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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 2728 | . . 3 β’ (oppgβπΊ) = (oppgβπΊ) | |
5 | gsumzinv.g | . . . 4 β’ (π β πΊ β Grp) | |
6 | 5 | grpmndd 18917 | . . 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 18957 | . . . . 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 6752 | . . . 4 β’ ((πΌ:π΅βΆπ΅ β§ πΉ:π΄βΆπ΅) β (πΌ β πΉ):π΄βΆπ΅) | |
13 | 10, 11, 12 | syl2anc 582 | . . 3 β’ (π β (πΌ β πΉ):π΄βΆπ΅) |
14 | 4, 8 | invoppggim 19328 | . . . . . 6 β’ (πΊ β Grp β πΌ β (πΊ GrpIso (oppgβπΊ))) |
15 | gimghm 19232 | . . . . . 6 β’ (πΌ β (πΊ GrpIso (oppgβπΊ)) β πΌ β (πΊ GrpHom (oppgβπΊ))) | |
16 | ghmmhm 19194 | . . . . . 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 2728 | . . . . . 6 β’ (Cntzβ(oppgβπΊ)) = (Cntzβ(oppgβπΊ)) | |
20 | 3, 19 | cntzmhm2 19307 | . . . . 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 582 | . . . 4 β’ (π β (πΌ β ran πΉ) β ((Cntzβ(oppgβπΊ))β(πΌ β ran πΉ))) |
22 | rnco2 6262 | . . . 4 β’ ran (πΌ β πΉ) = (πΌ β ran πΉ) | |
23 | 22 | fveq2i 6905 | . . . . 5 β’ (πβran (πΌ β πΉ)) = (πβ(πΌ β ran πΉ)) |
24 | 4, 3 | oppgcntz 19332 | . . . . 5 β’ (πβ(πΌ β ran πΉ)) = ((Cntzβ(oppgβπΊ))β(πΌ β ran πΉ)) |
25 | 23, 24 | eqtri 2756 | . . . 4 β’ (πβran (πΌ β πΉ)) = ((Cntzβ(oppgβπΊ))β(πΌ β ran πΉ)) |
26 | 21, 22, 25 | 3sstr4g 4027 | . . 3 β’ (π β ran (πΌ β πΉ) β (πβran (πΌ β πΉ))) |
27 | 2 | fvexi 6916 | . . . . 5 β’ 0 β V |
28 | 27 | a1i 11 | . . . 4 β’ (π β 0 β V) |
29 | 1 | fvexi 6916 | . . . . 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 18970 | . . . . 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 9436 | . . 3 β’ (π β (πΌ β πΉ) finSupp 0 ) |
35 | 1, 2, 3, 4, 6, 7, 13, 26, 34 | gsumzoppg 19913 | . 2 β’ (π β ((oppgβπΊ) Ξ£g (πΌ β πΉ)) = (πΊ Ξ£g (πΌ β πΉ))) |
36 | 4 | oppgmnd 19322 | . . . 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 19906 | . 2 β’ (π β ((oppgβπΊ) Ξ£g (πΌ β πΉ)) = (πΌβ(πΊ Ξ£g πΉ))) |
39 | 35, 38 | eqtr3d 2770 | 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 3473 β wss 3949 class class class wbr 5152 ran crn 5683 β cima 5685 β ccom 5686 βΆwf 6549 βcfv 6553 (class class class)co 7426 finSupp cfsupp 9395 Basecbs 17189 0gc0g 17430 Ξ£g cgsu 17431 Mndcmnd 18703 MndHom cmhm 18747 Grpcgrp 18904 invgcminusg 18905 GrpHom cghm 19181 GrpIso cgim 19225 Cntzccntz 19280 oppgcoppg 19310 |
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 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
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-iin 5003 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 7879 df-1st 8001 df-2nd 8002 df-supp 8174 df-tpos 8240 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-map 8855 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-fsupp 9396 df-oi 9543 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-n0 12513 df-z 12599 df-uz 12863 df-fz 13527 df-fzo 13670 df-seq 14009 df-hash 14332 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-0g 17432 df-gsum 17433 df-mre 17575 df-mrc 17576 df-acs 17578 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-mhm 18749 df-submnd 18750 df-grp 18907 df-minusg 18908 df-ghm 19182 df-gim 19227 df-cntz 19282 df-oppg 19311 df-cmn 19751 |
This theorem is referenced by: dprdfinv 19990 |
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