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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > gsumvsmul1 | Structured version Visualization version GIF version |
Description: Pull a scalar multiplication out of a sum of vectors. This theorem properly generalizes gsummulc1 20254, since every ring is a left module over itself. (Contributed by Thierry Arnoux, 12-Jun-2023.) |
Ref | Expression |
---|---|
gsumvsmul1.b | β’ π΅ = (Baseβπ ) |
gsumvsmul1.s | β’ π = (Scalarβπ ) |
gsumvsmul1.k | β’ πΎ = (Baseβπ) |
gsumvsmul1.z | β’ 0 = (0gβπ) |
gsumvsmul1.t | β’ Β· = ( Β·π βπ ) |
gsumvsmul1.r | β’ (π β π β LMod) |
gsumvsmul1.1 | β’ (π β π β CMnd) |
gsumvsmul1.a | β’ (π β π΄ β π) |
gsumvsmul1.x | β’ (π β π β π΅) |
gsumvsmul1.y | β’ ((π β§ π β π΄) β π β πΎ) |
gsumvsmul1.n | β’ (π β (π β π΄ β¦ π) finSupp 0 ) |
Ref | Expression |
---|---|
gsumvsmul1 | β’ (π β (π Ξ£g (π β π΄ β¦ (π Β· π))) = ((π Ξ£g (π β π΄ β¦ π)) Β· π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumvsmul1.k | . 2 β’ πΎ = (Baseβπ) | |
2 | gsumvsmul1.z | . 2 β’ 0 = (0gβπ) | |
3 | gsumvsmul1.1 | . 2 β’ (π β π β CMnd) | |
4 | gsumvsmul1.r | . . 3 β’ (π β π β LMod) | |
5 | lmodcmn 20795 | . . 3 β’ (π β LMod β π β CMnd) | |
6 | cmnmnd 19754 | . . 3 β’ (π β CMnd β π β Mnd) | |
7 | 4, 5, 6 | 3syl 18 | . 2 β’ (π β π β Mnd) |
8 | gsumvsmul1.a | . 2 β’ (π β π΄ β π) | |
9 | gsumvsmul1.x | . . . 4 β’ (π β π β π΅) | |
10 | gsumvsmul1.b | . . . . 5 β’ π΅ = (Baseβπ ) | |
11 | gsumvsmul1.s | . . . . 5 β’ π = (Scalarβπ ) | |
12 | gsumvsmul1.t | . . . . 5 β’ Β· = ( Β·π βπ ) | |
13 | 10, 11, 12, 1 | lmodvslmhm 32807 | . . . 4 β’ ((π β LMod β§ π β π΅) β (π₯ β πΎ β¦ (π₯ Β· π)) β (π GrpHom π )) |
14 | 4, 9, 13 | syl2anc 582 | . . 3 β’ (π β (π₯ β πΎ β¦ (π₯ Β· π)) β (π GrpHom π )) |
15 | ghmmhm 19182 | . . 3 β’ ((π₯ β πΎ β¦ (π₯ Β· π)) β (π GrpHom π ) β (π₯ β πΎ β¦ (π₯ Β· π)) β (π MndHom π )) | |
16 | 14, 15 | syl 17 | . 2 β’ (π β (π₯ β πΎ β¦ (π₯ Β· π)) β (π MndHom π )) |
17 | gsumvsmul1.y | . 2 β’ ((π β§ π β π΄) β π β πΎ) | |
18 | gsumvsmul1.n | . 2 β’ (π β (π β π΄ β¦ π) finSupp 0 ) | |
19 | oveq1 7422 | . 2 β’ (π₯ = π β (π₯ Β· π) = (π Β· π)) | |
20 | oveq1 7422 | . 2 β’ (π₯ = (π Ξ£g (π β π΄ β¦ π)) β (π₯ Β· π) = ((π Ξ£g (π β π΄ β¦ π)) Β· π)) | |
21 | 1, 2, 3, 7, 8, 16, 17, 18, 19, 20 | gsummhm2 19896 | 1 β’ (π β (π Ξ£g (π β π΄ β¦ (π Β· π))) = ((π Ξ£g (π β π΄ β¦ π)) Β· π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 class class class wbr 5143 β¦ cmpt 5226 βcfv 6542 (class class class)co 7415 finSupp cfsupp 9383 Basecbs 17177 Scalarcsca 17233 Β·π cvsca 17234 0gc0g 17418 Ξ£g cgsu 17419 Mndcmnd 18691 MndHom cmhm 18735 GrpHom cghm 19169 CMndccmn 19737 LModclmod 20745 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-map 8843 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fsupp 9384 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-n0 12501 df-z 12587 df-uz 12851 df-fz 13515 df-fzo 13658 df-seq 13997 df-hash 14320 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-plusg 17243 df-0g 17420 df-gsum 17421 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18737 df-grp 18895 df-minusg 18896 df-ghm 19170 df-cntz 19270 df-cmn 19739 df-abl 19740 df-mgp 20077 df-ur 20124 df-ring 20177 df-lmod 20747 |
This theorem is referenced by: (None) |
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