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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 7372 | . . 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 482 | . . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 𝐺 ∈ CMnd) |
7 | nn0ex 12427 | . . . . 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 8793 | . . . . . 6 ⊢ (𝐹 ∈ (𝐵 ↑m ℕ0) → 𝐹:ℕ0⟶𝐵) | |
11 | 9, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹:ℕ0⟶𝐵) |
12 | 11 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 𝐹:ℕ0⟶𝐵) |
13 | 4 | fvexi 6860 | . . . . . 6 ⊢ 0 ∈ V |
14 | 13 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 0 ∈ V) |
15 | 9 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 𝐹 ∈ (𝐵 ↑m ℕ0)) |
16 | fsfnn0gsumfsffz.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ℕ0) | |
17 | 16 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 𝑆 ∈ ℕ0) |
18 | simpr 486 | . . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) | |
19 | 14, 15, 17, 18 | suppssfz 13908 | . . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → (𝐹 supp 0 ) ⊆ (0...𝑆)) |
20 | elmapfun 8810 | . . . . . . . 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 1129 | . . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (𝐵 ↑m ℕ0) ∧ Fun 𝐹 ∧ 0 ∈ V)) |
24 | fzfid 13887 | . . . . . . 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 9328 | . . . . . 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 684 | . . . . 5 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) ⊆ (0...𝑆)) → 𝐹 finSupp 0 ) |
28 | 19, 27 | syldan 592 | . . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → 𝐹 finSupp 0 ) |
29 | 3, 4, 6, 8, 12, 19, 28 | gsumres 19698 | . . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → (𝐺 Σg (𝐹 ↾ (0...𝑆))) = (𝐺 Σg 𝐹)) |
30 | 2, 29 | eqtr2id 2786 | . 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 0 )) → (𝐺 Σg 𝐹) = (𝐺 Σg 𝐻)) |
31 | 30 | ex 414 | 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 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3061 Vcvv 3447 ⊆ wss 3914 class class class wbr 5109 ↾ cres 5639 Fun wfun 6494 ⟶wf 6496 ‘cfv 6500 (class class class)co 7361 supp csupp 8096 ↑m cmap 8771 Fincfn 8889 finSupp cfsupp 9311 0cc0 11059 < clt 11197 ℕ0cn0 12421 ...cfz 13433 Basecbs 17091 0gc0g 17329 Σg cgsu 17330 CMndccmn 19570 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-oi 9454 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 df-fzo 13577 df-seq 13916 df-hash 14240 df-0g 17331 df-gsum 17332 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-cntz 19105 df-cmn 19572 |
This theorem is referenced by: nn0gsumfz 19769 gsummptnn0fz 19771 |
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