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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 6826 | . . 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 472 | . . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 𝐺 ∈ CMnd) |
7 | nn0ex 11511 | . . . . 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 8048 | . . . . . 6 ⊢ (𝐹 ∈ (𝐵 ↑𝑚 ℕ0) → 𝐹:ℕ0⟶𝐵) | |
11 | 9, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹:ℕ0⟶𝐵) |
12 | 11 | adantr 472 | . . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 𝐹:ℕ0⟶𝐵) |
13 | fvex 6364 | . . . . . . 7 ⊢ (0g‘𝐺) ∈ V | |
14 | 4, 13 | eqeltri 2836 | . . . . . 6 ⊢ 0 ∈ V |
15 | 14 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 0 ∈ V) |
16 | 9 | adantr 472 | . . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 𝐹 ∈ (𝐵 ↑𝑚 ℕ0)) |
17 | fsfnn0gsumfsffz.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ℕ0) | |
18 | 17 | adantr 472 | . . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 𝑆 ∈ ℕ0) |
19 | simpr 479 | . . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) | |
20 | 15, 16, 18, 19 | suppssfz 13009 | . . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → (𝐹 supp 0 ) ⊆ (0...𝑆)) |
21 | elmapfun 8050 | . . . . . . . . 9 ⊢ (𝐹 ∈ (𝐵 ↑𝑚 ℕ0) → Fun 𝐹) | |
22 | 9, 21 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → Fun 𝐹) |
23 | 14 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 0 ∈ V) |
24 | 9, 22, 23 | 3jca 1123 | . . . . . . 7 ⊢ (𝜑 → (𝐹 ∈ (𝐵 ↑𝑚 ℕ0) ∧ Fun 𝐹 ∧ 0 ∈ V)) |
25 | 24 | adantr 472 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) ⊆ (0...𝑆)) → (𝐹 ∈ (𝐵 ↑𝑚 ℕ0) ∧ Fun 𝐹 ∧ 0 ∈ V)) |
26 | fzfid 12987 | . . . . . . 7 ⊢ (𝜑 → (0...𝑆) ∈ Fin) | |
27 | 26 | anim1i 593 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) ⊆ (0...𝑆)) → ((0...𝑆) ∈ Fin ∧ (𝐹 supp 0 ) ⊆ (0...𝑆))) |
28 | suppssfifsupp 8458 | . . . . . 6 ⊢ (((𝐹 ∈ (𝐵 ↑𝑚 ℕ0) ∧ Fun 𝐹 ∧ 0 ∈ V) ∧ ((0...𝑆) ∈ Fin ∧ (𝐹 supp 0 ) ⊆ (0...𝑆))) → 𝐹 finSupp 0 ) | |
29 | 25, 27, 28 | syl2anc 696 | . . . . 5 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) ⊆ (0...𝑆)) → 𝐹 finSupp 0 ) |
30 | 20, 29 | syldan 488 | . . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 𝐹 finSupp 0 ) |
31 | 3, 4, 6, 8, 12, 20, 30 | gsumres 18535 | . . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → (𝐺 Σg (𝐹 ↾ (0...𝑆))) = (𝐺 Σg 𝐹)) |
32 | 2, 31 | syl5req 2808 | . 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → (𝐺 Σg 𝐹) = (𝐺 Σg 𝐻)) |
33 | 32 | ex 449 | 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 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2140 ∀wral 3051 Vcvv 3341 ⊆ wss 3716 class class class wbr 4805 ↾ cres 5269 Fun wfun 6044 ⟶wf 6046 ‘cfv 6050 (class class class)co 6815 supp csupp 7465 ↑𝑚 cmap 8026 Fincfn 8124 finSupp cfsupp 8443 0cc0 10149 < clt 10287 ℕ0cn0 11505 ...cfz 12540 Basecbs 16080 0gc0g 16323 Σg cgsu 16324 CMndccmn 18414 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1989 ax-6 2055 ax-7 2091 ax-8 2142 ax-9 2149 ax-10 2169 ax-11 2184 ax-12 2197 ax-13 2392 ax-ext 2741 ax-rep 4924 ax-sep 4934 ax-nul 4942 ax-pow 4993 ax-pr 5056 ax-un 7116 ax-cnex 10205 ax-resscn 10206 ax-1cn 10207 ax-icn 10208 ax-addcl 10209 ax-addrcl 10210 ax-mulcl 10211 ax-mulrcl 10212 ax-mulcom 10213 ax-addass 10214 ax-mulass 10215 ax-distr 10216 ax-i2m1 10217 ax-1ne0 10218 ax-1rid 10219 ax-rnegex 10220 ax-rrecex 10221 ax-cnre 10222 ax-pre-lttri 10223 ax-pre-lttrn 10224 ax-pre-ltadd 10225 ax-pre-mulgt0 10226 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2048 df-eu 2612 df-mo 2613 df-clab 2748 df-cleq 2754 df-clel 2757 df-nfc 2892 df-ne 2934 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3343 df-sbc 3578 df-csb 3676 df-dif 3719 df-un 3721 df-in 3723 df-ss 3730 df-pss 3732 df-nul 4060 df-if 4232 df-pw 4305 df-sn 4323 df-pr 4325 df-tp 4327 df-op 4329 df-uni 4590 df-int 4629 df-iun 4675 df-br 4806 df-opab 4866 df-mpt 4883 df-tr 4906 df-id 5175 df-eprel 5180 df-po 5188 df-so 5189 df-fr 5226 df-se 5227 df-we 5228 df-xp 5273 df-rel 5274 df-cnv 5275 df-co 5276 df-dm 5277 df-rn 5278 df-res 5279 df-ima 5280 df-pred 5842 df-ord 5888 df-on 5889 df-lim 5890 df-suc 5891 df-iota 6013 df-fun 6052 df-fn 6053 df-f 6054 df-f1 6055 df-fo 6056 df-f1o 6057 df-fv 6058 df-isom 6059 df-riota 6776 df-ov 6818 df-oprab 6819 df-mpt2 6820 df-om 7233 df-1st 7335 df-2nd 7336 df-supp 7466 df-wrecs 7578 df-recs 7639 df-rdg 7677 df-1o 7731 df-oadd 7735 df-er 7914 df-map 8028 df-en 8125 df-dom 8126 df-sdom 8127 df-fin 8128 df-fsupp 8444 df-oi 8583 df-card 8976 df-pnf 10289 df-mnf 10290 df-xr 10291 df-ltxr 10292 df-le 10293 df-sub 10481 df-neg 10482 df-nn 11234 df-n0 11506 df-z 11591 df-uz 11901 df-fz 12541 df-fzo 12681 df-seq 13017 df-hash 13333 df-0g 16325 df-gsum 16326 df-mgm 17464 df-sgrp 17506 df-mnd 17517 df-cntz 17971 df-cmn 18416 |
This theorem is referenced by: nn0gsumfz 18601 gsummptnn0fz 18603 |
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