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Mathbox for Alexander van der Vekens |
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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 2726 | . . . 4 ⊢ (𝑘 ∈ 𝐴 ↦ 𝑋) = (𝑘 ∈ 𝐴 ↦ 𝑋) | |
9 | 6, 7, 8 | fmptdf 7123 | . . 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 19922 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐶)) + (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐷)))) |
14 | ssun1 4170 | . . . . . 6 ⊢ 𝐶 ⊆ (𝐶 ∪ 𝐷) | |
15 | 14, 12 | sseqtrrid 4032 | . . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
16 | 15 | resmptd 6041 | . . . 4 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐶) = (𝑘 ∈ 𝐶 ↦ 𝑋)) |
17 | 16 | oveq2d 7432 | . . 3 ⊢ (𝜑 → (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐶)) = (𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑋))) |
18 | ssun2 4171 | . . . . . 6 ⊢ 𝐷 ⊆ (𝐶 ∪ 𝐷) | |
19 | 18, 12 | sseqtrrid 4032 | . . . . 5 ⊢ (𝜑 → 𝐷 ⊆ 𝐴) |
20 | 19 | resmptd 6041 | . . . 4 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐷) = (𝑘 ∈ 𝐷 ↦ 𝑋)) |
21 | 20 | oveq2d 7432 | . . 3 ⊢ (𝜑 → (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ 𝑋) ↾ 𝐷)) = (𝐺 Σg (𝑘 ∈ 𝐷 ↦ 𝑋))) |
22 | 17, 21 | oveq12d 7434 | . 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 2766 | 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 394 = wceq 1534 Ⅎwnf 1778 ∈ wcel 2099 ∪ cun 3944 ∩ cin 3945 ∅c0 4322 class class class wbr 5145 ↦ cmpt 5228 ↾ cres 5676 ‘cfv 6546 (class class class)co 7416 finSupp cfsupp 9398 Basecbs 17208 +gcplusg 17261 0gc0g 17449 Σg cgsu 17450 CMndccmn 19774 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-iin 4996 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7995 df-2nd 7996 df-supp 8167 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9399 df-oi 9546 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-nn 12259 df-2 12321 df-n0 12519 df-z 12605 df-uz 12869 df-fz 13533 df-fzo 13676 df-seq 14016 df-hash 14343 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-0g 17451 df-gsum 17452 df-mre 17594 df-mrc 17595 df-acs 17597 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-submnd 18769 df-cntz 19307 df-cmn 19776 |
This theorem is referenced by: gsumdifsndf 47594 |
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