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Mirrors > Home > MPE Home > Th. List > rlmlsm | Structured version Visualization version GIF version |
Description: Subgroup sum of the ring module. (Contributed by Thierry Arnoux, 9-Apr-2024.) |
Ref | Expression |
---|---|
rlmlsm | β’ (π β π β (LSSumβπ ) = (LSSumβ(ringLModβπ ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . 3 β’ (Baseβπ ) = (Baseβπ ) | |
2 | eqid 2736 | . . 3 β’ (+gβπ ) = (+gβπ ) | |
3 | eqid 2736 | . . 3 β’ (LSSumβπ ) = (LSSumβπ ) | |
4 | 1, 2, 3 | lsmfval 19340 | . 2 β’ (π β π β (LSSumβπ ) = (π‘ β π« (Baseβπ ), π’ β π« (Baseβπ ) β¦ ran (π₯ β π‘, π¦ β π’ β¦ (π₯(+gβπ )π¦)))) |
5 | fvex 6839 | . . 3 β’ (ringLModβπ ) β V | |
6 | rlmbas 20572 | . . . 4 β’ (Baseβπ ) = (Baseβ(ringLModβπ )) | |
7 | rlmplusg 20573 | . . . 4 β’ (+gβπ ) = (+gβ(ringLModβπ )) | |
8 | eqid 2736 | . . . 4 β’ (LSSumβ(ringLModβπ )) = (LSSumβ(ringLModβπ )) | |
9 | 6, 7, 8 | lsmfval 19340 | . . 3 β’ ((ringLModβπ ) β V β (LSSumβ(ringLModβπ )) = (π‘ β π« (Baseβπ ), π’ β π« (Baseβπ ) β¦ ran (π₯ β π‘, π¦ β π’ β¦ (π₯(+gβπ )π¦)))) |
10 | 5, 9 | mp1i 13 | . 2 β’ (π β π β (LSSumβ(ringLModβπ )) = (π‘ β π« (Baseβπ ), π’ β π« (Baseβπ ) β¦ ran (π₯ β π‘, π¦ β π’ β¦ (π₯(+gβπ )π¦)))) |
11 | 4, 10 | eqtr4d 2779 | 1 β’ (π β π β (LSSumβπ ) = (LSSumβ(ringLModβπ ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1540 β wcel 2105 Vcvv 3441 π« cpw 4548 ran crn 5622 βcfv 6480 (class class class)co 7338 β cmpo 7340 Basecbs 17010 +gcplusg 17060 LSSumclsm 19336 ringLModcrglmod 20538 |
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 2707 ax-rep 5230 ax-sep 5244 ax-nul 5251 ax-pow 5309 ax-pr 5373 ax-un 7651 ax-cnex 11029 ax-resscn 11030 ax-1cn 11031 ax-icn 11032 ax-addcl 11033 ax-addrcl 11034 ax-mulcl 11035 ax-mulrcl 11036 ax-mulcom 11037 ax-addass 11038 ax-mulass 11039 ax-distr 11040 ax-i2m1 11041 ax-1ne0 11042 ax-1rid 11043 ax-rnegex 11044 ax-rrecex 11045 ax-cnre 11046 ax-pre-lttri 11047 ax-pre-lttrn 11048 ax-pre-ltadd 11049 ax-pre-mulgt0 11050 |
This theorem depends on definitions: df-bi 206 df-an 397 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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4271 df-if 4475 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4854 df-iun 4944 df-br 5094 df-opab 5156 df-mpt 5177 df-tr 5211 df-id 5519 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5576 df-we 5578 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6239 df-ord 6306 df-on 6307 df-lim 6308 df-suc 6309 df-iota 6432 df-fun 6482 df-fn 6483 df-f 6484 df-f1 6485 df-fo 6486 df-f1o 6487 df-fv 6488 df-riota 7294 df-ov 7341 df-oprab 7342 df-mpo 7343 df-om 7782 df-1st 7900 df-2nd 7901 df-frecs 8168 df-wrecs 8199 df-recs 8273 df-rdg 8312 df-er 8570 df-en 8806 df-dom 8807 df-sdom 8808 df-pnf 11113 df-mnf 11114 df-xr 11115 df-ltxr 11116 df-le 11117 df-sub 11309 df-neg 11310 df-nn 12076 df-2 12138 df-3 12139 df-4 12140 df-5 12141 df-6 12142 df-7 12143 df-8 12144 df-sets 16963 df-slot 16981 df-ndx 16993 df-base 17011 df-plusg 17073 df-sca 17076 df-vsca 17077 df-ip 17078 df-lsm 19338 df-sra 20541 df-rgmod 20542 |
This theorem is referenced by: lsmidllsp 31885 |
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