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Mirrors > Home > MPE Home > Th. List > gsummptfidmsplitres | Structured version Visualization version GIF version |
Description: Split a group sum expressed as mapping with a finite domain into two parts using restrictions. (Contributed by AV, 23-Jul-2019.) |
Ref | Expression |
---|---|
gsummptfidmsplit.b | ⊢ 𝐵 = (Base‘𝐺) |
gsummptfidmsplit.p | ⊢ + = (+g‘𝐺) |
gsummptfidmsplit.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsummptfidmsplit.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
gsummptfidmsplit.y | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑌 ∈ 𝐵) |
gsummptfidmsplit.i | ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) |
gsummptfidmsplit.u | ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷)) |
gsummptfidmsplitres.f | ⊢ 𝐹 = (𝑘 ∈ 𝐴 ↦ 𝑌) |
Ref | Expression |
---|---|
gsummptfidmsplitres | ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ 𝐶)) + (𝐺 Σg (𝐹 ↾ 𝐷)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsummptfidmsplit.b | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
2 | eqid 2726 | . 2 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
3 | gsummptfidmsplit.p | . 2 ⊢ + = (+g‘𝐺) | |
4 | gsummptfidmsplit.g | . 2 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
5 | gsummptfidmsplit.a | . 2 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
6 | gsummptfidmsplit.y | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑌 ∈ 𝐵) | |
7 | gsummptfidmsplitres.f | . . 3 ⊢ 𝐹 = (𝑘 ∈ 𝐴 ↦ 𝑌) | |
8 | 6, 7 | fmptd 7119 | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
9 | fvexd 6907 | . . 3 ⊢ (𝜑 → (0g‘𝐺) ∈ V) | |
10 | 7, 5, 6, 9 | fsuppmptdm 9411 | . 2 ⊢ (𝜑 → 𝐹 finSupp (0g‘𝐺)) |
11 | gsummptfidmsplit.i | . 2 ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) | |
12 | gsummptfidmsplit.u | . 2 ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷)) | |
13 | 1, 2, 3, 4, 5, 8, 10, 11, 12 | gsumsplit 19921 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ 𝐶)) + (𝐺 Σg (𝐹 ↾ 𝐷)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 Vcvv 3464 ∪ cun 3946 ∩ cin 3947 ∅c0 4324 ↦ cmpt 5228 ↾ cres 5676 ‘cfv 6545 (class class class)co 7415 Fincfn 8965 Basecbs 17207 +gcplusg 17260 0gc0g 17448 Σg cgsu 17449 CMndccmn 19773 |
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 7737 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
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 3365 df-reu 3366 df-rab 3421 df-v 3466 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3968 df-nul 4325 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4908 df-int 4949 df-iun 4997 df-iin 4998 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 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6497 df-fun 6547 df-fn 6548 df-f 6549 df-f1 6550 df-fo 6551 df-f1o 6552 df-fv 6553 df-isom 6554 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7681 df-om 7868 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-2o 8488 df-er 8725 df-en 8966 df-dom 8967 df-sdom 8968 df-fin 8969 df-fsupp 9398 df-oi 9545 df-card 9974 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12258 df-2 12320 df-n0 12518 df-z 12604 df-uz 12868 df-fz 13532 df-fzo 13675 df-seq 14015 df-hash 14342 df-sets 17160 df-slot 17178 df-ndx 17190 df-base 17208 df-ress 17237 df-plusg 17273 df-0g 17450 df-gsum 17451 df-mre 17593 df-mrc 17594 df-acs 17596 df-mgm 18627 df-sgrp 18706 df-mnd 18722 df-submnd 18768 df-cntz 19306 df-cmn 19775 |
This theorem is referenced by: gsumpr 19948 mdetralt 22597 |
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