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Mirrors > Home > MPE Home > Th. List > gsumvsmul | Structured version Visualization version GIF version |
Description: Pull a scalar multiplication out of a sum of vectors. This theorem properly generalizes gsummulc2 20036, since every ring is a left module over itself. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.) (Revised by AV, 10-Jul-2019.) |
Ref | Expression |
---|---|
gsumvsmul.b | β’ π΅ = (Baseβπ ) |
gsumvsmul.s | β’ π = (Scalarβπ ) |
gsumvsmul.k | β’ πΎ = (Baseβπ) |
gsumvsmul.z | β’ 0 = (0gβπ ) |
gsumvsmul.p | β’ + = (+gβπ ) |
gsumvsmul.t | β’ Β· = ( Β·π βπ ) |
gsumvsmul.r | β’ (π β π β LMod) |
gsumvsmul.a | β’ (π β π΄ β π) |
gsumvsmul.x | β’ (π β π β πΎ) |
gsumvsmul.y | β’ ((π β§ π β π΄) β π β π΅) |
gsumvsmul.n | β’ (π β (π β π΄ β¦ π) finSupp 0 ) |
Ref | Expression |
---|---|
gsumvsmul | β’ (π β (π Ξ£g (π β π΄ β¦ (π Β· π))) = (π Β· (π Ξ£g (π β π΄ β¦ π)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumvsmul.b | . 2 β’ π΅ = (Baseβπ ) | |
2 | gsumvsmul.z | . 2 β’ 0 = (0gβπ ) | |
3 | gsumvsmul.r | . . 3 β’ (π β π β LMod) | |
4 | lmodcmn 20385 | . . 3 β’ (π β LMod β π β CMnd) | |
5 | 3, 4 | syl 17 | . 2 β’ (π β π β CMnd) |
6 | cmnmnd 19584 | . . 3 β’ (π β CMnd β π β Mnd) | |
7 | 5, 6 | syl 17 | . 2 β’ (π β π β Mnd) |
8 | gsumvsmul.a | . 2 β’ (π β π΄ β π) | |
9 | gsumvsmul.x | . . . 4 β’ (π β π β πΎ) | |
10 | gsumvsmul.s | . . . . 5 β’ π = (Scalarβπ ) | |
11 | gsumvsmul.t | . . . . 5 β’ Β· = ( Β·π βπ ) | |
12 | gsumvsmul.k | . . . . 5 β’ πΎ = (Baseβπ) | |
13 | 1, 10, 11, 12 | lmodvsghm 20398 | . . . 4 β’ ((π β LMod β§ π β πΎ) β (π¦ β π΅ β¦ (π Β· π¦)) β (π GrpHom π )) |
14 | 3, 9, 13 | syl2anc 585 | . . 3 β’ (π β (π¦ β π΅ β¦ (π Β· π¦)) β (π GrpHom π )) |
15 | ghmmhm 19023 | . . 3 β’ ((π¦ β π΅ β¦ (π Β· π¦)) β (π GrpHom π ) β (π¦ β π΅ β¦ (π Β· π¦)) β (π MndHom π )) | |
16 | 14, 15 | syl 17 | . 2 β’ (π β (π¦ β π΅ β¦ (π Β· π¦)) β (π MndHom π )) |
17 | gsumvsmul.y | . 2 β’ ((π β§ π β π΄) β π β π΅) | |
18 | gsumvsmul.n | . 2 β’ (π β (π β π΄ β¦ π) finSupp 0 ) | |
19 | oveq2 7366 | . 2 β’ (π¦ = π β (π Β· π¦) = (π Β· π)) | |
20 | oveq2 7366 | . 2 β’ (π¦ = (π Ξ£g (π β π΄ β¦ π)) β (π Β· π¦) = (π Β· (π Ξ£g (π β π΄ β¦ π)))) | |
21 | 1, 2, 5, 7, 8, 16, 17, 18, 19, 20 | gsummhm2 19721 | 1 β’ (π β (π Ξ£g (π β π΄ β¦ (π Β· π))) = (π Β· (π Ξ£g (π β π΄ β¦ π)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 class class class wbr 5106 β¦ cmpt 5189 βcfv 6497 (class class class)co 7358 finSupp cfsupp 9308 Basecbs 17088 +gcplusg 17138 Scalarcsca 17141 Β·π cvsca 17142 0gc0g 17326 Ξ£g cgsu 17327 Mndcmnd 18561 MndHom cmhm 18604 GrpHom cghm 19010 CMndccmn 19567 LModclmod 20336 |
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 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-fzo 13574 df-seq 13913 df-hash 14237 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-0g 17328 df-gsum 17329 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-mhm 18606 df-grp 18756 df-minusg 18757 df-ghm 19011 df-cntz 19102 df-cmn 19569 df-abl 19570 df-mgp 19902 df-ur 19919 df-ring 19971 df-lmod 20338 |
This theorem is referenced by: frlmup1 21220 lincscm 46597 lincresunit3lem2 46647 |
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