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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 21620 | . . . 4 β’ (π β (SubRingβπ ) β π΅ β (SubRingβπ)) |
8 | subrgsubg 20242 | . . . . 5 β’ (π΅ β (SubRingβπ) β π΅ β (SubGrpβπ)) | |
9 | subgsubm 18955 | . . . . 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 2733 | . . 3 β’ (π βΎs π΅) = (π βΎs π΅) | |
14 | 1, 11, 12, 13 | gsumsubm 18650 | . 2 β’ (π β (π Ξ£g πΉ) = ((π βΎs π΅) Ξ£g πΉ)) |
15 | 12, 1 | fexd 7178 | . . 3 β’ (π β πΉ β V) |
16 | ovexd 7393 | . . 3 β’ (π β (π βΎs π΅) β V) | |
17 | 5 | fvexi 6857 | . . . 4 β’ π β V |
18 | 17 | a1i 11 | . . 3 β’ (π β π β V) |
19 | eqid 2733 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
20 | 6 | oveq2i 7369 | . . . . 5 β’ (π βΎs π΅) = (π βΎs (Baseβπ)) |
21 | 3, 4, 5, 19, 2, 20 | ressply1bas 21616 | . . . 4 β’ (π β (Baseβπ) = (Baseβ(π βΎs π΅))) |
22 | 21 | eqcomd 2739 | . . 3 β’ (π β (Baseβ(π βΎs π΅)) = (Baseβπ)) |
23 | 13 | subrgring 20239 | . . . . 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 19980 | . . . 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 484 | . . . . 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 21616 | . . . . . . . . . 10 β’ (π β π΅ = (Baseβ(π βΎs π΅))) |
29 | 28 | eqcomd 2739 | . . . . . . . . 9 β’ (π β (Baseβ(π βΎs π΅)) = π΅) |
30 | 29 | eleq2d 2820 | . . . . . . . 8 β’ (π β (π β (Baseβ(π βΎs π΅)) β π β π΅)) |
31 | 30 | biimpcd 249 | . . . . . . 7 β’ (π β (Baseβ(π βΎs π΅)) β (π β π β π΅)) |
32 | 31 | adantr 482 | . . . . . 6 β’ ((π β (Baseβ(π βΎs π΅)) β§ π‘ β (Baseβ(π βΎs π΅))) β (π β π β π΅)) |
33 | 32 | impcom 409 | . . . . 5 β’ ((π β§ (π β (Baseβ(π βΎs π΅)) β§ π‘ β (Baseβ(π βΎs π΅)))) β π β π΅) |
34 | 29 | eleq2d 2820 | . . . . . . . 8 β’ (π β (π‘ β (Baseβ(π βΎs π΅)) β π‘ β π΅)) |
35 | 34 | biimpcd 249 | . . . . . . 7 β’ (π‘ β (Baseβ(π βΎs π΅)) β (π β π‘ β π΅)) |
36 | 35 | adantl 483 | . . . . . 6 β’ ((π β (Baseβ(π βΎs π΅)) β§ π‘ β (Baseβ(π βΎs π΅))) β (π β π‘ β π΅)) |
37 | 36 | impcom 409 | . . . . 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 21617 | . . . . 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 2739 | . . 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 6673 | . . 3 β’ (π β Fun πΉ) |
42 | 12 | frnd 6677 | . . . 4 β’ (π β ran πΉ β π΅) |
43 | 42, 28 | sseqtrd 3985 | . . 3 β’ (π β ran πΉ β (Baseβ(π βΎs π΅))) |
44 | 15, 16, 18, 22, 26, 40, 41, 43 | gsummgmpropd 18541 | . 2 β’ (π β ((π βΎs π΅) Ξ£g πΉ) = (π Ξ£g πΉ)) |
45 | 14, 44 | eqtrd 2773 | 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 397 = wceq 1542 β wcel 2107 Vcvv 3444 ran crn 5635 βΆwf 6493 βcfv 6497 (class class class)co 7358 Basecbs 17088 βΎs cress 17117 +gcplusg 17138 Ξ£g cgsu 17327 Mgmcmgm 18500 SubMndcsubmnd 18605 SubGrpcsubg 18927 Ringcrg 19969 SubRingcsubrg 20232 Poly1cpl1 21564 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-ofr 7619 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-pm 8771 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 df-sup 9383 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-fzo 13574 df-seq 13913 df-hash 14237 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-sca 17154 df-vsca 17155 df-ip 17156 df-tset 17157 df-ple 17158 df-ds 17160 df-hom 17162 df-cco 17163 df-0g 17328 df-gsum 17329 df-prds 17334 df-pws 17336 df-mre 17471 df-mrc 17472 df-acs 17474 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-mhm 18606 df-submnd 18607 df-grp 18756 df-minusg 18757 df-mulg 18878 df-subg 18930 df-ghm 19011 df-cntz 19102 df-cmn 19569 df-abl 19570 df-mgp 19902 df-ur 19919 df-ring 19971 df-subrg 20234 df-psr 21327 df-mpl 21329 df-opsr 21331 df-psr1 21567 df-ply1 21569 |
This theorem is referenced by: (None) |
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