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 21774 | . . . . . . 7 β’ (π β NzRing β π = (Scalarβπ)) |
| 3 | 2 | adantr 481 | . . . . . 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 7423 | . . . 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 20294 | . . . . . . 7 β’ (π β NzRing β π β Ring) | |
| 7 | 6 | adantr 481 | . . . . . 6 β’ ((π β NzRing β§ πΉ β β0) β π β Ring) |
| 8 | 1 | ply1lmod 21773 | . . . . . 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 2732 | . . . . . . 7 β’ (Baseβπ) = (Baseβπ) | |
| 12 | 10, 11 | mgpbas 19992 | . . . . . 6 β’ (Baseβπ) = (Baseβπ) |
| 13 | deg1pw.e | . . . . . 6 β’ β = (.gβπ) | |
| 14 | 1 | ply1ring 21769 | . . . . . . 7 β’ (π β Ring β π β Ring) |
| 15 | 10 | ringmgp 20061 | . . . . . . 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 485 | . . . . . 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 21740 | . . . . . . 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 18974 | . . . . 5 β’ ((π β NzRing β§ πΉ β β0) β (πΉ β π) β (Baseβπ)) |
| 22 | eqid 2732 | . . . . . 6 β’ (Scalarβπ) = (Scalarβπ) | |
| 23 | eqid 2732 | . . . . . 6 β’ ( Β·π βπ) = ( Β·π βπ) | |
| 24 | eqid 2732 | . . . . . 6 β’ (1rβ(Scalarβπ)) = (1rβ(Scalarβπ)) | |
| 25 | 11, 22, 23, 24 | lmodvs1 20499 | . . . . 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 584 | . . . 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 2772 | . . 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 2732 | . . . . 5 β’ (Baseβπ ) = (Baseβπ ) | |
| 30 | eqid 2732 | . . . . 5 β’ (1rβπ ) = (1rβπ ) | |
| 31 | 29, 30 | ringidcl 20082 | . . . 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 2732 | . . . . 5 β’ (0gβπ ) = (0gβπ ) | |
| 34 | 30, 33 | nzrnz 20293 | . . . 4 β’ (π β NzRing β (1rβπ ) β (0gβπ )) |
| 35 | 34 | adantr 481 | . . 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 25635 | . . 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 1375 | . 2 β’ ((π β NzRing β§ πΉ β β0) β (π·β((1rβπ )( Β·π βπ)(πΉ β π))) = πΉ) |
| 39 | 28, 38 | eqtr3d 2774 | 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 396 = wceq 1541 β wcel 2106 β wne 2940 βcfv 6543 (class class class)co 7408 β0cn0 12471 Basecbs 17143 Scalarcsca 17199 Β·π cvsca 17200 0gc0g 17384 Mndcmnd 18624 .gcmg 18949 mulGrpcmgp 19986 1rcur 20003 Ringcrg 20055 NzRingcnzr 20290 LModclmod 20470 var1cv1 21699 Poly1cpl1 21700 deg1 cdg1 25568 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-ofr 7670 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-sup 9436 df-oi 9504 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13484 df-fzo 13627 df-seq 13966 df-hash 14290 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 18560 df-sgrp 18609 df-mnd 18625 df-mhm 18670 df-submnd 18671 df-grp 18821 df-minusg 18822 df-sbg 18823 df-mulg 18950 df-subg 19002 df-ghm 19089 df-cntz 19180 df-cmn 19649 df-abl 19650 df-mgp 19987 df-ur 20004 df-ring 20057 df-cring 20058 df-nzr 20291 df-subrg 20316 df-lmod 20472 df-lss 20542 df-cnfld 20944 df-psr 21461 df-mvr 21462 df-mpl 21463 df-opsr 21465 df-psr1 21703 df-vr1 21704 df-ply1 21705 df-coe1 21706 df-mdeg 25569 df-deg1 25570 |
| This theorem is referenced by: ply1remlem 25679 lgsqrlem4 26849 idomrootle 41927 |
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