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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 13884 | . 2 ⊢ (𝜑 → (0...𝑁) ∈ Fin) | |
5 | gsummptfzsplitl.y | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑌 ∈ 𝐵) | |
6 | incom 4162 | . . . 4 ⊢ ((1...𝑁) ∩ {0}) = ({0} ∩ (1...𝑁)) | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → ((1...𝑁) ∩ {0}) = ({0} ∩ (1...𝑁))) |
8 | 1e0p1 12665 | . . . . . 6 ⊢ 1 = (0 + 1) | |
9 | 8 | oveq1i 7368 | . . . . 5 ⊢ (1...𝑁) = ((0 + 1)...𝑁) |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝜑 → (1...𝑁) = ((0 + 1)...𝑁)) |
11 | 10 | ineq2d 4173 | . . 3 ⊢ (𝜑 → ({0} ∩ (1...𝑁)) = ({0} ∩ ((0 + 1)...𝑁))) |
12 | gsummptfzsplit.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
13 | elnn0uz 12813 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (ℤ≥‘0)) | |
14 | 13 | biimpi 215 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (ℤ≥‘0)) |
15 | fzpreddisj 13496 | . . . 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 2777 | . 2 ⊢ (𝜑 → ((1...𝑁) ∩ {0}) = ∅) |
18 | fzpred 13495 | . . . 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 4114 | . . . 4 ⊢ ({0} ∪ ((0 + 1)...𝑁)) = (((0 + 1)...𝑁) ∪ {0}) | |
21 | 0p1e1 12280 | . . . . . 6 ⊢ (0 + 1) = 1 | |
22 | 21 | oveq1i 7368 | . . . . 5 ⊢ ((0 + 1)...𝑁) = (1...𝑁) |
23 | 22 | uneq1i 4120 | . . . 4 ⊢ (((0 + 1)...𝑁) ∪ {0}) = ((1...𝑁) ∪ {0}) |
24 | 20, 23 | eqtri 2761 | . . 3 ⊢ ({0} ∪ ((0 + 1)...𝑁)) = ((1...𝑁) ∪ {0}) |
25 | 19, 24 | eqtrdi 2789 | . 2 ⊢ (𝜑 → (0...𝑁) = ((1...𝑁) ∪ {0})) |
26 | 1, 2, 3, 4, 5, 17, 25 | gsummptfidmsplit 19712 | 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 397 = wceq 1542 ∈ wcel 2107 ∪ cun 3909 ∩ cin 3910 ∅c0 4283 {csn 4587 ↦ cmpt 5189 ‘cfv 6497 (class class class)co 7358 0cc0 11056 1c1 11057 + caddc 11059 ℕ0cn0 12418 ℤ≥cuz 12768 ...cfz 13430 Basecbs 17088 +gcplusg 17138 Σg cgsu 17327 CMndccmn 19567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-fzo 13574 df-seq 13913 df-hash 14237 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-0g 17328 df-gsum 17329 df-mre 17471 df-mrc 17472 df-acs 17474 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-submnd 18607 df-cntz 19102 df-cmn 19569 |
This theorem is referenced by: srgbinomlem4 19965 chfacfscmulgsum 22225 chfacfpmmulgsum 22229 cpmadugsumlemF 22241 freshmansdream 32116 |
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