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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 22125 | . . . 4 β’ (π β (SubRingβπ ) β π΅ β (SubRingβπ)) |
8 | subrgsubg 20498 | . . . . 5 β’ (π΅ β (SubRingβπ) β π΅ β (SubGrpβπ)) | |
9 | subgsubm 19087 | . . . . 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 2727 | . . 3 β’ (π βΎs π΅) = (π βΎs π΅) | |
14 | 1, 11, 12, 13 | gsumsubm 18772 | . 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 7449 | . . 3 β’ (π β (π βΎs π΅) β V) | |
17 | 5 | fvexi 6905 | . . . 4 β’ π β V |
18 | 17 | a1i 11 | . . 3 β’ (π β π β V) |
19 | eqid 2727 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
20 | 6 | oveq2i 7425 | . . . . 5 β’ (π βΎs π΅) = (π βΎs (Baseβπ)) |
21 | 3, 4, 5, 19, 2, 20 | ressply1bas 22121 | . . . 4 β’ (π β (Baseβπ) = (Baseβ(π βΎs π΅))) |
22 | 21 | eqcomd 2733 | . . 3 β’ (π β (Baseβ(π βΎs π΅)) = (Baseβπ)) |
23 | 13 | subrgring 20495 | . . . . 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 20168 | . . . 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 482 | . . . . 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 22121 | . . . . . . . . . 10 β’ (π β π΅ = (Baseβ(π βΎs π΅))) |
29 | 28 | eqcomd 2733 | . . . . . . . . 9 β’ (π β (Baseβ(π βΎs π΅)) = π΅) |
30 | 29 | eleq2d 2814 | . . . . . . . 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 480 | . . . . . 6 β’ ((π β (Baseβ(π βΎs π΅)) β§ π‘ β (Baseβ(π βΎs π΅))) β (π β π β π΅)) |
33 | 32 | impcom 407 | . . . . 5 β’ ((π β§ (π β (Baseβ(π βΎs π΅)) β§ π‘ β (Baseβ(π βΎs π΅)))) β π β π΅) |
34 | 29 | eleq2d 2814 | . . . . . . . 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 481 | . . . . . 6 β’ ((π β (Baseβ(π βΎs π΅)) β§ π‘ β (Baseβ(π βΎs π΅))) β (π β π‘ β π΅)) |
37 | 36 | impcom 407 | . . . . 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 22122 | . . . . 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 836 | . . . 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 2733 | . . 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 6720 | . . 3 β’ (π β Fun πΉ) |
42 | 12 | frnd 6724 | . . . 4 β’ (π β ran πΉ β π΅) |
43 | 42, 28 | sseqtrd 4018 | . . 3 β’ (π β ran πΉ β (Baseβ(π βΎs π΅))) |
44 | 15, 16, 18, 22, 26, 40, 41, 43 | gsummgmpropd 18626 | . 2 β’ (π β ((π βΎs π΅) Ξ£g πΉ) = (π Ξ£g πΉ)) |
45 | 14, 44 | eqtrd 2767 | 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 395 = wceq 1534 β wcel 2099 Vcvv 3469 ran crn 5673 βΆwf 6538 βcfv 6542 (class class class)co 7414 Basecbs 17165 βΎs cress 17194 +gcplusg 17218 Ξ£g cgsu 17407 Mgmcmgm 18583 SubMndcsubmnd 18724 SubGrpcsubg 19059 Ringcrg 20157 SubRingcsubrg 20488 Poly1cpl1 22070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 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 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7677 df-ofr 7678 df-om 7863 df-1st 7985 df-2nd 7986 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-map 8836 df-pm 8837 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9376 df-sup 9451 df-oi 9519 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-uz 12839 df-fz 13503 df-fzo 13646 df-seq 13985 df-hash 14308 df-struct 17101 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-hom 17242 df-cco 17243 df-0g 17408 df-gsum 17409 df-prds 17414 df-pws 17416 df-mre 17551 df-mrc 17552 df-acs 17554 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-mhm 18725 df-submnd 18726 df-grp 18878 df-minusg 18879 df-mulg 19008 df-subg 19062 df-ghm 19152 df-cntz 19252 df-cmn 19721 df-abl 19722 df-mgp 20059 df-rng 20077 df-ur 20106 df-ring 20159 df-subrng 20465 df-subrg 20490 df-psr 21822 df-mpl 21824 df-opsr 21826 df-psr1 22073 df-ply1 22075 |
This theorem is referenced by: (None) |
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