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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 13974 | . 2 ⊢ (𝜑 → (0...𝑁) ∈ Fin) | |
5 | gsummptfzsplitl.y | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑌 ∈ 𝐵) | |
6 | incom 4199 | . . . 4 ⊢ ((1...𝑁) ∩ {0}) = ({0} ∩ (1...𝑁)) | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → ((1...𝑁) ∩ {0}) = ({0} ∩ (1...𝑁))) |
8 | 1e0p1 12752 | . . . . . 6 ⊢ 1 = (0 + 1) | |
9 | 8 | oveq1i 7429 | . . . . 5 ⊢ (1...𝑁) = ((0 + 1)...𝑁) |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝜑 → (1...𝑁) = ((0 + 1)...𝑁)) |
11 | 10 | ineq2d 4210 | . . 3 ⊢ (𝜑 → ({0} ∩ (1...𝑁)) = ({0} ∩ ((0 + 1)...𝑁))) |
12 | gsummptfzsplit.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
13 | elnn0uz 12900 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (ℤ≥‘0)) | |
14 | 13 | biimpi 215 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (ℤ≥‘0)) |
15 | fzpreddisj 13585 | . . . 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 2769 | . 2 ⊢ (𝜑 → ((1...𝑁) ∩ {0}) = ∅) |
18 | fzpred 13584 | . . . 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 4150 | . . . 4 ⊢ ({0} ∪ ((0 + 1)...𝑁)) = (((0 + 1)...𝑁) ∪ {0}) | |
21 | 0p1e1 12367 | . . . . . 6 ⊢ (0 + 1) = 1 | |
22 | 21 | oveq1i 7429 | . . . . 5 ⊢ ((0 + 1)...𝑁) = (1...𝑁) |
23 | 22 | uneq1i 4156 | . . . 4 ⊢ (((0 + 1)...𝑁) ∪ {0}) = ((1...𝑁) ∪ {0}) |
24 | 20, 23 | eqtri 2753 | . . 3 ⊢ ({0} ∪ ((0 + 1)...𝑁)) = ((1...𝑁) ∪ {0}) |
25 | 19, 24 | eqtrdi 2781 | . 2 ⊢ (𝜑 → (0...𝑁) = ((1...𝑁) ∪ {0})) |
26 | 1, 2, 3, 4, 5, 17, 25 | gsummptfidmsplit 19897 | 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 394 = wceq 1533 ∈ wcel 2098 ∪ cun 3942 ∩ cin 3943 ∅c0 4322 {csn 4630 ↦ cmpt 5232 ‘cfv 6549 (class class class)co 7419 0cc0 11140 1c1 11141 + caddc 11143 ℕ0cn0 12505 ℤ≥cuz 12855 ...cfz 13519 Basecbs 17183 +gcplusg 17236 Σg cgsu 17425 CMndccmn 19747 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-oi 9535 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-n0 12506 df-z 12592 df-uz 12856 df-fz 13520 df-fzo 13663 df-seq 14003 df-hash 14326 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-0g 17426 df-gsum 17427 df-mre 17569 df-mrc 17570 df-acs 17572 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18744 df-cntz 19280 df-cmn 19749 |
This theorem is referenced by: srgbinomlem4 20181 freshmansdream 21525 chfacfscmulgsum 22806 chfacfpmmulgsum 22810 cpmadugsumlemF 22822 |
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