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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 | ⊢ 𝐵 = (Base‘𝐺) |
gsumcl.z | ⊢ 0 = (0g‘𝐺) |
gsumcl.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsumcl.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
gsumcl.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
gsumcl.w | ⊢ (𝜑 → 𝐹 finSupp 0 ) |
gsumf1o.h | ⊢ (𝜑 → 𝐻:𝐶–1-1-onto→𝐴) |
Ref | Expression |
---|---|
gsumf1o | ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumcl.b | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsumcl.z | . 2 ⊢ 0 = (0g‘𝐺) | |
3 | eqid 2825 | . 2 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺) | |
4 | gsumcl.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
5 | cmnmnd 18561 | . . 3 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
7 | gsumcl.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
8 | gsumcl.f | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
9 | 1, 3, 4, 8 | cntzcmnf 18601 | . 2 ⊢ (𝜑 → ran 𝐹 ⊆ ((Cntz‘𝐺)‘ran 𝐹)) |
10 | gsumcl.w | . 2 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
11 | gsumf1o.h | . 2 ⊢ (𝜑 → 𝐻:𝐶–1-1-onto→𝐴) | |
12 | 1, 2, 3, 6, 7, 8, 9, 10, 11 | gsumzf1o 18666 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 class class class wbr 4873 ∘ ccom 5346 ⟶wf 6119 –1-1-onto→wf1o 6122 ‘cfv 6123 (class class class)co 6905 finSupp cfsupp 8544 Basecbs 16222 0gc0g 16453 Σg cgsu 16454 Mndcmnd 17647 Cntzccntz 18098 CMndccmn 18546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-supp 7560 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-fsupp 8545 df-oi 8684 df-card 9078 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-n0 11619 df-z 11705 df-uz 11969 df-fz 12620 df-fzo 12761 df-seq 13096 df-hash 13411 df-0g 16455 df-gsum 16456 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-cntz 18100 df-cmn 18548 |
This theorem is referenced by: gsummptshft 18689 gsummptf1o 18715 gsummptfif1o 18720 gsum2dlem2 18723 gsumcom2 18727 psrass1lem 19738 psrcom 19770 psropprmul 19968 coe1mul2 19999 ply1coe 20026 tsmsf1o 22318 lgseisenlem3 25515 gsummpt2d 30315 |
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