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Mirrors > Home > MPE Home > Th. List > gsummptnn0fz | Structured version Visualization version GIF version |
Description: A final group sum over a function over the nonnegative integers (given as mapping) is equal to a final group sum over a finite interval of nonnegative integers. (Contributed by AV, 10-Oct-2019.) (Revised by AV, 3-Jul-2022.) |
Ref | Expression |
---|---|
gsummptnn0fz.b | ⊢ 𝐵 = (Base‘𝐺) |
gsummptnn0fz.0 | ⊢ 0 = (0g‘𝐺) |
gsummptnn0fz.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsummptnn0fz.f | ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵) |
gsummptnn0fz.s | ⊢ (𝜑 → 𝑆 ∈ ℕ0) |
gsummptnn0fz.u | ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (𝑆 < 𝑘 → 𝐶 = 0 )) |
Ref | Expression |
---|---|
gsummptnn0fz | ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ ℕ0 ↦ 𝐶)) = (𝐺 Σg (𝑘 ∈ (0...𝑆) ↦ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsummptnn0fz.u | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (𝑆 < 𝑘 → 𝐶 = 0 )) | |
2 | nfv 1917 | . . . . 5 ⊢ Ⅎ𝑥(𝑆 < 𝑘 → 𝐶 = 0 ) | |
3 | nfv 1917 | . . . . . 6 ⊢ Ⅎ𝑘 𝑆 < 𝑥 | |
4 | nfcsb1v 3915 | . . . . . . 7 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐶 | |
5 | 4 | nfeq1 2918 | . . . . . 6 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐶 = 0 |
6 | 3, 5 | nfim 1899 | . . . . 5 ⊢ Ⅎ𝑘(𝑆 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 ) |
7 | breq2 5146 | . . . . . 6 ⊢ (𝑘 = 𝑥 → (𝑆 < 𝑘 ↔ 𝑆 < 𝑥)) | |
8 | csbeq1a 3904 | . . . . . . 7 ⊢ (𝑘 = 𝑥 → 𝐶 = ⦋𝑥 / 𝑘⦌𝐶) | |
9 | 8 | eqeq1d 2734 | . . . . . 6 ⊢ (𝑘 = 𝑥 → (𝐶 = 0 ↔ ⦋𝑥 / 𝑘⦌𝐶 = 0 )) |
10 | 7, 9 | imbi12d 344 | . . . . 5 ⊢ (𝑘 = 𝑥 → ((𝑆 < 𝑘 → 𝐶 = 0 ) ↔ (𝑆 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 ))) |
11 | 2, 6, 10 | cbvralw 3303 | . . . 4 ⊢ (∀𝑘 ∈ ℕ0 (𝑆 < 𝑘 → 𝐶 = 0 ) ↔ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 )) |
12 | 1, 11 | sylib 217 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 )) |
13 | simpr 485 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → 𝑥 ∈ ℕ0) | |
14 | gsummptnn0fz.f | . . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵) | |
15 | 14 | anim1ci 616 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → (𝑥 ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵)) |
16 | rspcsbela 4432 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵) → ⦋𝑥 / 𝑘⦌𝐶 ∈ 𝐵) | |
17 | 15, 16 | syl 17 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → ⦋𝑥 / 𝑘⦌𝐶 ∈ 𝐵) |
18 | 13, 17 | jca 512 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → (𝑥 ∈ ℕ0 ∧ ⦋𝑥 / 𝑘⦌𝐶 ∈ 𝐵)) |
19 | 18 | adantr 481 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ⦋𝑥 / 𝑘⦌𝐶 = 0 ) → (𝑥 ∈ ℕ0 ∧ ⦋𝑥 / 𝑘⦌𝐶 ∈ 𝐵)) |
20 | eqid 2732 | . . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 ↦ 𝐶) = (𝑘 ∈ ℕ0 ↦ 𝐶) | |
21 | 20 | fvmpts 6988 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℕ0 ∧ ⦋𝑥 / 𝑘⦌𝐶 ∈ 𝐵) → ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) = ⦋𝑥 / 𝑘⦌𝐶) |
22 | 19, 21 | syl 17 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ⦋𝑥 / 𝑘⦌𝐶 = 0 ) → ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) = ⦋𝑥 / 𝑘⦌𝐶) |
23 | simpr 485 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ⦋𝑥 / 𝑘⦌𝐶 = 0 ) → ⦋𝑥 / 𝑘⦌𝐶 = 0 ) | |
24 | 22, 23 | eqtrd 2772 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ⦋𝑥 / 𝑘⦌𝐶 = 0 ) → ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) = 0 ) |
25 | 24 | ex 413 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → (⦋𝑥 / 𝑘⦌𝐶 = 0 → ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) = 0 )) |
26 | 25 | imim2d 57 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → ((𝑆 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 ) → (𝑆 < 𝑥 → ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) = 0 ))) |
27 | 26 | ralimdva 3167 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 ) → ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) = 0 ))) |
28 | 12, 27 | mpd 15 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) = 0 )) |
29 | gsummptnn0fz.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
30 | gsummptnn0fz.0 | . . 3 ⊢ 0 = (0g‘𝐺) | |
31 | gsummptnn0fz.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
32 | 20 | fmpt 7095 | . . . . 5 ⊢ (∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵 ↔ (𝑘 ∈ ℕ0 ↦ 𝐶):ℕ0⟶𝐵) |
33 | 14, 32 | sylib 217 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐶):ℕ0⟶𝐵) |
34 | 29 | fvexi 6893 | . . . . . 6 ⊢ 𝐵 ∈ V |
35 | nn0ex 12462 | . . . . . 6 ⊢ ℕ0 ∈ V | |
36 | 34, 35 | pm3.2i 471 | . . . . 5 ⊢ (𝐵 ∈ V ∧ ℕ0 ∈ V) |
37 | elmapg 8818 | . . . . 5 ⊢ ((𝐵 ∈ V ∧ ℕ0 ∈ V) → ((𝑘 ∈ ℕ0 ↦ 𝐶) ∈ (𝐵 ↑m ℕ0) ↔ (𝑘 ∈ ℕ0 ↦ 𝐶):ℕ0⟶𝐵)) | |
38 | 36, 37 | mp1i 13 | . . . 4 ⊢ (𝜑 → ((𝑘 ∈ ℕ0 ↦ 𝐶) ∈ (𝐵 ↑m ℕ0) ↔ (𝑘 ∈ ℕ0 ↦ 𝐶):ℕ0⟶𝐵)) |
39 | 33, 38 | mpbird 256 | . . 3 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐶) ∈ (𝐵 ↑m ℕ0)) |
40 | gsummptnn0fz.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ ℕ0) | |
41 | fz0ssnn0 13580 | . . . . 5 ⊢ (0...𝑆) ⊆ ℕ0 | |
42 | resmpt 6028 | . . . . 5 ⊢ ((0...𝑆) ⊆ ℕ0 → ((𝑘 ∈ ℕ0 ↦ 𝐶) ↾ (0...𝑆)) = (𝑘 ∈ (0...𝑆) ↦ 𝐶)) | |
43 | 41, 42 | ax-mp 5 | . . . 4 ⊢ ((𝑘 ∈ ℕ0 ↦ 𝐶) ↾ (0...𝑆)) = (𝑘 ∈ (0...𝑆) ↦ 𝐶) |
44 | 43 | eqcomi 2741 | . . 3 ⊢ (𝑘 ∈ (0...𝑆) ↦ 𝐶) = ((𝑘 ∈ ℕ0 ↦ 𝐶) ↾ (0...𝑆)) |
45 | 29, 30, 31, 39, 40, 44 | fsfnn0gsumfsffz 19812 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) = 0 ) → (𝐺 Σg (𝑘 ∈ ℕ0 ↦ 𝐶)) = (𝐺 Σg (𝑘 ∈ (0...𝑆) ↦ 𝐶)))) |
46 | 28, 45 | mpd 15 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ ℕ0 ↦ 𝐶)) = (𝐺 Σg (𝑘 ∈ (0...𝑆) ↦ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 Vcvv 3474 ⦋csb 3890 ⊆ wss 3945 class class class wbr 5142 ↦ cmpt 5225 ↾ cres 5672 ⟶wf 6529 ‘cfv 6533 (class class class)co 7394 ↑m cmap 8805 0cc0 11094 < clt 11232 ℕ0cn0 12456 ...cfz 13468 Basecbs 17128 0gc0g 17369 Σg cgsu 17370 CMndccmn 19614 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-cnex 11150 ax-resscn 11151 ax-1cn 11152 ax-icn 11153 ax-addcl 11154 ax-addrcl 11155 ax-mulcl 11156 ax-mulrcl 11157 ax-mulcom 11158 ax-addass 11159 ax-mulass 11160 ax-distr 11161 ax-i2m1 11162 ax-1ne0 11163 ax-1rid 11164 ax-rnegex 11165 ax-rrecex 11166 ax-cnre 11167 ax-pre-lttri 11168 ax-pre-lttrn 11169 ax-pre-ltadd 11170 ax-pre-mulgt0 11171 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-se 5626 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7350 df-ov 7397 df-oprab 7398 df-mpo 7399 df-om 7840 df-1st 7959 df-2nd 7960 df-supp 8131 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-rdg 8394 df-1o 8450 df-er 8688 df-map 8807 df-en 8925 df-dom 8926 df-sdom 8927 df-fin 8928 df-fsupp 9347 df-oi 9489 df-card 9918 df-pnf 11234 df-mnf 11235 df-xr 11236 df-ltxr 11237 df-le 11238 df-sub 11430 df-neg 11431 df-nn 12197 df-n0 12457 df-z 12543 df-uz 12807 df-fz 13469 df-fzo 13612 df-seq 13951 df-hash 14275 df-0g 17371 df-gsum 17372 df-mgm 18545 df-sgrp 18594 df-mnd 18605 df-cntz 19149 df-cmn 19616 |
This theorem is referenced by: gsummptnn0fzfv 19816 telgsums 19822 gsummoncoe1 21759 pmatcollpwfi 22215 mp2pm2mplem4 22242 |
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