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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 2732 | . 2 β’ (CntzβπΊ) = (CntzβπΊ) | |
4 | gsumcl.g | . . 3 β’ (π β πΊ β CMnd) | |
5 | cmnmnd 19664 | . . 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 19712 | . 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 19779 | 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 1541 β wcel 2106 class class class wbr 5148 β ccom 5680 βΆwf 6539 β1-1-ontoβwf1o 6542 βcfv 6543 (class class class)co 7408 finSupp cfsupp 9360 Basecbs 17143 0gc0g 17384 Ξ£g cgsu 17385 Mndcmnd 18624 Cntzccntz 19178 CMndccmn 19647 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-oi 9504 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13484 df-fzo 13627 df-seq 13966 df-hash 14290 df-0g 17386 df-gsum 17387 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-cntz 19180 df-cmn 19649 |
This theorem is referenced by: gsumreidx 19784 gsummptshft 19803 gsummptf1o 19830 gsummptfif1o 19835 gsum2dlem2 19838 gsumcom2 19842 psrass1lemOLD 21492 psrass1lem 21495 psrcom 21528 psropprmul 21759 coe1mul2 21790 ply1coe 21819 tsmsf1o 23648 lgseisenlem3 26877 gsummpt2d 32196 gsumpart 32202 evlselv 41161 |
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