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Mirrors > Home > MPE Home > Th. List > gsummhm | Structured version Visualization version GIF version |
Description: Apply a group homomorphism to a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 6-Jun-2019.) |
Ref | Expression |
---|---|
gsummhm.b | β’ π΅ = (BaseβπΊ) |
gsummhm.z | β’ 0 = (0gβπΊ) |
gsummhm.g | β’ (π β πΊ β CMnd) |
gsummhm.h | β’ (π β π» β Mnd) |
gsummhm.a | β’ (π β π΄ β π) |
gsummhm.k | β’ (π β πΎ β (πΊ MndHom π»)) |
gsummhm.f | β’ (π β πΉ:π΄βΆπ΅) |
gsummhm.w | β’ (π β πΉ finSupp 0 ) |
Ref | Expression |
---|---|
gsummhm | β’ (π β (π» Ξ£g (πΎ β πΉ)) = (πΎβ(πΊ Ξ£g πΉ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsummhm.b | . 2 β’ π΅ = (BaseβπΊ) | |
2 | eqid 2728 | . 2 β’ (CntzβπΊ) = (CntzβπΊ) | |
3 | gsummhm.g | . . 3 β’ (π β πΊ β CMnd) | |
4 | cmnmnd 19759 | . . 3 β’ (πΊ β CMnd β πΊ β Mnd) | |
5 | 3, 4 | syl 17 | . 2 β’ (π β πΊ β Mnd) |
6 | gsummhm.h | . 2 β’ (π β π» β Mnd) | |
7 | gsummhm.a | . 2 β’ (π β π΄ β π) | |
8 | gsummhm.k | . 2 β’ (π β πΎ β (πΊ MndHom π»)) | |
9 | gsummhm.f | . 2 β’ (π β πΉ:π΄βΆπ΅) | |
10 | 1, 2, 3, 9 | cntzcmnf 19807 | . 2 β’ (π β ran πΉ β ((CntzβπΊ)βran πΉ)) |
11 | gsummhm.z | . 2 β’ 0 = (0gβπΊ) | |
12 | gsummhm.w | . 2 β’ (π β πΉ finSupp 0 ) | |
13 | 1, 2, 5, 6, 7, 8, 9, 10, 11, 12 | gsumzmhm 19899 | 1 β’ (π β (π» Ξ£g (πΎ β πΉ)) = (πΎβ(πΊ Ξ£g πΉ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 class class class wbr 5152 β ccom 5686 βΆwf 6549 βcfv 6553 (class class class)co 7426 finSupp cfsupp 9393 Basecbs 17187 0gc0g 17428 Ξ£g cgsu 17429 Mndcmnd 18701 MndHom cmhm 18745 Cntzccntz 19273 CMndccmn 19742 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-fzo 13668 df-seq 14007 df-hash 14330 df-0g 17430 df-gsum 17431 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-cntz 19275 df-cmn 19744 |
This theorem is referenced by: gsummhm2 19901 gsummptmhm 19902 gsuminv 19908 evlslem2 22032 tsmsmhm 24070 plypf1 26166 amgmlem 26942 selvvvval 41849 evlselv 41851 amgmwlem 48313 amgmlemALT 48314 |
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