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Mirrors > Home > MPE Home > Th. List > gsumsubm | Structured version Visualization version GIF version |
Description: Evaluate a group sum in a submonoid. (Contributed by Mario Carneiro, 19-Dec-2014.) |
Ref | Expression |
---|---|
gsumsubm.a | β’ (π β π΄ β π) |
gsumsubm.s | β’ (π β π β (SubMndβπΊ)) |
gsumsubm.f | β’ (π β πΉ:π΄βΆπ) |
gsumsubm.h | β’ π» = (πΊ βΎs π) |
Ref | Expression |
---|---|
gsumsubm | β’ (π β (πΊ Ξ£g πΉ) = (π» Ξ£g πΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . 2 β’ (BaseβπΊ) = (BaseβπΊ) | |
2 | eqid 2726 | . 2 β’ (+gβπΊ) = (+gβπΊ) | |
3 | gsumsubm.h | . 2 β’ π» = (πΊ βΎs π) | |
4 | gsumsubm.s | . . 3 β’ (π β π β (SubMndβπΊ)) | |
5 | submrcl 18727 | . . 3 β’ (π β (SubMndβπΊ) β πΊ β Mnd) | |
6 | 4, 5 | syl 17 | . 2 β’ (π β πΊ β Mnd) |
7 | gsumsubm.a | . 2 β’ (π β π΄ β π) | |
8 | 1 | submss 18734 | . . 3 β’ (π β (SubMndβπΊ) β π β (BaseβπΊ)) |
9 | 4, 8 | syl 17 | . 2 β’ (π β π β (BaseβπΊ)) |
10 | gsumsubm.f | . 2 β’ (π β πΉ:π΄βΆπ) | |
11 | eqid 2726 | . . . 4 β’ (0gβπΊ) = (0gβπΊ) | |
12 | 11 | subm0cl 18736 | . . 3 β’ (π β (SubMndβπΊ) β (0gβπΊ) β π) |
13 | 4, 12 | syl 17 | . 2 β’ (π β (0gβπΊ) β π) |
14 | 1, 2, 11 | mndlrid 18686 | . . 3 β’ ((πΊ β Mnd β§ π₯ β (BaseβπΊ)) β (((0gβπΊ)(+gβπΊ)π₯) = π₯ β§ (π₯(+gβπΊ)(0gβπΊ)) = π₯)) |
15 | 6, 14 | sylan 579 | . 2 β’ ((π β§ π₯ β (BaseβπΊ)) β (((0gβπΊ)(+gβπΊ)π₯) = π₯ β§ (π₯(+gβπΊ)(0gβπΊ)) = π₯)) |
16 | 1, 2, 3, 6, 7, 9, 10, 13, 15 | gsumress 18615 | 1 β’ (π β (πΊ Ξ£g πΉ) = (π» Ξ£g πΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β wss 3943 βΆwf 6533 βcfv 6537 (class class class)co 7405 Basecbs 17153 βΎs cress 17182 +gcplusg 17206 0gc0g 17394 Ξ£g cgsu 17395 Mndcmnd 18667 SubMndcsubmnd 18712 |
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-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-iun 4992 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-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-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 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-seq 13973 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-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 |
This theorem is referenced by: gsumzsubmcl 19838 resspsrmul 21879 gsumply1subr 22107 tsmssubm 24002 amgmlem 26877 lgseisenlem4 27266 ply1degltdimlem 33225 fedgmullem1 33232 sge0tsms 45665 amgmwlem 48120 amgmlemALT 48121 |
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