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Mirrors > Home > MPE Home > Th. List > gsummptfzsplitl | 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 left. (Contributed by AV, 7-Nov-2019.) |
Ref | Expression |
---|---|
gsummptfzsplit.b | ⊢ 𝐵 = (Base‘𝐺) |
gsummptfzsplit.p | ⊢ + = (+g‘𝐺) |
gsummptfzsplit.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsummptfzsplit.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
gsummptfzsplitl.y | ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
gsummptfzsplitl | ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (0...𝑁) ↦ 𝑌)) = ((𝐺 Σg (𝑘 ∈ (1...𝑁) ↦ 𝑌)) + (𝐺 Σg (𝑘 ∈ {0} ↦ 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 13342 | . 2 ⊢ (𝜑 → (0...𝑁) ∈ Fin) | |
5 | gsummptfzsplitl.y | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑌 ∈ 𝐵) | |
6 | incom 4178 | . . . 4 ⊢ ((1...𝑁) ∩ {0}) = ({0} ∩ (1...𝑁)) | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → ((1...𝑁) ∩ {0}) = ({0} ∩ (1...𝑁))) |
8 | 1e0p1 12141 | . . . . . 6 ⊢ 1 = (0 + 1) | |
9 | 8 | oveq1i 7166 | . . . . 5 ⊢ (1...𝑁) = ((0 + 1)...𝑁) |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝜑 → (1...𝑁) = ((0 + 1)...𝑁)) |
11 | 10 | ineq2d 4189 | . . 3 ⊢ (𝜑 → ({0} ∩ (1...𝑁)) = ({0} ∩ ((0 + 1)...𝑁))) |
12 | gsummptfzsplit.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
13 | elnn0uz 12284 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (ℤ≥‘0)) | |
14 | 13 | biimpi 218 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (ℤ≥‘0)) |
15 | fzpreddisj 12957 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘0) → ({0} ∩ ((0 + 1)...𝑁)) = ∅) | |
16 | 12, 14, 15 | 3syl 18 | . . 3 ⊢ (𝜑 → ({0} ∩ ((0 + 1)...𝑁)) = ∅) |
17 | 7, 11, 16 | 3eqtrd 2860 | . 2 ⊢ (𝜑 → ((1...𝑁) ∩ {0}) = ∅) |
18 | fzpred 12956 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘0) → (0...𝑁) = ({0} ∪ ((0 + 1)...𝑁))) | |
19 | 12, 14, 18 | 3syl 18 | . . 3 ⊢ (𝜑 → (0...𝑁) = ({0} ∪ ((0 + 1)...𝑁))) |
20 | uncom 4129 | . . . 4 ⊢ ({0} ∪ ((0 + 1)...𝑁)) = (((0 + 1)...𝑁) ∪ {0}) | |
21 | 0p1e1 11760 | . . . . . 6 ⊢ (0 + 1) = 1 | |
22 | 21 | oveq1i 7166 | . . . . 5 ⊢ ((0 + 1)...𝑁) = (1...𝑁) |
23 | 22 | uneq1i 4135 | . . . 4 ⊢ (((0 + 1)...𝑁) ∪ {0}) = ((1...𝑁) ∪ {0}) |
24 | 20, 23 | eqtri 2844 | . . 3 ⊢ ({0} ∪ ((0 + 1)...𝑁)) = ((1...𝑁) ∪ {0}) |
25 | 19, 24 | syl6eq 2872 | . 2 ⊢ (𝜑 → (0...𝑁) = ((1...𝑁) ∪ {0})) |
26 | 1, 2, 3, 4, 5, 17, 25 | gsummptfidmsplit 19050 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (0...𝑁) ↦ 𝑌)) = ((𝐺 Σg (𝑘 ∈ (1...𝑁) ↦ 𝑌)) + (𝐺 Σg (𝑘 ∈ {0} ↦ 𝑌)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∪ cun 3934 ∩ cin 3935 ∅c0 4291 {csn 4567 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 0cc0 10537 1c1 10538 + caddc 10540 ℕ0cn0 11898 ℤ≥cuz 12244 ...cfz 12893 Basecbs 16483 +gcplusg 16565 Σg cgsu 16714 CMndccmn 18906 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-0g 16715 df-gsum 16716 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-cntz 18447 df-cmn 18908 |
This theorem is referenced by: srgbinomlem4 19293 chfacfscmulgsum 21468 chfacfpmmulgsum 21472 cpmadugsumlemF 21484 freshmansdream 30859 |
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