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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 13344 | . 2 ⊢ (𝜑 → (0...𝑁) ∈ Fin) | |
5 | gsummptfzsplitl.y | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑌 ∈ 𝐵) | |
6 | incom 4180 | . . . 4 ⊢ ((1...𝑁) ∩ {0}) = ({0} ∩ (1...𝑁)) | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → ((1...𝑁) ∩ {0}) = ({0} ∩ (1...𝑁))) |
8 | 1e0p1 12143 | . . . . . 6 ⊢ 1 = (0 + 1) | |
9 | 8 | oveq1i 7168 | . . . . 5 ⊢ (1...𝑁) = ((0 + 1)...𝑁) |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝜑 → (1...𝑁) = ((0 + 1)...𝑁)) |
11 | 10 | ineq2d 4191 | . . 3 ⊢ (𝜑 → ({0} ∩ (1...𝑁)) = ({0} ∩ ((0 + 1)...𝑁))) |
12 | gsummptfzsplit.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
13 | elnn0uz 12286 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (ℤ≥‘0)) | |
14 | 13 | biimpi 218 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (ℤ≥‘0)) |
15 | fzpreddisj 12959 | . . . 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 2862 | . 2 ⊢ (𝜑 → ((1...𝑁) ∩ {0}) = ∅) |
18 | fzpred 12958 | . . . 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 4131 | . . . 4 ⊢ ({0} ∪ ((0 + 1)...𝑁)) = (((0 + 1)...𝑁) ∪ {0}) | |
21 | 0p1e1 11762 | . . . . . 6 ⊢ (0 + 1) = 1 | |
22 | 21 | oveq1i 7168 | . . . . 5 ⊢ ((0 + 1)...𝑁) = (1...𝑁) |
23 | 22 | uneq1i 4137 | . . . 4 ⊢ (((0 + 1)...𝑁) ∪ {0}) = ((1...𝑁) ∪ {0}) |
24 | 20, 23 | eqtri 2846 | . . 3 ⊢ ({0} ∪ ((0 + 1)...𝑁)) = ((1...𝑁) ∪ {0}) |
25 | 19, 24 | syl6eq 2874 | . 2 ⊢ (𝜑 → (0...𝑁) = ((1...𝑁) ∪ {0})) |
26 | 1, 2, 3, 4, 5, 17, 25 | gsummptfidmsplit 19052 | 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 3936 ∩ cin 3937 ∅c0 4293 {csn 4569 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 0cc0 10539 1c1 10540 + caddc 10542 ℕ0cn0 11900 ℤ≥cuz 12246 ...cfz 12895 Basecbs 16485 +gcplusg 16567 Σg cgsu 16716 CMndccmn 18908 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-0g 16717 df-gsum 16718 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-cntz 18449 df-cmn 18910 |
This theorem is referenced by: srgbinomlem4 19295 chfacfscmulgsum 21470 chfacfpmmulgsum 21474 cpmadugsumlemF 21486 freshmansdream 30861 |
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