![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > gsumsplit | Structured version Visualization version GIF version |
Description: Split a group sum into two parts. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 5-Jun-2019.) |
Ref | Expression |
---|---|
gsumsplit.b | ⊢ 𝐵 = (Base‘𝐺) |
gsumsplit.z | ⊢ 0 = (0g‘𝐺) |
gsumsplit.p | ⊢ + = (+g‘𝐺) |
gsumsplit.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsumsplit.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
gsumsplit.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
gsumsplit.w | ⊢ (𝜑 → 𝐹 finSupp 0 ) |
gsumsplit.i | ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) |
gsumsplit.u | ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷)) |
Ref | Expression |
---|---|
gsumsplit | ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ 𝐶)) + (𝐺 Σg (𝐹 ↾ 𝐷)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumsplit.b | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsumsplit.z | . 2 ⊢ 0 = (0g‘𝐺) | |
3 | gsumsplit.p | . 2 ⊢ + = (+g‘𝐺) | |
4 | eqid 2725 | . 2 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺) | |
5 | gsumsplit.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
6 | cmnmnd 19764 | . . 3 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
8 | gsumsplit.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
9 | gsumsplit.f | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
10 | 1, 4, 5, 9 | cntzcmnf 19812 | . 2 ⊢ (𝜑 → ran 𝐹 ⊆ ((Cntz‘𝐺)‘ran 𝐹)) |
11 | gsumsplit.w | . 2 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
12 | gsumsplit.i | . 2 ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) | |
13 | gsumsplit.u | . 2 ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷)) | |
14 | 1, 2, 3, 4, 7, 8, 9, 10, 11, 12, 13 | gsumzsplit 19894 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ 𝐶)) + (𝐺 Σg (𝐹 ↾ 𝐷)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∪ cun 3942 ∩ cin 3943 ∅c0 4322 class class class wbr 5149 ↾ cres 5680 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 finSupp cfsupp 9387 Basecbs 17183 +gcplusg 17236 0gc0g 17424 Σg cgsu 17425 Mndcmnd 18697 Cntzccntz 19278 CMndccmn 19747 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 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-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-oi 9535 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-n0 12506 df-z 12592 df-uz 12856 df-fz 13520 df-fzo 13663 df-seq 14003 df-hash 14326 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-0g 17426 df-gsum 17427 df-mre 17569 df-mrc 17570 df-acs 17572 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18744 df-cntz 19280 df-cmn 19749 |
This theorem is referenced by: gsumsplit2 19896 gsummptfidmsplitres 19898 gsum2dlem2 19938 islindf4 21789 tmdgsum 24043 xrge0gsumle 24793 amgm 26968 wilthlem2 27046 rprmdvdsprod 33346 gsumesum 33809 selvvvval 41953 evlselv 41955 gsumsplit2f 47428 |
Copyright terms: Public domain | W3C validator |