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Mirrors > Home > MPE Home > Th. List > gsumply1subr | Structured version Visualization version GIF version |
Description: Evaluate a group sum in a polynomial ring over a subring. (Contributed by AV, 22-Sep-2019.) (Proof shortened by AV, 31-Jan-2020.) |
Ref | Expression |
---|---|
subrgply1.s | β’ π = (Poly1βπ ) |
subrgply1.h | β’ π» = (π βΎs π) |
subrgply1.u | β’ π = (Poly1βπ») |
subrgply1.b | β’ π΅ = (Baseβπ) |
gsumply1subr.s | β’ (π β π β (SubRingβπ )) |
gsumply1subr.a | β’ (π β π΄ β π) |
gsumply1subr.f | β’ (π β πΉ:π΄βΆπ΅) |
Ref | Expression |
---|---|
gsumply1subr | β’ (π β (π Ξ£g πΉ) = (π Ξ£g πΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumply1subr.a | . . 3 β’ (π β π΄ β π) | |
2 | gsumply1subr.s | . . . 4 β’ (π β π β (SubRingβπ )) | |
3 | subrgply1.s | . . . . 5 β’ π = (Poly1βπ ) | |
4 | subrgply1.h | . . . . 5 β’ π» = (π βΎs π) | |
5 | subrgply1.u | . . . . 5 β’ π = (Poly1βπ») | |
6 | subrgply1.b | . . . . 5 β’ π΅ = (Baseβπ) | |
7 | 3, 4, 5, 6 | subrgply1 22155 | . . . 4 β’ (π β (SubRingβπ ) β π΅ β (SubRingβπ)) |
8 | subrgsubg 20515 | . . . . 5 β’ (π΅ β (SubRingβπ) β π΅ β (SubGrpβπ)) | |
9 | subgsubm 19102 | . . . . 5 β’ (π΅ β (SubGrpβπ) β π΅ β (SubMndβπ)) | |
10 | 8, 9 | syl 17 | . . . 4 β’ (π΅ β (SubRingβπ) β π΅ β (SubMndβπ)) |
11 | 2, 7, 10 | 3syl 18 | . . 3 β’ (π β π΅ β (SubMndβπ)) |
12 | gsumply1subr.f | . . 3 β’ (π β πΉ:π΄βΆπ΅) | |
13 | eqid 2725 | . . 3 β’ (π βΎs π΅) = (π βΎs π΅) | |
14 | 1, 11, 12, 13 | gsumsubm 18786 | . 2 β’ (π β (π Ξ£g πΉ) = ((π βΎs π΅) Ξ£g πΉ)) |
15 | 12, 1 | fexd 7233 | . . 3 β’ (π β πΉ β V) |
16 | ovexd 7448 | . . 3 β’ (π β (π βΎs π΅) β V) | |
17 | 5 | fvexi 6904 | . . . 4 β’ π β V |
18 | 17 | a1i 11 | . . 3 β’ (π β π β V) |
19 | eqid 2725 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
20 | 6 | oveq2i 7424 | . . . . 5 β’ (π βΎs π΅) = (π βΎs (Baseβπ)) |
21 | 3, 4, 5, 19, 2, 20 | ressply1bas 22151 | . . . 4 β’ (π β (Baseβπ) = (Baseβ(π βΎs π΅))) |
22 | 21 | eqcomd 2731 | . . 3 β’ (π β (Baseβ(π βΎs π΅)) = (Baseβπ)) |
23 | 13 | subrgring 20512 | . . . . 5 β’ (π΅ β (SubRingβπ) β (π βΎs π΅) β Ring) |
24 | 7, 23 | syl 17 | . . . 4 β’ (π β (SubRingβπ ) β (π βΎs π΅) β Ring) |
25 | ringmgm 20183 | . . . 4 β’ ((π βΎs π΅) β Ring β (π βΎs π΅) β Mgm) | |
26 | 2, 24, 25 | 3syl 18 | . . 3 β’ (π β (π βΎs π΅) β Mgm) |
27 | simpl 481 | . . . . 5 β’ ((π β§ (π β (Baseβ(π βΎs π΅)) β§ π‘ β (Baseβ(π βΎs π΅)))) β π) | |
28 | 3, 4, 5, 6, 2, 13 | ressply1bas 22151 | . . . . . . . . . 10 β’ (π β π΅ = (Baseβ(π βΎs π΅))) |
29 | 28 | eqcomd 2731 | . . . . . . . . 9 β’ (π β (Baseβ(π βΎs π΅)) = π΅) |
30 | 29 | eleq2d 2811 | . . . . . . . 8 β’ (π β (π β (Baseβ(π βΎs π΅)) β π β π΅)) |
31 | 30 | biimpcd 248 | . . . . . . 7 β’ (π β (Baseβ(π βΎs π΅)) β (π β π β π΅)) |
32 | 31 | adantr 479 | . . . . . 6 β’ ((π β (Baseβ(π βΎs π΅)) β§ π‘ β (Baseβ(π βΎs π΅))) β (π β π β π΅)) |
33 | 32 | impcom 406 | . . . . 5 β’ ((π β§ (π β (Baseβ(π βΎs π΅)) β§ π‘ β (Baseβ(π βΎs π΅)))) β π β π΅) |
34 | 29 | eleq2d 2811 | . . . . . . . 8 β’ (π β (π‘ β (Baseβ(π βΎs π΅)) β π‘ β π΅)) |
35 | 34 | biimpcd 248 | . . . . . . 7 β’ (π‘ β (Baseβ(π βΎs π΅)) β (π β π‘ β π΅)) |
36 | 35 | adantl 480 | . . . . . 6 β’ ((π β (Baseβ(π βΎs π΅)) β§ π‘ β (Baseβ(π βΎs π΅))) β (π β π‘ β π΅)) |
37 | 36 | impcom 406 | . . . . 5 β’ ((π β§ (π β (Baseβ(π βΎs π΅)) β§ π‘ β (Baseβ(π βΎs π΅)))) β π‘ β π΅) |
38 | 3, 4, 5, 6, 2, 13 | ressply1add 22152 | . . . . 5 β’ ((π β§ (π β π΅ β§ π‘ β π΅)) β (π (+gβπ)π‘) = (π (+gβ(π βΎs π΅))π‘)) |
39 | 27, 33, 37, 38 | syl12anc 835 | . . . 4 β’ ((π β§ (π β (Baseβ(π βΎs π΅)) β§ π‘ β (Baseβ(π βΎs π΅)))) β (π (+gβπ)π‘) = (π (+gβ(π βΎs π΅))π‘)) |
40 | 39 | eqcomd 2731 | . . 3 β’ ((π β§ (π β (Baseβ(π βΎs π΅)) β§ π‘ β (Baseβ(π βΎs π΅)))) β (π (+gβ(π βΎs π΅))π‘) = (π (+gβπ)π‘)) |
41 | 12 | ffund 6721 | . . 3 β’ (π β Fun πΉ) |
42 | 12 | frnd 6725 | . . . 4 β’ (π β ran πΉ β π΅) |
43 | 42, 28 | sseqtrd 4014 | . . 3 β’ (π β ran πΉ β (Baseβ(π βΎs π΅))) |
44 | 15, 16, 18, 22, 26, 40, 41, 43 | gsummgmpropd 18635 | . 2 β’ (π β ((π βΎs π΅) Ξ£g πΉ) = (π Ξ£g πΉ)) |
45 | 14, 44 | eqtrd 2765 | 1 β’ (π β (π Ξ£g πΉ) = (π Ξ£g πΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 Vcvv 3463 ran crn 5674 βΆwf 6539 βcfv 6543 (class class class)co 7413 Basecbs 17174 βΎs cress 17203 +gcplusg 17227 Ξ£g cgsu 17416 Mgmcmgm 18592 SubMndcsubmnd 18733 SubGrpcsubg 19074 Ringcrg 20172 SubRingcsubrg 20505 Poly1cpl1 22099 |
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 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
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 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-ofr 7680 df-om 7866 df-1st 7987 df-2nd 7988 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9381 df-sup 9460 df-oi 9528 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-fz 13512 df-fzo 13655 df-seq 13994 df-hash 14317 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-hom 17251 df-cco 17252 df-0g 17417 df-gsum 17418 df-prds 17423 df-pws 17425 df-mre 17560 df-mrc 17561 df-acs 17563 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-mhm 18734 df-submnd 18735 df-grp 18892 df-minusg 18893 df-mulg 19023 df-subg 19077 df-ghm 19167 df-cntz 19267 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-subrng 20482 df-subrg 20507 df-psr 21841 df-mpl 21843 df-opsr 21845 df-psr1 22102 df-ply1 22104 |
This theorem is referenced by: (None) |
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