Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > deg1pwle | Structured version Visualization version GIF version | ||
| Description: Limiting degree of a variable power. (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 |
|---|---|
| deg1pwle | β’ ((π β Ring β§ πΉ β β0) β (π·β(πΉ β π)) β€ πΉ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | deg1pw.p | . . . . 5 β’ π = (Poly1βπ ) | |
| 2 | 1 | ply1lmod 21773 | . . . 4 β’ (π β Ring β π β LMod) |
| 3 | deg1pw.x | . . . . 5 β’ π = (var1βπ ) | |
| 4 | deg1pw.n | . . . . 5 β’ π = (mulGrpβπ) | |
| 5 | deg1pw.e | . . . . 5 β’ β = (.gβπ) | |
| 6 | eqid 2732 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
| 7 | 1, 3, 4, 5, 6 | ply1moncl 21792 | . . . 4 β’ ((π β Ring β§ πΉ β β0) β (πΉ β π) β (Baseβπ)) |
| 8 | eqid 2732 | . . . . 5 β’ (Scalarβπ) = (Scalarβπ) | |
| 9 | eqid 2732 | . . . . 5 β’ ( Β·π βπ) = ( Β·π βπ) | |
| 10 | eqid 2732 | . . . . 5 β’ (1rβ(Scalarβπ)) = (1rβ(Scalarβπ)) | |
| 11 | 6, 8, 9, 10 | lmodvs1 20499 | . . . 4 β’ ((π β LMod β§ (πΉ β π) β (Baseβπ)) β ((1rβ(Scalarβπ))( Β·π βπ)(πΉ β π)) = (πΉ β π)) |
| 12 | 2, 7, 11 | syl2an2r 683 | . . 3 β’ ((π β Ring β§ πΉ β β0) β ((1rβ(Scalarβπ))( Β·π βπ)(πΉ β π)) = (πΉ β π)) |
| 13 | 12 | fveq2d 6895 | . 2 β’ ((π β Ring β§ πΉ β β0) β (π·β((1rβ(Scalarβπ))( Β·π βπ)(πΉ β π))) = (π·β(πΉ β π))) |
| 14 | simpl 483 | . . 3 β’ ((π β Ring β§ πΉ β β0) β π β Ring) | |
| 15 | 1 | ply1sca 21774 | . . . . . 6 β’ (π β Ring β π = (Scalarβπ)) |
| 16 | 15 | fveq2d 6895 | . . . . 5 β’ (π β Ring β (1rβπ ) = (1rβ(Scalarβπ))) |
| 17 | eqid 2732 | . . . . . 6 β’ (Baseβπ ) = (Baseβπ ) | |
| 18 | eqid 2732 | . . . . . 6 β’ (1rβπ ) = (1rβπ ) | |
| 19 | 17, 18 | ringidcl 20082 | . . . . 5 β’ (π β Ring β (1rβπ ) β (Baseβπ )) |
| 20 | 16, 19 | eqeltrrd 2834 | . . . 4 β’ (π β Ring β (1rβ(Scalarβπ)) β (Baseβπ )) |
| 21 | 20 | adantr 481 | . . 3 β’ ((π β Ring β§ πΉ β β0) β (1rβ(Scalarβπ)) β (Baseβπ )) |
| 22 | simpr 485 | . . 3 β’ ((π β Ring β§ πΉ β β0) β πΉ β β0) | |
| 23 | deg1pw.d | . . . 4 β’ π· = ( deg1 βπ ) | |
| 24 | 23, 17, 1, 3, 9, 4, 5 | deg1tmle 25634 | . . 3 β’ ((π β Ring β§ (1rβ(Scalarβπ)) β (Baseβπ ) β§ πΉ β β0) β (π·β((1rβ(Scalarβπ))( Β·π βπ)(πΉ β π))) β€ πΉ) |
| 25 | 14, 21, 22, 24 | syl3anc 1371 | . 2 β’ ((π β Ring β§ πΉ β β0) β (π·β((1rβ(Scalarβπ))( Β·π βπ)(πΉ β π))) β€ πΉ) |
| 26 | 13, 25 | eqbrtrrd 5172 | 1 β’ ((π β Ring β§ πΉ β β0) β (π·β(πΉ β π)) β€ πΉ) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 class class class wbr 5148 βcfv 6543 (class class class)co 7408 β€ cle 11248 β0cn0 12471 Basecbs 17143 Scalarcsca 17199 Β·π cvsca 17200 .gcmg 18949 mulGrpcmgp 19986 1rcur 20003 Ringcrg 20055 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-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: ply1degltdimlem 32702 ply1degltdim 32703 hbtlem4 41858 |
| Copyright terms: Public domain | W3C validator |