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Mirrors > Home > MPE Home > Th. List > gsumf1o | Structured version Visualization version GIF version |
Description: Re-index a finite group sum using a bijection. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 3-Jun-2019.) |
Ref | Expression |
---|---|
gsumcl.b | β’ π΅ = (BaseβπΊ) |
gsumcl.z | β’ 0 = (0gβπΊ) |
gsumcl.g | β’ (π β πΊ β CMnd) |
gsumcl.a | β’ (π β π΄ β π) |
gsumcl.f | β’ (π β πΉ:π΄βΆπ΅) |
gsumcl.w | β’ (π β πΉ finSupp 0 ) |
gsumf1o.h | β’ (π β π»:πΆβ1-1-ontoβπ΄) |
Ref | Expression |
---|---|
gsumf1o | β’ (π β (πΊ Ξ£g πΉ) = (πΊ Ξ£g (πΉ β π»))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumcl.b | . 2 β’ π΅ = (BaseβπΊ) | |
2 | gsumcl.z | . 2 β’ 0 = (0gβπΊ) | |
3 | eqid 2727 | . 2 β’ (CntzβπΊ) = (CntzβπΊ) | |
4 | gsumcl.g | . . 3 β’ (π β πΊ β CMnd) | |
5 | cmnmnd 19745 | . . 3 β’ (πΊ β CMnd β πΊ β Mnd) | |
6 | 4, 5 | syl 17 | . 2 β’ (π β πΊ β Mnd) |
7 | gsumcl.a | . 2 β’ (π β π΄ β π) | |
8 | gsumcl.f | . 2 β’ (π β πΉ:π΄βΆπ΅) | |
9 | 1, 3, 4, 8 | cntzcmnf 19793 | . 2 β’ (π β ran πΉ β ((CntzβπΊ)βran πΉ)) |
10 | gsumcl.w | . 2 β’ (π β πΉ finSupp 0 ) | |
11 | gsumf1o.h | . 2 β’ (π β π»:πΆβ1-1-ontoβπ΄) | |
12 | 1, 2, 3, 6, 7, 8, 9, 10, 11 | gsumzf1o 19860 | 1 β’ (π β (πΊ Ξ£g πΉ) = (πΊ Ξ£g (πΉ β π»))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 class class class wbr 5142 β ccom 5676 βΆwf 6538 β1-1-ontoβwf1o 6541 βcfv 6542 (class class class)co 7414 finSupp cfsupp 9379 Basecbs 17173 0gc0g 17414 Ξ£g cgsu 17415 Mndcmnd 18687 Cntzccntz 19259 CMndccmn 19728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 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 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 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 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9380 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-n0 12497 df-z 12583 df-uz 12847 df-fz 13511 df-fzo 13654 df-seq 13993 df-hash 14316 df-0g 17416 df-gsum 17417 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-cntz 19261 df-cmn 19730 |
This theorem is referenced by: gsumreidx 19865 gsummptshft 19884 gsummptf1o 19911 gsummptfif1o 19916 gsum2dlem2 19919 gsumcom2 19923 psrass1lemOLD 21867 psrass1lem 21870 psrcom 21904 psdmul 22083 psropprmul 22149 coe1mul2 22181 ply1coe 22210 tsmsf1o 24042 lgseisenlem3 27303 gsummpt2d 32757 gsumpart 32763 evlselv 41792 |
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