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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 7430 | . . 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 479 | . . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 𝐺 ∈ CMnd) |
7 | nn0ex 12511 | . . . . 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 8868 | . . . . . 6 ⊢ (𝐹 ∈ (𝐵 ↑m ℕ0) → 𝐹:ℕ0⟶𝐵) | |
11 | 9, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹:ℕ0⟶𝐵) |
12 | 11 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 𝐹:ℕ0⟶𝐵) |
13 | 4 | fvexi 6910 | . . . . . 6 ⊢ 0 ∈ V |
14 | 13 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 0 ∈ V) |
15 | 9 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 𝐹 ∈ (𝐵 ↑m ℕ0)) |
16 | fsfnn0gsumfsffz.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ℕ0) | |
17 | 16 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 𝑆 ∈ ℕ0) |
18 | simpr 483 | . . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) | |
19 | 14, 15, 17, 18 | suppssfz 13995 | . . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → (𝐹 supp 0 ) ⊆ (0...𝑆)) |
20 | elmapfun 8885 | . . . . . . . 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 1125 | . . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (𝐵 ↑m ℕ0) ∧ Fun 𝐹 ∧ 0 ∈ V)) |
24 | fzfid 13974 | . . . . . . 7 ⊢ (𝜑 → (0...𝑆) ∈ Fin) | |
25 | 24 | anim1i 613 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) ⊆ (0...𝑆)) → ((0...𝑆) ∈ Fin ∧ (𝐹 supp 0 ) ⊆ (0...𝑆))) |
26 | suppssfifsupp 9405 | . . . . . 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 589 | . . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 𝐹 finSupp 0 ) |
29 | 3, 4, 6, 8, 12, 19, 28 | gsumres 19880 | . . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → (𝐺 Σg (𝐹 ↾ (0...𝑆))) = (𝐺 Σg 𝐹)) |
30 | 2, 29 | eqtr2id 2778 | . 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → (𝐺 Σg 𝐹) = (𝐺 Σg 𝐻)) |
31 | 30 | ex 411 | 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 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3050 Vcvv 3461 ⊆ wss 3944 class class class wbr 5149 ↾ cres 5680 Fun wfun 6543 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 supp csupp 8165 ↑m cmap 8845 Fincfn 8964 finSupp cfsupp 9387 0cc0 11140 < clt 11280 ℕ0cn0 12505 ...cfz 13519 Basecbs 17183 0gc0g 17424 Σg cgsu 17425 CMndccmn 19747 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-oi 9535 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-n0 12506 df-z 12592 df-uz 12856 df-fz 13520 df-fzo 13663 df-seq 14003 df-hash 14326 df-0g 17426 df-gsum 17427 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-cntz 19280 df-cmn 19749 |
This theorem is referenced by: nn0gsumfz 19951 gsummptnn0fz 19953 |
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