Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > gsummptfzsplit | Structured version Visualization version GIF version |
Description: Split a group sum expressed as mapping with a finite set of sequential integers as domain into two parts, extracting a singleton from the right. (Contributed by AV, 25-Oct-2019.) |
Ref | Expression |
---|---|
gsummptfzsplit.b | ⊢ 𝐵 = (Base‘𝐺) |
gsummptfzsplit.p | ⊢ + = (+g‘𝐺) |
gsummptfzsplit.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsummptfzsplit.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
gsummptfzsplit.y | ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
gsummptfzsplit | ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (0...(𝑁 + 1)) ↦ 𝑌)) = ((𝐺 Σg (𝑘 ∈ (0...𝑁) ↦ 𝑌)) + (𝐺 Σg (𝑘 ∈ {(𝑁 + 1)} ↦ 𝑌)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsummptfzsplit.b | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsummptfzsplit.p | . 2 ⊢ + = (+g‘𝐺) | |
3 | gsummptfzsplit.g | . 2 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
4 | fzfid 13739 | . 2 ⊢ (𝜑 → (0...(𝑁 + 1)) ∈ Fin) | |
5 | gsummptfzsplit.y | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → 𝑌 ∈ 𝐵) | |
6 | fzp1disj 13361 | . . 3 ⊢ ((0...𝑁) ∩ {(𝑁 + 1)}) = ∅ | |
7 | 6 | a1i 11 | . 2 ⊢ (𝜑 → ((0...𝑁) ∩ {(𝑁 + 1)}) = ∅) |
8 | gsummptfzsplit.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
9 | elnn0uz 12669 | . . . 4 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (ℤ≥‘0)) | |
10 | 8, 9 | sylib 217 | . . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0)) |
11 | fzsuc 13349 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘0) → (0...(𝑁 + 1)) = ((0...𝑁) ∪ {(𝑁 + 1)})) | |
12 | 10, 11 | syl 17 | . 2 ⊢ (𝜑 → (0...(𝑁 + 1)) = ((0...𝑁) ∪ {(𝑁 + 1)})) |
13 | 1, 2, 3, 4, 5, 7, 12 | gsummptfidmsplit 19576 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (0...(𝑁 + 1)) ↦ 𝑌)) = ((𝐺 Σg (𝑘 ∈ (0...𝑁) ↦ 𝑌)) + (𝐺 Σg (𝑘 ∈ {(𝑁 + 1)} ↦ 𝑌)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1539 ∈ wcel 2104 ∪ cun 3890 ∩ cin 3891 ∅c0 4262 {csn 4565 ↦ cmpt 5164 ‘cfv 6458 (class class class)co 7307 0cc0 10917 1c1 10918 + caddc 10920 ℕ0cn0 12279 ℤ≥cuz 12628 ...cfz 13285 Basecbs 16957 +gcplusg 17007 Σg cgsu 17196 CMndccmn 19431 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-iin 4934 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-se 5556 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-isom 6467 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-of 7565 df-om 7745 df-1st 7863 df-2nd 7864 df-supp 8009 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-er 8529 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-fsupp 9173 df-oi 9313 df-card 9741 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-nn 12020 df-2 12082 df-n0 12280 df-z 12366 df-uz 12629 df-fz 13286 df-fzo 13429 df-seq 13768 df-hash 14091 df-sets 16910 df-slot 16928 df-ndx 16940 df-base 16958 df-ress 16987 df-plusg 17020 df-0g 17197 df-gsum 17198 df-mre 17340 df-mrc 17341 df-acs 17343 df-mgm 18371 df-sgrp 18420 df-mnd 18431 df-submnd 18476 df-cntz 18968 df-cmn 19433 |
This theorem is referenced by: srgbinomlem3 19823 pmatcollpw3fi1lem1 21980 chfacfscmulgsum 22054 chfacfpmmulgsum 22058 cpmadugsumlemF 22070 freshmansdream 31529 |
Copyright terms: Public domain | W3C validator |