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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 1910 | . . . . 5 ⊢ Ⅎ𝑥(𝑆 < 𝑘 → 𝐶 = 0 ) | |
3 | nfv 1910 | . . . . . 6 ⊢ Ⅎ𝑘 𝑆 < 𝑥 | |
4 | nfcsb1v 3914 | . . . . . . 7 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐶 | |
5 | 4 | nfeq1 2913 | . . . . . 6 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐶 = 0 |
6 | 3, 5 | nfim 1892 | . . . . 5 ⊢ Ⅎ𝑘(𝑆 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 ) |
7 | breq2 5146 | . . . . . 6 ⊢ (𝑘 = 𝑥 → (𝑆 < 𝑘 ↔ 𝑆 < 𝑥)) | |
8 | csbeq1a 3903 | . . . . . . 7 ⊢ (𝑘 = 𝑥 → 𝐶 = ⦋𝑥 / 𝑘⦌𝐶) | |
9 | 8 | eqeq1d 2729 | . . . . . 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 3298 | . . . 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 484 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → 𝑥 ∈ ℕ0) | |
14 | gsummptnn0fz.f | . . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵) | |
15 | 14 | anim1ci 615 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → (𝑥 ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵)) |
16 | rspcsbela 4431 | . . . . . . . . . . 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 2727 | . . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 ↦ 𝐶) = (𝑘 ∈ ℕ0 ↦ 𝐶) | |
21 | 20 | fvmpts 7002 | . . . . . . . 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 2767 | . . . . . 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 3162 | . . 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 7114 | . . . . 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 6905 | . . . . . 6 ⊢ 𝐵 ∈ V |
35 | nn0ex 12494 | . . . . . 6 ⊢ ℕ0 ∈ V | |
36 | 34, 35 | pm3.2i 470 | . . . . 5 ⊢ (𝐵 ∈ V ∧ ℕ0 ∈ V) |
37 | elmapg 8847 | . . . . 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 13614 | . . . . 5 ⊢ (0...𝑆) ⊆ ℕ0 | |
42 | resmpt 6035 | . . . . 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 2736 | . . 3 ⊢ (𝑘 ∈ (0...𝑆) ↦ 𝐶) = ((𝑘 ∈ ℕ0 ↦ 𝐶) ↾ (0...𝑆)) |
45 | 29, 30, 31, 39, 40, 44 | fsfnn0gsumfsffz 19922 | . 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 395 = wceq 1534 ∈ wcel 2099 ∀wral 3056 Vcvv 3469 ⦋csb 3889 ⊆ wss 3944 class class class wbr 5142 ↦ cmpt 5225 ↾ cres 5674 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ↑m cmap 8834 0cc0 11124 < clt 11264 ℕ0cn0 12488 ...cfz 13502 Basecbs 17165 0gc0g 17406 Σg cgsu 17407 CMndccmn 19719 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-map 8836 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9376 df-oi 9519 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-n0 12489 df-z 12575 df-uz 12839 df-fz 13503 df-fzo 13646 df-seq 13985 df-hash 14308 df-0g 17408 df-gsum 17409 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-cntz 19252 df-cmn 19721 |
This theorem is referenced by: gsummptnn0fzfv 19926 telgsums 19932 gsummoncoe1 22201 pmatcollpwfi 22658 mp2pm2mplem4 22685 |
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