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Mirrors > Home > MPE Home > Th. List > gsumzsubmcl | Structured version Visualization version GIF version |
Description: Closure of a group sum in a submonoid. (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 3-Jun-2019.) |
Ref | Expression |
---|---|
gsumzsubmcl.0 | β’ 0 = (0gβπΊ) |
gsumzsubmcl.z | β’ π = (CntzβπΊ) |
gsumzsubmcl.g | β’ (π β πΊ β Mnd) |
gsumzsubmcl.a | β’ (π β π΄ β π) |
gsumzsubmcl.s | β’ (π β π β (SubMndβπΊ)) |
gsumzsubmcl.f | β’ (π β πΉ:π΄βΆπ) |
gsumzsubmcl.c | β’ (π β ran πΉ β (πβran πΉ)) |
gsumzsubmcl.w | β’ (π β πΉ finSupp 0 ) |
Ref | Expression |
---|---|
gsumzsubmcl | β’ (π β (πΊ Ξ£g πΉ) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2725 | . . 3 β’ (Baseβ(πΊ βΎs π)) = (Baseβ(πΊ βΎs π)) | |
2 | eqid 2725 | . . 3 β’ (0gβ(πΊ βΎs π)) = (0gβ(πΊ βΎs π)) | |
3 | eqid 2725 | . . 3 β’ (Cntzβ(πΊ βΎs π)) = (Cntzβ(πΊ βΎs π)) | |
4 | gsumzsubmcl.s | . . . 4 β’ (π β π β (SubMndβπΊ)) | |
5 | eqid 2725 | . . . . 5 β’ (πΊ βΎs π) = (πΊ βΎs π) | |
6 | 5 | submmnd 18767 | . . . 4 β’ (π β (SubMndβπΊ) β (πΊ βΎs π) β Mnd) |
7 | 4, 6 | syl 17 | . . 3 β’ (π β (πΊ βΎs π) β Mnd) |
8 | gsumzsubmcl.a | . . 3 β’ (π β π΄ β π) | |
9 | gsumzsubmcl.f | . . . 4 β’ (π β πΉ:π΄βΆπ) | |
10 | 5 | submbas 18768 | . . . . . 6 β’ (π β (SubMndβπΊ) β π = (Baseβ(πΊ βΎs π))) |
11 | 4, 10 | syl 17 | . . . . 5 β’ (π β π = (Baseβ(πΊ βΎs π))) |
12 | 11 | feq3d 6703 | . . . 4 β’ (π β (πΉ:π΄βΆπ β πΉ:π΄βΆ(Baseβ(πΊ βΎs π)))) |
13 | 9, 12 | mpbid 231 | . . 3 β’ (π β πΉ:π΄βΆ(Baseβ(πΊ βΎs π))) |
14 | gsumzsubmcl.c | . . . . 5 β’ (π β ran πΉ β (πβran πΉ)) | |
15 | 9 | frnd 6724 | . . . . 5 β’ (π β ran πΉ β π) |
16 | 14, 15 | ssind 4227 | . . . 4 β’ (π β ran πΉ β ((πβran πΉ) β© π)) |
17 | gsumzsubmcl.z | . . . . . 6 β’ π = (CntzβπΊ) | |
18 | 5, 17, 3 | resscntz 19286 | . . . . 5 β’ ((π β (SubMndβπΊ) β§ ran πΉ β π) β ((Cntzβ(πΊ βΎs π))βran πΉ) = ((πβran πΉ) β© π)) |
19 | 4, 15, 18 | syl2anc 582 | . . . 4 β’ (π β ((Cntzβ(πΊ βΎs π))βran πΉ) = ((πβran πΉ) β© π)) |
20 | 16, 19 | sseqtrrd 4014 | . . 3 β’ (π β ran πΉ β ((Cntzβ(πΊ βΎs π))βran πΉ)) |
21 | gsumzsubmcl.w | . . . 4 β’ (π β πΉ finSupp 0 ) | |
22 | gsumzsubmcl.0 | . . . . . 6 β’ 0 = (0gβπΊ) | |
23 | 5, 22 | subm0 18769 | . . . . 5 β’ (π β (SubMndβπΊ) β 0 = (0gβ(πΊ βΎs π))) |
24 | 4, 23 | syl 17 | . . . 4 β’ (π β 0 = (0gβ(πΊ βΎs π))) |
25 | 21, 24 | breqtrd 5169 | . . 3 β’ (π β πΉ finSupp (0gβ(πΊ βΎs π))) |
26 | 1, 2, 3, 7, 8, 13, 20, 25 | gsumzcl 19868 | . 2 β’ (π β ((πΊ βΎs π) Ξ£g πΉ) β (Baseβ(πΊ βΎs π))) |
27 | 8, 4, 9, 5 | gsumsubm 18789 | . 2 β’ (π β (πΊ Ξ£g πΉ) = ((πΊ βΎs π) Ξ£g πΉ)) |
28 | 26, 27, 11 | 3eltr4d 2840 | 1 β’ (π β (πΊ Ξ£g πΉ) β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β© cin 3939 β wss 3940 class class class wbr 5143 ran crn 5673 βΆwf 6538 βcfv 6542 (class class class)co 7415 finSupp cfsupp 9383 Basecbs 17177 βΎs cress 17206 0gc0g 17418 Ξ£g cgsu 17419 Mndcmnd 18691 SubMndcsubmnd 18736 Cntzccntz 19268 |
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 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
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 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fsupp 9384 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-n0 12501 df-z 12587 df-uz 12851 df-fz 13515 df-fzo 13658 df-seq 13997 df-hash 14320 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-0g 17420 df-gsum 17421 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-submnd 18738 df-cntz 19270 |
This theorem is referenced by: gsumsubmcl 19876 gsumzadd 19879 dprdfadd 19979 dprdfeq0 19981 dprdlub 19985 |
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