Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| 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 22164 | . . . . . . 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 6895 | . . . . 5 β’ ((π β NzRing β§ πΉ β β0) β (1rβπ ) = (1rβ(Scalarβπ))) |
| 5 | 4 | oveq1d 7429 | . . . 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 20448 | . . . . . . 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 22163 | . . . . . 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 2727 | . . . . . . 7 β’ (Baseβπ) = (Baseβπ) | |
| 12 | 10, 11 | mgpbas 20073 | . . . . . 6 β’ (Baseβπ) = (Baseβπ) |
| 13 | deg1pw.e | . . . . . 6 β’ β = (.gβπ) | |
| 14 | 1 | ply1ring 22159 | . . . . . . 7 β’ (π β Ring β π β Ring) |
| 15 | 10 | ringmgp 20172 | . . . . . . 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 22129 | . . . . . . 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 19043 | . . . . 5 β’ ((π β NzRing β§ πΉ β β0) β (πΉ β π) β (Baseβπ)) |
| 22 | eqid 2727 | . . . . . 6 β’ (Scalarβπ) = (Scalarβπ) | |
| 23 | eqid 2727 | . . . . . 6 β’ ( Β·π βπ) = ( Β·π βπ) | |
| 24 | eqid 2727 | . . . . . 6 β’ (1rβ(Scalarβπ)) = (1rβ(Scalarβπ)) | |
| 25 | 11, 22, 23, 24 | lmodvs1 20766 | . . . . 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 2767 | . . 3 β’ ((π β NzRing β§ πΉ β β0) β ((1rβπ )( Β·π βπ)(πΉ β π)) = (πΉ β π)) |
| 28 | 27 | fveq2d 6895 | . 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 2727 | . . . . 5 β’ (Baseβπ ) = (Baseβπ ) | |
| 30 | eqid 2727 | . . . . 5 β’ (1rβπ ) = (1rβπ ) | |
| 31 | 29, 30 | ringidcl 20195 | . . . 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 2727 | . . . . 5 β’ (0gβπ ) = (0gβπ ) | |
| 34 | 30, 33 | nzrnz 20447 | . . . 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 26047 | . . 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 1373 | . 2 β’ ((π β NzRing β§ πΉ β β0) β (π·β((1rβπ )( Β·π βπ)(πΉ β π))) = πΉ) |
| 39 | 28, 38 | eqtr3d 2769 | 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 1534 β wcel 2099 β wne 2935 βcfv 6542 (class class class)co 7414 β0cn0 12496 Basecbs 17173 Scalarcsca 17229 Β·π cvsca 17230 0gc0g 17414 Mndcmnd 18687 .gcmg 19016 mulGrpcmgp 20067 1rcur 20114 Ringcrg 20166 NzRingcnzr 20444 LModclmod 20736 var1cv1 22088 Poly1cpl1 22089 deg1 cdg1 25980 |
| 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 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 ax-addf 11211 |
| 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 7679 df-ofr 7680 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 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 9380 df-sup 9459 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-fz 13511 df-fzo 13654 df-seq 13993 df-hash 14316 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-starv 17241 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-hom 17250 df-cco 17251 df-0g 17416 df-gsum 17417 df-prds 17422 df-pws 17424 df-mre 17559 df-mrc 17560 df-acs 17562 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-mhm 18733 df-submnd 18734 df-grp 18886 df-minusg 18887 df-sbg 18888 df-mulg 19017 df-subg 19071 df-ghm 19161 df-cntz 19261 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-cring 20169 df-nzr 20445 df-subrng 20476 df-subrg 20501 df-lmod 20738 df-lss 20809 df-cnfld 21273 df-psr 21835 df-mvr 21836 df-mpl 21837 df-opsr 21839 df-psr1 22092 df-vr1 22093 df-ply1 22094 df-coe1 22095 df-mdeg 25981 df-deg1 25982 |
| This theorem is referenced by: ply1remlem 26092 idomrootle 26100 lgsqrlem4 27275 aks6d1c2lem4 41582 aks6d1c5lem3 41592 aks6d1c6lem1 41626 |
| Copyright terms: Public domain | W3C validator |