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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 | β’ π΅ = (BaseβπΊ) |
gsumsplit.z | β’ 0 = (0gβπΊ) |
gsumsplit.p | β’ + = (+gβπΊ) |
gsumsplit.g | β’ (π β πΊ β CMnd) |
gsumsplit.a | β’ (π β π΄ β π) |
gsumsplit.f | β’ (π β πΉ:π΄βΆπ΅) |
gsumsplit.w | β’ (π β πΉ finSupp 0 ) |
gsumsplit.i | β’ (π β (πΆ β© π·) = β ) |
gsumsplit.u | β’ (π β π΄ = (πΆ βͺ π·)) |
Ref | Expression |
---|---|
gsumsplit | β’ (π β (πΊ Ξ£g πΉ) = ((πΊ Ξ£g (πΉ βΎ πΆ)) + (πΊ Ξ£g (πΉ βΎ π·)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumsplit.b | . 2 β’ π΅ = (BaseβπΊ) | |
2 | gsumsplit.z | . 2 β’ 0 = (0gβπΊ) | |
3 | gsumsplit.p | . 2 β’ + = (+gβπΊ) | |
4 | eqid 2737 | . 2 β’ (CntzβπΊ) = (CntzβπΊ) | |
5 | gsumsplit.g | . . 3 β’ (π β πΊ β CMnd) | |
6 | cmnmnd 19586 | . . 3 β’ (πΊ β CMnd β πΊ β Mnd) | |
7 | 5, 6 | syl 17 | . 2 β’ (π β πΊ β Mnd) |
8 | gsumsplit.a | . 2 β’ (π β π΄ β π) | |
9 | gsumsplit.f | . 2 β’ (π β πΉ:π΄βΆπ΅) | |
10 | 1, 4, 5, 9 | cntzcmnf 19630 | . 2 β’ (π β ran πΉ β ((CntzβπΊ)βran πΉ)) |
11 | gsumsplit.w | . 2 β’ (π β πΉ finSupp 0 ) | |
12 | gsumsplit.i | . 2 β’ (π β (πΆ β© π·) = β ) | |
13 | gsumsplit.u | . 2 β’ (π β π΄ = (πΆ βͺ π·)) | |
14 | 1, 2, 3, 4, 7, 8, 9, 10, 11, 12, 13 | gsumzsplit 19711 | 1 β’ (π β (πΊ Ξ£g πΉ) = ((πΊ Ξ£g (πΉ βΎ πΆ)) + (πΊ Ξ£g (πΉ βΎ π·)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βͺ cun 3913 β© cin 3914 β c0 4287 class class class wbr 5110 βΎ cres 5640 βΆwf 6497 βcfv 6501 (class class class)co 7362 finSupp cfsupp 9312 Basecbs 17090 +gcplusg 17140 0gc0g 17328 Ξ£g cgsu 17329 Mndcmnd 18563 Cntzccntz 19102 CMndccmn 19569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-iin 4962 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-om 7808 df-1st 7926 df-2nd 7927 df-supp 8098 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9313 df-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-n0 12421 df-z 12507 df-uz 12771 df-fz 13432 df-fzo 13575 df-seq 13914 df-hash 14238 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-0g 17330 df-gsum 17331 df-mre 17473 df-mrc 17474 df-acs 17476 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-submnd 18609 df-cntz 19104 df-cmn 19571 |
This theorem is referenced by: gsumsplit2 19713 gsummptfidmsplitres 19715 gsum2dlem2 19755 islindf4 21260 tmdgsum 23462 xrge0gsumle 24212 amgm 26356 wilthlem2 26434 gsumesum 32698 gsumsplit2f 46188 |
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