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Mirrors > Home > MPE Home > Th. List > fsfnn0gsumfsffz | Structured version Visualization version GIF version |
Description: Replacing a finitely supported function over the nonnegative integers by a function over a finite set of sequential integers in a finite group sum. (Contributed by AV, 9-Oct-2019.) |
Ref | Expression |
---|---|
nn0gsumfz.b | ⊢ 𝐵 = (Base‘𝐺) |
nn0gsumfz.0 | ⊢ 0 = (0g‘𝐺) |
nn0gsumfz.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
nn0gsumfz.f | ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m ℕ0)) |
fsfnn0gsumfsffz.s | ⊢ (𝜑 → 𝑆 ∈ ℕ0) |
fsfnn0gsumfsffz.h | ⊢ 𝐻 = (𝐹 ↾ (0...𝑆)) |
Ref | Expression |
---|---|
fsfnn0gsumfsffz | ⊢ (𝜑 → (∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 ) → (𝐺 Σg 𝐹) = (𝐺 Σg 𝐻))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsfnn0gsumfsffz.h | . . . 4 ⊢ 𝐻 = (𝐹 ↾ (0...𝑆)) | |
2 | 1 | oveq2i 7166 | . . 3 ⊢ (𝐺 Σg 𝐻) = (𝐺 Σg (𝐹 ↾ (0...𝑆))) |
3 | nn0gsumfz.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
4 | nn0gsumfz.0 | . . . 4 ⊢ 0 = (0g‘𝐺) | |
5 | nn0gsumfz.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
6 | 5 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 𝐺 ∈ CMnd) |
7 | nn0ex 11902 | . . . . 5 ⊢ ℕ0 ∈ V | |
8 | 7 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → ℕ0 ∈ V) |
9 | nn0gsumfz.f | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m ℕ0)) | |
10 | elmapi 8427 | . . . . . 6 ⊢ (𝐹 ∈ (𝐵 ↑m ℕ0) → 𝐹:ℕ0⟶𝐵) | |
11 | 9, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹:ℕ0⟶𝐵) |
12 | 11 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 𝐹:ℕ0⟶𝐵) |
13 | 4 | fvexi 6683 | . . . . . 6 ⊢ 0 ∈ V |
14 | 13 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 0 ∈ V) |
15 | 9 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 𝐹 ∈ (𝐵 ↑m ℕ0)) |
16 | fsfnn0gsumfsffz.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ℕ0) | |
17 | 16 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 𝑆 ∈ ℕ0) |
18 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) | |
19 | 14, 15, 17, 18 | suppssfz 13361 | . . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → (𝐹 supp 0 ) ⊆ (0...𝑆)) |
20 | elmapfun 8429 | . . . . . . . 8 ⊢ (𝐹 ∈ (𝐵 ↑m ℕ0) → Fun 𝐹) | |
21 | 9, 20 | syl 17 | . . . . . . 7 ⊢ (𝜑 → Fun 𝐹) |
22 | 13 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ V) |
23 | 9, 21, 22 | 3jca 1124 | . . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (𝐵 ↑m ℕ0) ∧ Fun 𝐹 ∧ 0 ∈ V)) |
24 | fzfid 13340 | . . . . . . 7 ⊢ (𝜑 → (0...𝑆) ∈ Fin) | |
25 | 24 | anim1i 616 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) ⊆ (0...𝑆)) → ((0...𝑆) ∈ Fin ∧ (𝐹 supp 0 ) ⊆ (0...𝑆))) |
26 | suppssfifsupp 8847 | . . . . . 6 ⊢ (((𝐹 ∈ (𝐵 ↑m ℕ0) ∧ Fun 𝐹 ∧ 0 ∈ V) ∧ ((0...𝑆) ∈ Fin ∧ (𝐹 supp 0 ) ⊆ (0...𝑆))) → 𝐹 finSupp 0 ) | |
27 | 23, 25, 26 | syl2an2r 683 | . . . . 5 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) ⊆ (0...𝑆)) → 𝐹 finSupp 0 ) |
28 | 19, 27 | syldan 593 | . . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 𝐹 finSupp 0 ) |
29 | 3, 4, 6, 8, 12, 19, 28 | gsumres 19032 | . . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → (𝐺 Σg (𝐹 ↾ (0...𝑆))) = (𝐺 Σg 𝐹)) |
30 | 2, 29 | syl5req 2869 | . 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → (𝐺 Σg 𝐹) = (𝐺 Σg 𝐻)) |
31 | 30 | ex 415 | 1 ⊢ (𝜑 → (∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 ) → (𝐺 Σg 𝐹) = (𝐺 Σg 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 ⊆ wss 3935 class class class wbr 5065 ↾ cres 5556 Fun wfun 6348 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 supp csupp 7829 ↑m cmap 8405 Fincfn 8508 finSupp cfsupp 8832 0cc0 10536 < clt 10674 ℕ0cn0 11896 ...cfz 12891 Basecbs 16482 0gc0g 16712 Σg cgsu 16713 CMndccmn 18905 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-fzo 13033 df-seq 13369 df-hash 13690 df-0g 16714 df-gsum 16715 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-cntz 18446 df-cmn 18907 |
This theorem is referenced by: nn0gsumfz 19103 gsummptnn0fz 19105 |
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