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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 2738 | . 2 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺) | |
5 | gsumsplit.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
6 | cmnmnd 19033 | . . 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 19077 | . 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 19159 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ 𝐶)) + (𝐺 Σg (𝐹 ↾ 𝐷)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 ∪ cun 3839 ∩ cin 3840 ∅c0 4209 class class class wbr 5027 ↾ cres 5521 ⟶wf 6329 ‘cfv 6333 (class class class)co 7164 finSupp cfsupp 8899 Basecbs 16579 +gcplusg 16661 0gc0g 16809 Σg cgsu 16810 Mndcmnd 18020 Cntzccntz 18556 CMndccmn 19017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-iin 4881 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-isom 6342 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-of 7419 df-om 7594 df-1st 7707 df-2nd 7708 df-supp 7850 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-fsupp 8900 df-oi 9040 df-card 9434 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-2 11772 df-n0 11970 df-z 12056 df-uz 12318 df-fz 12975 df-fzo 13118 df-seq 13454 df-hash 13776 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-0g 16811 df-gsum 16812 df-mre 16953 df-mrc 16954 df-acs 16956 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-submnd 18066 df-cntz 18558 df-cmn 19019 |
This theorem is referenced by: gsumsplit2 19161 gsummptfidmsplitres 19163 gsum2dlem2 19203 islindf4 20647 tmdgsum 22839 xrge0gsumle 23578 amgm 25720 wilthlem2 25798 gsumesum 31589 gsumsplit2f 44892 |
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