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Mirrors > Home > MPE Home > Th. List > Mathboxes > mnringvscadOLD | Structured version Visualization version GIF version |
Description: Obsolete version of mnringvscad 42740 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 17193 | . 2 β’ Β·π = Slot 6 | |
3 | 6nn 12280 | . 2 β’ 6 β β | |
4 | 3re 12271 | . . . 4 β’ 3 β β | |
5 | 3lt6 12374 | . . . 4 β’ 3 < 6 | |
6 | 4, 5 | gtneii 11305 | . . 3 β’ 6 β 3 |
7 | mulrndx 17217 | . . 3 β’ (.rβndx) = 3 | |
8 | 6, 7 | neeqtrri 3013 | . 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 42726 | 1 β’ (π β ( Β·π βπ) = ( Β·π βπΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 βcfv 6529 (class class class)co 7390 3c3 12247 6c6 12250 ndxcnx 17105 Basecbs 17123 .rcmulr 17177 Β·π cvsca 17180 freeLMod cfrlm 21229 MndRing cmnring 42722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7705 ax-cnex 11145 ax-resscn 11146 ax-1cn 11147 ax-icn 11148 ax-addcl 11149 ax-addrcl 11150 ax-mulcl 11151 ax-mulrcl 11152 ax-mulcom 11153 ax-addass 11154 ax-mulass 11155 ax-distr 11156 ax-i2m1 11157 ax-1ne0 11158 ax-1rid 11159 ax-rnegex 11160 ax-rrecex 11161 ax-cnre 11162 ax-pre-lttri 11163 ax-pre-lttrn 11164 ax-pre-ltadd 11165 ax-pre-mulgt0 11166 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3430 df-v 3472 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4520 df-pw 4595 df-sn 4620 df-pr 4622 df-op 4626 df-uni 4899 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6286 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6531 df-fn 6532 df-f 6533 df-f1 6534 df-fo 6535 df-f1o 6536 df-fv 6537 df-riota 7346 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7836 df-2nd 7955 df-frecs 8245 df-wrecs 8276 df-recs 8350 df-rdg 8389 df-er 8683 df-en 8920 df-dom 8921 df-sdom 8922 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-le 11233 df-sub 11425 df-neg 11426 df-nn 12192 df-2 12254 df-3 12255 df-4 12256 df-5 12257 df-6 12258 df-sets 17076 df-slot 17094 df-ndx 17106 df-mulr 17190 df-vsca 17193 df-mnring 42723 |
This theorem is referenced by: (None) |
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