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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 8664 | . . . 4 ⊢ (𝐹 ∈ (𝐵 ↑m ℕ0) → 𝐹:ℕ0⟶𝐵) | |
6 | ffvelcdm 6987 | . . . . 5 ⊢ ((𝐹:ℕ0⟶𝐵 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ 𝐵) | |
7 | 6 | ex 414 | . . . 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 3139 | . 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 5085 | . . . . 5 ⊢ (𝑥 = 𝑘 → (𝑆 < 𝑥 ↔ 𝑆 < 𝑘)) | |
13 | fveqeq2 6809 | . . . . 5 ⊢ (𝑥 = 𝑘 → ((𝐹‘𝑥) = 0 ↔ (𝐹‘𝑘) = 0 )) | |
14 | 12, 13 | imbi12d 346 | . . . 4 ⊢ (𝑥 = 𝑘 → ((𝑆 < 𝑥 → (𝐹‘𝑥) = 0 ) ↔ (𝑆 < 𝑘 → (𝐹‘𝑘) = 0 ))) |
15 | 14 | cbvralvw 3222 | . . 3 ⊢ (∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 ) ↔ ∀𝑘 ∈ ℕ0 (𝑆 < 𝑘 → (𝐹‘𝑘) = 0 )) |
16 | 11, 15 | sylib 217 | . 2 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (𝑆 < 𝑘 → (𝐹‘𝑘) = 0 )) |
17 | 1, 2, 3, 9, 10, 16 | gsummptnn0fz 19628 | 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 1539 ∈ wcel 2104 ∀wral 3062 class class class wbr 5081 ↦ cmpt 5164 ⟶wf 6450 ‘cfv 6454 (class class class)co 7303 ↑m cmap 8642 0cc0 10913 < clt 11051 ℕ0cn0 12275 ...cfz 13281 Basecbs 16953 0gc0g 17191 Σg cgsu 17192 CMndccmn 19427 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7616 ax-cnex 10969 ax-resscn 10970 ax-1cn 10971 ax-icn 10972 ax-addcl 10973 ax-addrcl 10974 ax-mulcl 10975 ax-mulrcl 10976 ax-mulcom 10977 ax-addass 10978 ax-mulass 10979 ax-distr 10980 ax-i2m1 10981 ax-1ne0 10982 ax-1rid 10983 ax-rnegex 10984 ax-rrecex 10985 ax-cnre 10986 ax-pre-lttri 10987 ax-pre-lttrn 10988 ax-pre-ltadd 10989 ax-pre-mulgt0 10990 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5496 df-eprel 5502 df-po 5510 df-so 5511 df-fr 5551 df-se 5552 df-we 5553 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 df-ima 5609 df-pred 6213 df-ord 6280 df-on 6281 df-lim 6282 df-suc 6283 df-iota 6406 df-fun 6456 df-fn 6457 df-f 6458 df-f1 6459 df-fo 6460 df-f1o 6461 df-fv 6462 df-isom 6463 df-riota 7260 df-ov 7306 df-oprab 7307 df-mpo 7308 df-om 7741 df-1st 7859 df-2nd 7860 df-supp 8005 df-frecs 8124 df-wrecs 8155 df-recs 8229 df-rdg 8268 df-1o 8324 df-er 8525 df-map 8644 df-en 8761 df-dom 8762 df-sdom 8763 df-fin 8764 df-fsupp 9169 df-oi 9309 df-card 9737 df-pnf 11053 df-mnf 11054 df-xr 11055 df-ltxr 11056 df-le 11057 df-sub 11249 df-neg 11250 df-nn 12016 df-n0 12276 df-z 12362 df-uz 12625 df-fz 13282 df-fzo 13425 df-seq 13764 df-hash 14087 df-0g 17193 df-gsum 17194 df-mgm 18367 df-sgrp 18416 df-mnd 18427 df-cntz 18964 df-cmn 19429 |
This theorem is referenced by: (None) |
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