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Mirrors > Home > MPE Home > Th. List > Mathboxes > gsumsplit2f | Structured version Visualization version GIF version |
Description: Split a group sum into two parts. (Contributed by AV, 4-Sep-2019.) |
Ref | Expression |
---|---|
gsumsplit2f.n | ⊢ Ⅎ𝑘𝜑 |
gsumsplit2f.b | ⊢ 𝐵 = (Base‘𝐺) |
gsumsplit2f.z | ⊢ 0 = (0g‘𝐺) |
gsumsplit2f.p | ⊢ + = (+g‘𝐺) |
gsumsplit2f.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsumsplit2f.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
gsumsplit2f.f | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) |
gsumsplit2f.w | ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 ) |
gsumsplit2f.i | ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) |
gsumsplit2f.u | ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷)) |
Ref | Expression |
---|---|
gsumsplit2f | ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ 𝐷 ↦ 𝑋)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumsplit2f.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsumsplit2f.z | . . 3 ⊢ 0 = (0g‘𝐺) | |
3 | gsumsplit2f.p | . . 3 ⊢ + = (+g‘𝐺) | |
4 | gsumsplit2f.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
5 | gsumsplit2f.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
6 | gsumsplit2f.n | . . . 4 ⊢ Ⅎ𝑘𝜑 | |
7 | gsumsplit2f.f | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) | |
8 | eqid 2818 | . . . 4 ⊢ (𝑘 ∈ 𝐴 ↦ 𝑋) = (𝑘 ∈ 𝐴 ↦ 𝑋) | |
9 | 6, 7, 8 | fmptdf 6873 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋):𝐴⟶𝐵) |
10 | gsumsplit2f.w | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 ) | |
11 | gsumsplit2f.i | . . 3 ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) | |
12 | gsumsplit2f.u | . . 3 ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷)) | |
13 | 1, 2, 3, 4, 5, 9, 10, 11, 12 | gsumsplit 18977 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐶)) + (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐷)))) |
14 | ssun1 4145 | . . . . . 6 ⊢ 𝐶 ⊆ (𝐶 ∪ 𝐷) | |
15 | 14, 12 | sseqtrrid 4017 | . . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
16 | 15 | resmptd 5901 | . . . 4 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐶) = (𝑘 ∈ 𝐶 ↦ 𝑋)) |
17 | 16 | oveq2d 7161 | . . 3 ⊢ (𝜑 → (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐶)) = (𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑋))) |
18 | ssun2 4146 | . . . . . 6 ⊢ 𝐷 ⊆ (𝐶 ∪ 𝐷) | |
19 | 18, 12 | sseqtrrid 4017 | . . . . 5 ⊢ (𝜑 → 𝐷 ⊆ 𝐴) |
20 | 19 | resmptd 5901 | . . . 4 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐷) = (𝑘 ∈ 𝐷 ↦ 𝑋)) |
21 | 20 | oveq2d 7161 | . . 3 ⊢ (𝜑 → (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐷)) = (𝐺 Σg (𝑘 ∈ 𝐷 ↦ 𝑋))) |
22 | 17, 21 | oveq12d 7163 | . 2 ⊢ (𝜑 → ((𝐺 Σg ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐶)) + (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐷))) = ((𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ 𝐷 ↦ 𝑋)))) |
23 | 13, 22 | eqtrd 2853 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ 𝐷 ↦ 𝑋)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 Ⅎwnf 1775 ∈ wcel 2105 ∪ cun 3931 ∩ cin 3932 ∅c0 4288 class class class wbr 5057 ↦ cmpt 5137 ↾ cres 5550 ‘cfv 6348 (class class class)co 7145 finSupp cfsupp 8821 Basecbs 16471 +gcplusg 16553 0gc0g 16701 Σg cgsu 16702 CMndccmn 18835 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-seq 13358 df-hash 13679 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-0g 16703 df-gsum 16704 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-cntz 18385 df-cmn 18837 |
This theorem is referenced by: gsumdifsndf 43965 |
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