Nearby theorems
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| Mirrors > Home > MPE Home > Th. List > deg1pw | Structured version Visualization version GIF version | ||
| Description: Exact degree of a variable power over a nontrivial ring. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
| Ref | Expression |
|---|---|
| deg1pw.d | β’ π· = ( deg1 βπ ) |
| deg1pw.p | β’ π = (Poly1βπ ) |
| deg1pw.x | β’ π = (var1βπ ) |
| deg1pw.n | β’ π = (mulGrpβπ) |
| deg1pw.e | β’ β = (.gβπ) |
| Ref | Expression |
|---|---|
| deg1pw | β’ ((π β NzRing β§ πΉ β β0) β (π·β(πΉ β π)) = πΉ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | deg1pw.p | . . . . . . . 8 β’ π = (Poly1βπ ) | |
| 2 | 1 | ply1sca 22094 | . . . . . . 7 β’ (π β NzRing β π = (Scalarβπ)) |
| 3 | 2 | adantr 480 | . . . . . 6 β’ ((π β NzRing β§ πΉ β β0) β π = (Scalarβπ)) |
| 4 | 3 | fveq2d 6885 | . . . . 5 β’ ((π β NzRing β§ πΉ β β0) β (1rβπ ) = (1rβ(Scalarβπ))) |
| 5 | 4 | oveq1d 7416 | . . . 4 β’ ((π β NzRing β§ πΉ β β0) β ((1rβπ )( Β·π βπ)(πΉ β π)) = ((1rβ(Scalarβπ))( Β·π βπ)(πΉ β π))) |
| 6 | nzrring 20408 | . . . . . . 7 β’ (π β NzRing β π β Ring) | |
| 7 | 6 | adantr 480 | . . . . . 6 β’ ((π β NzRing β§ πΉ β β0) β π β Ring) |
| 8 | 1 | ply1lmod 22093 | . . . . . 6 β’ (π β Ring β π β LMod) |
| 9 | 7, 8 | syl 17 | . . . . 5 β’ ((π β NzRing β§ πΉ β β0) β π β LMod) |
| 10 | deg1pw.n | . . . . . . 7 β’ π = (mulGrpβπ) | |
| 11 | eqid 2724 | . . . . . . 7 β’ (Baseβπ) = (Baseβπ) | |
| 12 | 10, 11 | mgpbas 20035 | . . . . . 6 β’ (Baseβπ) = (Baseβπ) |
| 13 | deg1pw.e | . . . . . 6 β’ β = (.gβπ) | |
| 14 | 1 | ply1ring 22089 | . . . . . . 7 β’ (π β Ring β π β Ring) |
| 15 | 10 | ringmgp 20134 | . . . . . . 7 β’ (π β Ring β π β Mnd) |
| 16 | 7, 14, 15 | 3syl 18 | . . . . . 6 β’ ((π β NzRing β§ πΉ β β0) β π β Mnd) |
| 17 | simpr 484 | . . . . . 6 β’ ((π β NzRing β§ πΉ β β0) β πΉ β β0) | |
| 18 | deg1pw.x | . . . . . . . 8 β’ π = (var1βπ ) | |
| 19 | 18, 1, 11 | vr1cl 22059 | . . . . . . 7 β’ (π β Ring β π β (Baseβπ)) |
| 20 | 7, 19 | syl 17 | . . . . . 6 β’ ((π β NzRing β§ πΉ β β0) β π β (Baseβπ)) |
| 21 | 12, 13, 16, 17, 20 | mulgnn0cld 19012 | . . . . 5 β’ ((π β NzRing β§ πΉ β β0) β (πΉ β π) β (Baseβπ)) |
| 22 | eqid 2724 | . . . . . 6 β’ (Scalarβπ) = (Scalarβπ) | |
| 23 | eqid 2724 | . . . . . 6 β’ ( Β·π βπ) = ( Β·π βπ) | |
| 24 | eqid 2724 | . . . . . 6 β’ (1rβ(Scalarβπ)) = (1rβ(Scalarβπ)) | |
| 25 | 11, 22, 23, 24 | lmodvs1 20726 | . . . . 5 β’ ((π β LMod β§ (πΉ β π) β (Baseβπ)) β ((1rβ(Scalarβπ))( Β·π βπ)(πΉ β π)) = (πΉ β π)) |
| 26 | 9, 21, 25 | syl2anc 583 | . . . 4 β’ ((π β NzRing β§ πΉ β β0) β ((1rβ(Scalarβπ))( Β·π βπ)(πΉ β π)) = (πΉ β π)) |
| 27 | 5, 26 | eqtrd 2764 | . . 3 β’ ((π β NzRing β§ πΉ β β0) β ((1rβπ )( Β·π βπ)(πΉ β π)) = (πΉ β π)) |
| 28 | 27 | fveq2d 6885 | . 2 β’ ((π β NzRing β§ πΉ β β0) β (π·β((1rβπ )( Β·π βπ)(πΉ β π))) = (π·β(πΉ β π))) |
| 29 | eqid 2724 | . . . . 5 β’ (Baseβπ ) = (Baseβπ ) | |
| 30 | eqid 2724 | . . . . 5 β’ (1rβπ ) = (1rβπ ) | |
| 31 | 29, 30 | ringidcl 20155 | . . . 4 β’ (π β Ring β (1rβπ ) β (Baseβπ )) |
| 32 | 7, 31 | syl 17 | . . 3 β’ ((π β NzRing β§ πΉ β β0) β (1rβπ ) β (Baseβπ )) |
| 33 | eqid 2724 | . . . . 5 β’ (0gβπ ) = (0gβπ ) | |
| 34 | 30, 33 | nzrnz 20407 | . . . 4 β’ (π β NzRing β (1rβπ ) β (0gβπ )) |
| 35 | 34 | adantr 480 | . . 3 β’ ((π β NzRing β§ πΉ β β0) β (1rβπ ) β (0gβπ )) |
| 36 | deg1pw.d | . . . 4 β’ π· = ( deg1 βπ ) | |
| 37 | 36, 29, 1, 18, 23, 10, 13, 33 | deg1tm 25976 | . . 3 β’ ((π β Ring β§ ((1rβπ ) β (Baseβπ ) β§ (1rβπ ) β (0gβπ )) β§ πΉ β β0) β (π·β((1rβπ )( Β·π βπ)(πΉ β π))) = πΉ) |
| 38 | 7, 32, 35, 17, 37 | syl121anc 1372 | . 2 β’ ((π β NzRing β§ πΉ β β0) β (π·β((1rβπ )( Β·π βπ)(πΉ β π))) = πΉ) |
| 39 | 28, 38 | eqtr3d 2766 | 1 β’ ((π β NzRing β§ πΉ β β0) β (π·β(πΉ β π)) = πΉ) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β wne 2932 βcfv 6533 (class class class)co 7401 β0cn0 12469 Basecbs 17143 Scalarcsca 17199 Β·π cvsca 17200 0gc0g 17384 Mndcmnd 18657 .gcmg 18985 mulGrpcmgp 20029 1rcur 20076 Ringcrg 20128 NzRingcnzr 20404 LModclmod 20696 var1cv1 22018 Poly1cpl1 22019 deg1 cdg1 25909 |
| 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 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-ofr 7664 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-fzo 13625 df-seq 13964 df-hash 14288 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-0g 17386 df-gsum 17387 df-prds 17392 df-pws 17394 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mhm 18703 df-submnd 18704 df-grp 18856 df-minusg 18857 df-sbg 18858 df-mulg 18986 df-subg 19040 df-ghm 19129 df-cntz 19223 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-cring 20131 df-nzr 20405 df-subrng 20436 df-subrg 20461 df-lmod 20698 df-lss 20769 df-cnfld 21229 df-psr 21771 df-mvr 21772 df-mpl 21773 df-opsr 21775 df-psr1 22022 df-vr1 22023 df-ply1 22024 df-coe1 22025 df-mdeg 25910 df-deg1 25911 |
| This theorem is referenced by: ply1remlem 26020 lgsqrlem4 27198 idomrootle 42426 |
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