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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 1914 | . . . . 5 ⊢ Ⅎ𝑥(𝑆 < 𝑘 → 𝐶 = 0 ) | |
| 3 | nfv 1914 | . . . . . 6 ⊢ Ⅎ𝑘 𝑆 < 𝑥 | |
| 4 | nfcsb1v 3923 | . . . . . . 7 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐶 | |
| 5 | 4 | nfeq1 2921 | . . . . . 6 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐶 = 0 |
| 6 | 3, 5 | nfim 1896 | . . . . 5 ⊢ Ⅎ𝑘(𝑆 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 ) |
| 7 | breq2 5147 | . . . . . 6 ⊢ (𝑘 = 𝑥 → (𝑆 < 𝑘 ↔ 𝑆 < 𝑥)) | |
| 8 | csbeq1a 3913 | . . . . . . 7 ⊢ (𝑘 = 𝑥 → 𝐶 = ⦋𝑥 / 𝑘⦌𝐶) | |
| 9 | 8 | eqeq1d 2739 | . . . . . 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 3306 | . . . 4 ⊢ (∀𝑘 ∈ ℕ0 (𝑆 < 𝑘 → 𝐶 = 0 ) ↔ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 )) |
| 12 | 1, 11 | sylib 218 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 )) |
| 13 | simpr 484 | . . . . . . . . . 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 4438 | . . . . . . . . . . 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 511 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → (𝑥 ∈ ℕ0 ∧ ⦋𝑥 / 𝑘⦌𝐶 ∈ 𝐵)) |
| 19 | 18 | adantr 480 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ⦋𝑥 / 𝑘⦌𝐶 = 0 ) → (𝑥 ∈ ℕ0 ∧ ⦋𝑥 / 𝑘⦌𝐶 ∈ 𝐵)) |
| 20 | eqid 2737 | . . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 ↦ 𝐶) = (𝑘 ∈ ℕ0 ↦ 𝐶) | |
| 21 | 20 | fvmpts 7019 | . . . . . . . 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 484 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ⦋𝑥 / 𝑘⦌𝐶 = 0 ) → ⦋𝑥 / 𝑘⦌𝐶 = 0 ) | |
| 24 | 22, 23 | eqtrd 2777 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ⦋𝑥 / 𝑘⦌𝐶 = 0 ) → ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) = 0 ) |
| 25 | 24 | ex 412 | . . . . 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 7130 | . . . . 5 ⊢ (∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵 ↔ (𝑘 ∈ ℕ0 ↦ 𝐶):ℕ0⟶𝐵) |
| 33 | 14, 32 | sylib 218 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐶):ℕ0⟶𝐵) |
| 34 | 29 | fvexi 6920 | . . . . . 6 ⊢ 𝐵 ∈ V |
| 35 | nn0ex 12532 | . . . . . 6 ⊢ ℕ0 ∈ V | |
| 36 | 34, 35 | pm3.2i 470 | . . . . 5 ⊢ (𝐵 ∈ V ∧ ℕ0 ∈ V) |
| 37 | elmapg 8879 | . . . . 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 257 | . . 3 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐶) ∈ (𝐵 ↑m ℕ0)) |
| 40 | gsummptnn0fz.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ ℕ0) | |
| 41 | fz0ssnn0 13662 | . . . . 5 ⊢ (0...𝑆) ⊆ ℕ0 | |
| 42 | resmpt 6055 | . . . . 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 2746 | . . 3 ⊢ (𝑘 ∈ (0...𝑆) ↦ 𝐶) = ((𝑘 ∈ ℕ0 ↦ 𝐶) ↾ (0...𝑆)) |
| 45 | 29, 30, 31, 39, 40, 44 | fsfnn0gsumfsffz 20001 | . 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ⦋csb 3899 ⊆ wss 3951 class class class wbr 5143 ↦ cmpt 5225 ↾ cres 5687 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 0cc0 11155 < clt 11295 ℕ0cn0 12526 ...cfz 13547 Basecbs 17247 0gc0g 17484 Σg cgsu 17485 CMndccmn 19798 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-0g 17486 df-gsum 17487 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-cntz 19335 df-cmn 19800 |
| This theorem is referenced by: gsummptnn0fzfv 20005 telgsums 20011 gsummoncoe1 22312 pmatcollpwfi 22788 mp2pm2mplem4 22815 |
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