |
11 | mnringvscad.4 | . 2 β’ (π β π β π) | |
12 | mnringvscad.5 | . 2 β’ (π β π β π) | |
13 | 1, 2, 3, 8, 9, 10, 11, 12 | mnringnmulrdOLD 41865 | 1 β’ (π β ( Β·π βπ) = ( Β·π βπΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1539 β wcel 2104 βcfv 6454 (class class class)co 7303 3c3 12071 6c6 12074 ndxcnx 16935 Basecbs 16953 .rcmulr 17004 Β·π cvsca 17007 freeLMod cfrlm 20994 MndRing cmnring 41861 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7616 ax-cnex 10969 ax-resscn 10970 ax-1cn 10971 ax-icn 10972 ax-addcl 10973 ax-addrcl 10974 ax-mulcl 10975 ax-mulrcl 10976 ax-mulcom 10977 ax-addass 10978 ax-mulass 10979 ax-distr 10980 ax-i2m1 10981 ax-1ne0 10982 ax-1rid 10983 ax-rnegex 10984 ax-rrecex 10985 ax-cnre 10986 ax-pre-lttri 10987 ax-pre-lttrn 10988 ax-pre-ltadd 10989 ax-pre-mulgt0 10990 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3261 df-rab 3262 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5496 df-eprel 5502 df-po 5510 df-so 5511 df-fr 5551 df-we 5553 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 df-ima 5609 df-pred 6213 df-ord 6280 df-on 6281 df-lim 6282 df-suc 6283 df-iota 6406 df-fun 6456 df-fn 6457 df-f 6458 df-f1 6459 df-fo 6460 df-f1o 6461 df-fv 6462 df-riota 7260 df-ov 7306 df-oprab 7307 df-mpo 7308 df-om 7741 df-2nd 7860 df-frecs 8124 df-wrecs 8155 df-recs 8229 df-rdg 8268 df-er 8525 df-en 8761 df-dom 8762 df-sdom 8763 df-pnf 11053 df-mnf 11054 df-xr 11055 df-ltxr 11056 df-le 11057 df-sub 11249 df-neg 11250 df-nn 12016 df-2 12078 df-3 12079 df-4 12080 df-5 12081 df-6 12082 df-sets 16906 df-slot 16924 df-ndx 16936 df-mulr 17017 df-vsca 17020 df-mnring 41862 |
This theorem is referenced by: (None) |
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