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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 2726 | . 2 β’ (CntzβπΊ) = (CntzβπΊ) | |
5 | gsumsplit.g | . . 3 β’ (π β πΊ β CMnd) | |
6 | cmnmnd 19717 | . . 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 19765 | . 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 19847 | 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 1533 β wcel 2098 βͺ cun 3941 β© cin 3942 β c0 4317 class class class wbr 5141 βΎ cres 5671 βΆwf 6533 βcfv 6537 (class class class)co 7405 finSupp cfsupp 9363 Basecbs 17153 +gcplusg 17206 0gc0g 17394 Ξ£g cgsu 17395 Mndcmnd 18667 Cntzccntz 19231 CMndccmn 19700 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-fzo 13634 df-seq 13973 df-hash 14296 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-0g 17396 df-gsum 17397 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-cntz 19233 df-cmn 19702 |
This theorem is referenced by: gsumsplit2 19849 gsummptfidmsplitres 19851 gsum2dlem2 19891 islindf4 21733 tmdgsum 23954 xrge0gsumle 24704 amgm 26878 wilthlem2 26956 gsumesum 33587 selvvvval 41714 evlselv 41716 gsumsplit2f 47130 |
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