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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mnringbasedOLD | Structured version Visualization version GIF version |
Description: Obsolete version of mnringnmulrd 42563 as of 1-Nov-2024. The base set of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
mnringbased.1 | β’ πΉ = (π MndRing π) |
mnringbased.2 | β’ π΄ = (Baseβπ) |
mnringbased.3 | β’ π = (π freeLMod π΄) |
mnringbased.4 | β’ π΅ = (Baseβπ) |
mnringbased.5 | β’ (π β π β π) |
mnringbased.6 | β’ (π β π β π) |
Ref | Expression |
---|---|
mnringbasedOLD | β’ (π β π΅ = (BaseβπΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnringbased.4 | . 2 β’ π΅ = (Baseβπ) | |
2 | mnringbased.1 | . . 3 β’ πΉ = (π MndRing π) | |
3 | df-base 17091 | . . 3 β’ Base = Slot 1 | |
4 | 1nn 12171 | . . 3 β’ 1 β β | |
5 | 1re 11162 | . . . . 5 β’ 1 β β | |
6 | 1lt3 12333 | . . . . 5 β’ 1 < 3 | |
7 | 5, 6 | ltneii 11275 | . . . 4 β’ 1 β 3 |
8 | mulrndx 17181 | . . . 4 β’ (.rβndx) = 3 | |
9 | 7, 8 | neeqtrri 3018 | . . 3 β’ 1 β (.rβndx) |
10 | mnringbased.2 | . . 3 β’ π΄ = (Baseβπ) | |
11 | mnringbased.3 | . . 3 β’ π = (π freeLMod π΄) | |
12 | mnringbased.5 | . . 3 β’ (π β π β π) | |
13 | mnringbased.6 | . . 3 β’ (π β π β π) | |
14 | 2, 3, 4, 9, 10, 11, 12, 13 | mnringnmulrdOLD 42564 | . 2 β’ (π β (Baseβπ) = (BaseβπΉ)) |
15 | 1, 14 | eqtrid 2789 | 1 β’ (π β π΅ = (BaseβπΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βcfv 6501 (class class class)co 7362 1c1 11059 3c3 12216 ndxcnx 17072 Basecbs 17090 .rcmulr 17141 freeLMod cfrlm 21168 MndRing cmnring 42560 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-3 12224 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-mulr 17154 df-mnring 42561 |
This theorem is referenced by: (None) |
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