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Mirrors > Home > MPE Home > Th. List > gsumzinv | Structured version Visualization version GIF version |
Description: Inverse of a group sum. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 6-Jun-2019.) |
Ref | Expression |
---|---|
gsumzinv.b | β’ π΅ = (BaseβπΊ) |
gsumzinv.0 | β’ 0 = (0gβπΊ) |
gsumzinv.z | β’ π = (CntzβπΊ) |
gsumzinv.i | β’ πΌ = (invgβπΊ) |
gsumzinv.g | β’ (π β πΊ β Grp) |
gsumzinv.a | β’ (π β π΄ β π) |
gsumzinv.f | β’ (π β πΉ:π΄βΆπ΅) |
gsumzinv.c | β’ (π β ran πΉ β (πβran πΉ)) |
gsumzinv.n | β’ (π β πΉ finSupp 0 ) |
Ref | Expression |
---|---|
gsumzinv | β’ (π β (πΊ Ξ£g (πΌ β πΉ)) = (πΌβ(πΊ Ξ£g πΉ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumzinv.b | . . 3 β’ π΅ = (BaseβπΊ) | |
2 | gsumzinv.0 | . . 3 β’ 0 = (0gβπΊ) | |
3 | gsumzinv.z | . . 3 β’ π = (CntzβπΊ) | |
4 | eqid 2726 | . . 3 β’ (oppgβπΊ) = (oppgβπΊ) | |
5 | gsumzinv.g | . . . 4 β’ (π β πΊ β Grp) | |
6 | 5 | grpmndd 18876 | . . 3 β’ (π β πΊ β Mnd) |
7 | gsumzinv.a | . . 3 β’ (π β π΄ β π) | |
8 | gsumzinv.i | . . . . . 6 β’ πΌ = (invgβπΊ) | |
9 | 1, 8 | grpinvf 18916 | . . . . 5 β’ (πΊ β Grp β πΌ:π΅βΆπ΅) |
10 | 5, 9 | syl 17 | . . . 4 β’ (π β πΌ:π΅βΆπ΅) |
11 | gsumzinv.f | . . . 4 β’ (π β πΉ:π΄βΆπ΅) | |
12 | fco 6735 | . . . 4 β’ ((πΌ:π΅βΆπ΅ β§ πΉ:π΄βΆπ΅) β (πΌ β πΉ):π΄βΆπ΅) | |
13 | 10, 11, 12 | syl2anc 583 | . . 3 β’ (π β (πΌ β πΉ):π΄βΆπ΅) |
14 | 4, 8 | invoppggim 19279 | . . . . . 6 β’ (πΊ β Grp β πΌ β (πΊ GrpIso (oppgβπΊ))) |
15 | gimghm 19189 | . . . . . 6 β’ (πΌ β (πΊ GrpIso (oppgβπΊ)) β πΌ β (πΊ GrpHom (oppgβπΊ))) | |
16 | ghmmhm 19151 | . . . . . 6 β’ (πΌ β (πΊ GrpHom (oppgβπΊ)) β πΌ β (πΊ MndHom (oppgβπΊ))) | |
17 | 5, 14, 15, 16 | 4syl 19 | . . . . 5 β’ (π β πΌ β (πΊ MndHom (oppgβπΊ))) |
18 | gsumzinv.c | . . . . 5 β’ (π β ran πΉ β (πβran πΉ)) | |
19 | eqid 2726 | . . . . . 6 β’ (Cntzβ(oppgβπΊ)) = (Cntzβ(oppgβπΊ)) | |
20 | 3, 19 | cntzmhm2 19258 | . . . . 5 β’ ((πΌ β (πΊ MndHom (oppgβπΊ)) β§ ran πΉ β (πβran πΉ)) β (πΌ β ran πΉ) β ((Cntzβ(oppgβπΊ))β(πΌ β ran πΉ))) |
21 | 17, 18, 20 | syl2anc 583 | . . . 4 β’ (π β (πΌ β ran πΉ) β ((Cntzβ(oppgβπΊ))β(πΌ β ran πΉ))) |
22 | rnco2 6246 | . . . 4 β’ ran (πΌ β πΉ) = (πΌ β ran πΉ) | |
23 | 22 | fveq2i 6888 | . . . . 5 β’ (πβran (πΌ β πΉ)) = (πβ(πΌ β ran πΉ)) |
24 | 4, 3 | oppgcntz 19283 | . . . . 5 β’ (πβ(πΌ β ran πΉ)) = ((Cntzβ(oppgβπΊ))β(πΌ β ran πΉ)) |
25 | 23, 24 | eqtri 2754 | . . . 4 β’ (πβran (πΌ β πΉ)) = ((Cntzβ(oppgβπΊ))β(πΌ β ran πΉ)) |
26 | 21, 22, 25 | 3sstr4g 4022 | . . 3 β’ (π β ran (πΌ β πΉ) β (πβran (πΌ β πΉ))) |
27 | 2 | fvexi 6899 | . . . . 5 β’ 0 β V |
28 | 27 | a1i 11 | . . . 4 β’ (π β 0 β V) |
29 | 1 | fvexi 6899 | . . . . 5 β’ π΅ β V |
30 | 29 | a1i 11 | . . . 4 β’ (π β π΅ β V) |
31 | gsumzinv.n | . . . 4 β’ (π β πΉ finSupp 0 ) | |
32 | 2, 8 | grpinvid 18929 | . . . . 5 β’ (πΊ β Grp β (πΌβ 0 ) = 0 ) |
33 | 5, 32 | syl 17 | . . . 4 β’ (π β (πΌβ 0 ) = 0 ) |
34 | 28, 11, 10, 7, 30, 31, 33 | fsuppco2 9400 | . . 3 β’ (π β (πΌ β πΉ) finSupp 0 ) |
35 | 1, 2, 3, 4, 6, 7, 13, 26, 34 | gsumzoppg 19864 | . 2 β’ (π β ((oppgβπΊ) Ξ£g (πΌ β πΉ)) = (πΊ Ξ£g (πΌ β πΉ))) |
36 | 4 | oppgmnd 19273 | . . . 4 β’ (πΊ β Mnd β (oppgβπΊ) β Mnd) |
37 | 6, 36 | syl 17 | . . 3 β’ (π β (oppgβπΊ) β Mnd) |
38 | 1, 3, 6, 37, 7, 17, 11, 18, 2, 31 | gsumzmhm 19857 | . 2 β’ (π β ((oppgβπΊ) Ξ£g (πΌ β πΉ)) = (πΌβ(πΊ Ξ£g πΉ))) |
39 | 35, 38 | eqtr3d 2768 | 1 β’ (π β (πΊ Ξ£g (πΌ β πΉ)) = (πΌβ(πΊ Ξ£g πΉ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 Vcvv 3468 β wss 3943 class class class wbr 5141 ran crn 5670 β cima 5672 β ccom 5673 βΆwf 6533 βcfv 6537 (class class class)co 7405 finSupp cfsupp 9363 Basecbs 17153 0gc0g 17394 Ξ£g cgsu 17395 Mndcmnd 18667 MndHom cmhm 18711 Grpcgrp 18863 invgcminusg 18864 GrpHom cghm 19138 GrpIso cgim 19182 Cntzccntz 19231 oppgcoppg 19261 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-fzo 13634 df-seq 13973 df-hash 14296 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-0g 17396 df-gsum 17397 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-submnd 18714 df-grp 18866 df-minusg 18867 df-ghm 19139 df-gim 19184 df-cntz 19233 df-oppg 19262 df-cmn 19702 |
This theorem is referenced by: dprdfinv 19941 |
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