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Mirrors > Home > MPE Home > Th. List > gsummptnn0fzfv | Structured version Visualization version GIF version |
Description: A final group sum over a function over the nonnegative integers (given as mapping to its function values) is equal to a final group sum over a finite interval of nonnegative integers. (Contributed by AV, 10-Oct-2019.) |
Ref | Expression |
---|---|
gsummptnn0fzfv.b | ⊢ 𝐵 = (Base‘𝐺) |
gsummptnn0fzfv.0 | ⊢ 0 = (0g‘𝐺) |
gsummptnn0fzfv.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsummptnn0fzfv.f | ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m ℕ0)) |
gsummptnn0fzfv.s | ⊢ (𝜑 → 𝑆 ∈ ℕ0) |
gsummptnn0fzfv.u | ⊢ (𝜑 → ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) |
Ref | Expression |
---|---|
gsummptnn0fzfv | ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ ℕ0 ↦ (𝐹‘𝑘))) = (𝐺 Σg (𝑘 ∈ (0...𝑆) ↦ (𝐹‘𝑘)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsummptnn0fzfv.b | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsummptnn0fzfv.0 | . 2 ⊢ 0 = (0g‘𝐺) | |
3 | gsummptnn0fzfv.g | . 2 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
4 | gsummptnn0fzfv.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m ℕ0)) | |
5 | elmapi 8519 | . . . 4 ⊢ (𝐹 ∈ (𝐵 ↑m ℕ0) → 𝐹:ℕ0⟶𝐵) | |
6 | ffvelrn 6891 | . . . . 5 ⊢ ((𝐹:ℕ0⟶𝐵 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ 𝐵) | |
7 | 6 | ex 416 | . . . 4 ⊢ (𝐹:ℕ0⟶𝐵 → (𝑘 ∈ ℕ0 → (𝐹‘𝑘) ∈ 𝐵)) |
8 | 4, 5, 7 | 3syl 18 | . . 3 ⊢ (𝜑 → (𝑘 ∈ ℕ0 → (𝐹‘𝑘) ∈ 𝐵)) |
9 | 8 | ralrimiv 3097 | . 2 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (𝐹‘𝑘) ∈ 𝐵) |
10 | gsummptnn0fzfv.s | . 2 ⊢ (𝜑 → 𝑆 ∈ ℕ0) | |
11 | gsummptnn0fzfv.u | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) | |
12 | breq2 5047 | . . . . 5 ⊢ (𝑥 = 𝑘 → (𝑆 < 𝑥 ↔ 𝑆 < 𝑘)) | |
13 | fveqeq2 6715 | . . . . 5 ⊢ (𝑥 = 𝑘 → ((𝐹‘𝑥) = 0 ↔ (𝐹‘𝑘) = 0 )) | |
14 | 12, 13 | imbi12d 348 | . . . 4 ⊢ (𝑥 = 𝑘 → ((𝑆 < 𝑥 → (𝐹‘𝑥) = 0 ) ↔ (𝑆 < 𝑘 → (𝐹‘𝑘) = 0 ))) |
15 | 14 | cbvralvw 3351 | . . 3 ⊢ (∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 ) ↔ ∀𝑘 ∈ ℕ0 (𝑆 < 𝑘 → (𝐹‘𝑘) = 0 )) |
16 | 11, 15 | sylib 221 | . 2 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (𝑆 < 𝑘 → (𝐹‘𝑘) = 0 )) |
17 | 1, 2, 3, 9, 10, 16 | gsummptnn0fz 19343 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ ℕ0 ↦ (𝐹‘𝑘))) = (𝐺 Σg (𝑘 ∈ (0...𝑆) ↦ (𝐹‘𝑘)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ∀wral 3054 class class class wbr 5043 ↦ cmpt 5124 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 ↑m cmap 8497 0cc0 10712 < clt 10850 ℕ0cn0 12073 ...cfz 13078 Basecbs 16684 0gc0g 16916 Σg cgsu 16917 CMndccmn 19142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 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 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-map 8499 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-oi 9115 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-n0 12074 df-z 12160 df-uz 12422 df-fz 13079 df-fzo 13222 df-seq 13558 df-hash 13880 df-0g 16918 df-gsum 16919 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-cntz 18683 df-cmn 19144 |
This theorem is referenced by: (None) |
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