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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 7279 | . . 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 480 | . . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 𝐺 ∈ CMnd) |
7 | nn0ex 12222 | . . . . 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 8611 | . . . . . 6 ⊢ (𝐹 ∈ (𝐵 ↑m ℕ0) → 𝐹:ℕ0⟶𝐵) | |
11 | 9, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹:ℕ0⟶𝐵) |
12 | 11 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 𝐹:ℕ0⟶𝐵) |
13 | 4 | fvexi 6782 | . . . . . 6 ⊢ 0 ∈ V |
14 | 13 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 0 ∈ V) |
15 | 9 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 𝐹 ∈ (𝐵 ↑m ℕ0)) |
16 | fsfnn0gsumfsffz.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ℕ0) | |
17 | 16 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 𝑆 ∈ ℕ0) |
18 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) | |
19 | 14, 15, 17, 18 | suppssfz 13695 | . . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → (𝐹 supp 0 ) ⊆ (0...𝑆)) |
20 | elmapfun 8628 | . . . . . . . 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 1126 | . . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (𝐵 ↑m ℕ0) ∧ Fun 𝐹 ∧ 0 ∈ V)) |
24 | fzfid 13674 | . . . . . . 7 ⊢ (𝜑 → (0...𝑆) ∈ Fin) | |
25 | 24 | anim1i 614 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) ⊆ (0...𝑆)) → ((0...𝑆) ∈ Fin ∧ (𝐹 supp 0 ) ⊆ (0...𝑆))) |
26 | suppssfifsupp 9104 | . . . . . 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 681 | . . . . 5 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) ⊆ (0...𝑆)) → 𝐹 finSupp 0 ) |
28 | 19, 27 | syldan 590 | . . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 𝐹 finSupp 0 ) |
29 | 3, 4, 6, 8, 12, 19, 28 | gsumres 19495 | . . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → (𝐺 Σg (𝐹 ↾ (0...𝑆))) = (𝐺 Σg 𝐹)) |
30 | 2, 29 | eqtr2id 2792 | . 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → (𝐺 Σg 𝐹) = (𝐺 Σg 𝐻)) |
31 | 30 | ex 412 | 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 395 ∧ w3a 1085 = wceq 1541 ∈ wcel 2109 ∀wral 3065 Vcvv 3430 ⊆ wss 3891 class class class wbr 5078 ↾ cres 5590 Fun wfun 6424 ⟶wf 6426 ‘cfv 6430 (class class class)co 7268 supp csupp 7961 ↑m cmap 8589 Fincfn 8707 finSupp cfsupp 9089 0cc0 10855 < clt 10993 ℕ0cn0 12216 ...cfz 13221 Basecbs 16893 0gc0g 17131 Σg cgsu 17132 CMndccmn 19367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-se 5544 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-isom 6439 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-supp 7962 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-er 8472 df-map 8591 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-fsupp 9090 df-oi 9230 df-card 9681 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-n0 12217 df-z 12303 df-uz 12565 df-fz 13222 df-fzo 13365 df-seq 13703 df-hash 14026 df-0g 17133 df-gsum 17134 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-cntz 18904 df-cmn 19369 |
This theorem is referenced by: nn0gsumfz 19566 gsummptnn0fz 19568 |
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