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Mirrors > Home > MPE Home > Th. List > evl1gsumd | Structured version Visualization version Unicode version |
Description: Polynomial evaluation builder for a finite group sum of polynomials. (Contributed by AV, 17-Sep-2019.) |
Ref | Expression |
---|---|
evl1gsumd.q | eval1 |
evl1gsumd.p | Poly1 |
evl1gsumd.b | |
evl1gsumd.u | |
evl1gsumd.r | |
evl1gsumd.y | |
evl1gsumd.m | |
evl1gsumd.n |
Ref | Expression |
---|---|
evl1gsumd | g g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evl1gsumd.m | . 2 | |
2 | evl1gsumd.n | . . 3 | |
3 | raleq 3138 | . . . . . . 7 | |
4 | 3 | anbi2d 740 | . . . . . 6 |
5 | mpteq1 4737 | . . . . . . . . . 10 | |
6 | 5 | oveq2d 6666 | . . . . . . . . 9 g g |
7 | 6 | fveq2d 6195 | . . . . . . . 8 g g |
8 | 7 | fveq1d 6193 | . . . . . . 7 g g |
9 | mpteq1 4737 | . . . . . . . 8 | |
10 | 9 | oveq2d 6666 | . . . . . . 7 g g |
11 | 8, 10 | eqeq12d 2637 | . . . . . 6 g g g g |
12 | 4, 11 | imbi12d 334 | . . . . 5 g g g g |
13 | raleq 3138 | . . . . . . 7 | |
14 | 13 | anbi2d 740 | . . . . . 6 |
15 | mpteq1 4737 | . . . . . . . . . 10 | |
16 | 15 | oveq2d 6666 | . . . . . . . . 9 g g |
17 | 16 | fveq2d 6195 | . . . . . . . 8 g g |
18 | 17 | fveq1d 6193 | . . . . . . 7 g g |
19 | mpteq1 4737 | . . . . . . . 8 | |
20 | 19 | oveq2d 6666 | . . . . . . 7 g g |
21 | 18, 20 | eqeq12d 2637 | . . . . . 6 g g g g |
22 | 14, 21 | imbi12d 334 | . . . . 5 g g g g |
23 | raleq 3138 | . . . . . . 7 | |
24 | 23 | anbi2d 740 | . . . . . 6 |
25 | mpteq1 4737 | . . . . . . . . . 10 | |
26 | 25 | oveq2d 6666 | . . . . . . . . 9 g g |
27 | 26 | fveq2d 6195 | . . . . . . . 8 g g |
28 | 27 | fveq1d 6193 | . . . . . . 7 g g |
29 | mpteq1 4737 | . . . . . . . 8 | |
30 | 29 | oveq2d 6666 | . . . . . . 7 g g |
31 | 28, 30 | eqeq12d 2637 | . . . . . 6 g g g g |
32 | 24, 31 | imbi12d 334 | . . . . 5 g g g g |
33 | raleq 3138 | . . . . . . 7 | |
34 | 33 | anbi2d 740 | . . . . . 6 |
35 | mpteq1 4737 | . . . . . . . . . 10 | |
36 | 35 | oveq2d 6666 | . . . . . . . . 9 g g |
37 | 36 | fveq2d 6195 | . . . . . . . 8 g g |
38 | 37 | fveq1d 6193 | . . . . . . 7 g g |
39 | mpteq1 4737 | . . . . . . . 8 | |
40 | 39 | oveq2d 6666 | . . . . . . 7 g g |
41 | 38, 40 | eqeq12d 2637 | . . . . . 6 g g g g |
42 | 34, 41 | imbi12d 334 | . . . . 5 g g g g |
43 | mpt0 6021 | . . . . . . . . . . . . 13 | |
44 | 43 | oveq2i 6661 | . . . . . . . . . . . 12 g g |
45 | eqid 2622 | . . . . . . . . . . . . 13 | |
46 | 45 | gsum0 17278 | . . . . . . . . . . . 12 g |
47 | 44, 46 | eqtri 2644 | . . . . . . . . . . 11 g |
48 | 47 | fveq2i 6194 | . . . . . . . . . 10 g |
49 | evl1gsumd.r | . . . . . . . . . . . . . 14 | |
50 | crngring 18558 | . . . . . . . . . . . . . 14 | |
51 | 49, 50 | syl 17 | . . . . . . . . . . . . 13 |
52 | evl1gsumd.p | . . . . . . . . . . . . . 14 Poly1 | |
53 | eqid 2622 | . . . . . . . . . . . . . 14 algSc algSc | |
54 | eqid 2622 | . . . . . . . . . . . . . 14 | |
55 | 52, 53, 54, 45 | ply1scl0 19660 | . . . . . . . . . . . . 13 algSc |
56 | 51, 55 | syl 17 | . . . . . . . . . . . 12 algSc |
57 | 56 | eqcomd 2628 | . . . . . . . . . . 11 algSc |
58 | 57 | fveq2d 6195 | . . . . . . . . . 10 algSc |
59 | 48, 58 | syl5eq 2668 | . . . . . . . . 9 g algSc |
60 | 59 | fveq1d 6193 | . . . . . . . 8 g algSc |
61 | evl1gsumd.q | . . . . . . . . . 10 eval1 | |
62 | evl1gsumd.b | . . . . . . . . . 10 | |
63 | evl1gsumd.u | . . . . . . . . . 10 | |
64 | ringgrp 18552 | . . . . . . . . . . . 12 | |
65 | 51, 64 | syl 17 | . . . . . . . . . . 11 |
66 | 62, 54 | grpidcl 17450 | . . . . . . . . . . 11 |
67 | 65, 66 | syl 17 | . . . . . . . . . 10 |
68 | evl1gsumd.y | . . . . . . . . . 10 | |
69 | 61, 52, 62, 53, 63, 49, 67, 68 | evl1scad 19699 | . . . . . . . . 9 algSc algSc |
70 | 69 | simprd 479 | . . . . . . . 8 algSc |
71 | 60, 70 | eqtrd 2656 | . . . . . . 7 g |
72 | mpt0 6021 | . . . . . . . . 9 | |
73 | 72 | oveq2i 6661 | . . . . . . . 8 g g |
74 | 54 | gsum0 17278 | . . . . . . . 8 g |
75 | 73, 74 | eqtri 2644 | . . . . . . 7 g |
76 | 71, 75 | syl6eqr 2674 | . . . . . 6 g g |
77 | 76 | adantr 481 | . . . . 5 g g |
78 | 61, 52, 62, 63, 49, 68 | evl1gsumdlem 19720 | . . . . . . . 8 g g g g |
79 | 78 | 3expia 1267 | . . . . . . 7 g g g g |
80 | 79 | a2d 29 | . . . . . 6 g g g g |
81 | impexp 462 | . . . . . 6 g g g g | |
82 | impexp 462 | . . . . . 6 g g g g | |
83 | 80, 81, 82 | 3imtr4g 285 | . . . . 5 g g g g |
84 | 12, 22, 32, 42, 77, 83 | findcard2s 8201 | . . . 4 g g |
85 | 84 | expd 452 | . . 3 g g |
86 | 2, 85 | mpcom 38 | . 2 g g |
87 | 1, 86 | mpd 15 | 1 g g |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wa 384 wceq 1483 wcel 1990 wral 2912 cun 3572 c0 3915 csn 4177 cmpt 4729 cfv 5888 (class class class)co 6650 cfn 7955 cbs 15857 c0g 16100 g cgsu 16101 cgrp 17422 crg 18547 ccrg 18548 algSccascl 19311 Poly1cpl1 19547 eval1ce1 19679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-0g 16102 df-gsum 16103 df-prds 16108 df-pws 16110 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-ghm 17658 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-srg 18506 df-ring 18549 df-cring 18550 df-rnghom 18715 df-subrg 18778 df-lmod 18865 df-lss 18933 df-lsp 18972 df-assa 19312 df-asp 19313 df-ascl 19314 df-psr 19356 df-mvr 19357 df-mpl 19358 df-opsr 19360 df-evls 19506 df-evl 19507 df-psr1 19550 df-ply1 19552 df-evl1 19681 |
This theorem is referenced by: evl1gsumaddval 19723 |
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