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Mirrors > Home > MPE Home > Th. List > gsummptfif1o | Structured version Visualization version GIF version |
Description: Re-index a finite group sum as map, using a bijection. (Contributed by by AV, 23-Jul-2019.) |
Ref | Expression |
---|---|
gsummptcl.b | ⊢ 𝐵 = (Base‘𝐺) |
gsummptcl.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsummptcl.n | ⊢ (𝜑 → 𝑁 ∈ Fin) |
gsummptcl.e | ⊢ (𝜑 → ∀𝑖 ∈ 𝑁 𝑋 ∈ 𝐵) |
gsummptfif1o.f | ⊢ 𝐹 = (𝑖 ∈ 𝑁 ↦ 𝑋) |
gsummptfif1o.h | ⊢ (𝜑 → 𝐻:𝐶–1-1-onto→𝑁) |
Ref | Expression |
---|---|
gsummptfif1o | ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsummptcl.b | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
2 | eqid 2739 | . 2 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
3 | gsummptcl.g | . 2 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
4 | gsummptcl.n | . 2 ⊢ (𝜑 → 𝑁 ∈ Fin) | |
5 | gsummptcl.e | . . 3 ⊢ (𝜑 → ∀𝑖 ∈ 𝑁 𝑋 ∈ 𝐵) | |
6 | gsummptfif1o.f | . . . 4 ⊢ 𝐹 = (𝑖 ∈ 𝑁 ↦ 𝑋) | |
7 | 6 | fmpt 6897 | . . 3 ⊢ (∀𝑖 ∈ 𝑁 𝑋 ∈ 𝐵 ↔ 𝐹:𝑁⟶𝐵) |
8 | 5, 7 | sylib 221 | . 2 ⊢ (𝜑 → 𝐹:𝑁⟶𝐵) |
9 | fvexd 6702 | . . 3 ⊢ (𝜑 → (0g‘𝐺) ∈ V) | |
10 | 8, 4, 9 | fdmfifsupp 8929 | . 2 ⊢ (𝜑 → 𝐹 finSupp (0g‘𝐺)) |
11 | gsummptfif1o.h | . 2 ⊢ (𝜑 → 𝐻:𝐶–1-1-onto→𝑁) | |
12 | 1, 2, 3, 4, 8, 10, 11 | gsumf1o 19168 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ∀wral 3054 Vcvv 3400 ↦ cmpt 5120 ∘ ccom 5539 ⟶wf 6346 –1-1-onto→wf1o 6349 ‘cfv 6350 (class class class)co 7183 Fincfn 8568 Basecbs 16599 0gc0g 16829 Σg cgsu 16830 CMndccmn 19037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-om 7613 df-1st 7727 df-2nd 7728 df-supp 7870 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-1o 8144 df-er 8333 df-en 8569 df-dom 8570 df-sdom 8571 df-fin 8572 df-fsupp 8920 df-oi 9060 df-card 9454 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-nn 11730 df-n0 11990 df-z 12076 df-uz 12338 df-fz 12995 df-fzo 13138 df-seq 13474 df-hash 13796 df-0g 16831 df-gsum 16832 df-mgm 17981 df-sgrp 18030 df-mnd 18041 df-cntz 18578 df-cmn 19039 |
This theorem is referenced by: mdetleib2 21352 mdetralt 21372 |
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