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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > gsumzrsum | Structured version Visualization version GIF version |
Description: Relate a group sum on ℤring to a finite sum on the complex numbers. See also gsumfsum 21469. (Contributed by Thierry Arnoux, 5-Oct-2025.) |
Ref | Expression |
---|---|
gsumzrsum.1 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
gsumzrsum.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) |
Ref | Expression |
---|---|
gsumzrsum | ⊢ (𝜑 → (ℤring Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ𝑘 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnfldbas 21385 | . . 3 ⊢ ℂ = (Base‘ℂfld) | |
2 | cnfldadd 21387 | . . 3 ⊢ + = (+g‘ℂfld) | |
3 | df-zring 21475 | . . 3 ⊢ ℤring = (ℂfld ↾s ℤ) | |
4 | cnfldex 21384 | . . . 4 ⊢ ℂfld ∈ V | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → ℂfld ∈ V) |
6 | gsumzrsum.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
7 | zsscn 12618 | . . . 4 ⊢ ℤ ⊆ ℂ | |
8 | 7 | a1i 11 | . . 3 ⊢ (𝜑 → ℤ ⊆ ℂ) |
9 | gsumzrsum.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) | |
10 | 9 | fmpttd 7134 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℤ) |
11 | 0zd 12622 | . . 3 ⊢ (𝜑 → 0 ∈ ℤ) | |
12 | addlid 11441 | . . . . 5 ⊢ (𝑘 ∈ ℂ → (0 + 𝑘) = 𝑘) | |
13 | addrid 11438 | . . . . 5 ⊢ (𝑘 ∈ ℂ → (𝑘 + 0) = 𝑘) | |
14 | 12, 13 | jca 511 | . . . 4 ⊢ (𝑘 ∈ ℂ → ((0 + 𝑘) = 𝑘 ∧ (𝑘 + 0) = 𝑘)) |
15 | 14 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℂ) → ((0 + 𝑘) = 𝑘 ∧ (𝑘 + 0) = 𝑘)) |
16 | 1, 2, 3, 5, 6, 8, 10, 11, 15 | gsumress 18707 | . 2 ⊢ (𝜑 → (ℂfld Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) = (ℤring Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) |
17 | 9 | zcnd 12720 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
18 | 6, 17 | gsumfsum 21469 | . 2 ⊢ (𝜑 → (ℂfld Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ𝑘 ∈ 𝐴 𝐵) |
19 | 16, 18 | eqtr3d 2776 | 1 ⊢ (𝜑 → (ℤring Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ⊆ wss 3962 ↦ cmpt 5230 (class class class)co 7430 Fincfn 8983 ℂcc 11150 0cc0 11152 + caddc 11155 ℤcz 12610 Σcsu 15718 Σg cgsu 17486 ℂfldccnfld 21381 ℤringczring 21474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-rp 13032 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-sum 15719 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17487 df-gsum 17488 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 df-cntz 19347 df-cmn 19814 df-abl 19815 df-mgp 20152 df-ur 20199 df-ring 20252 df-cring 20253 df-cnfld 21382 df-zring 21475 |
This theorem is referenced by: gsummulgc2 33045 |
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