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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 | ⊢ 𝐷 = ( deg1 ‘𝑅) |
deg1pw.p | ⊢ 𝑃 = (Poly1‘𝑅) |
deg1pw.x | ⊢ 𝑋 = (var1‘𝑅) |
deg1pw.n | ⊢ 𝑁 = (mulGrp‘𝑃) |
deg1pw.e | ⊢ ↑ = (.g‘𝑁) |
Ref | Expression |
---|---|
deg1pwle | ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐹 ↑ 𝑋)) ≤ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | deg1pw.p | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | 1 | ply1lmod 21569 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod) |
3 | deg1pw.x | . . . . 5 ⊢ 𝑋 = (var1‘𝑅) | |
4 | deg1pw.n | . . . . 5 ⊢ 𝑁 = (mulGrp‘𝑃) | |
5 | deg1pw.e | . . . . 5 ⊢ ↑ = (.g‘𝑁) | |
6 | eqid 2736 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
7 | 1, 3, 4, 5, 6 | ply1moncl 21588 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ ℕ0) → (𝐹 ↑ 𝑋) ∈ (Base‘𝑃)) |
8 | eqid 2736 | . . . . 5 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃) | |
9 | eqid 2736 | . . . . 5 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃) | |
10 | eqid 2736 | . . . . 5 ⊢ (1r‘(Scalar‘𝑃)) = (1r‘(Scalar‘𝑃)) | |
11 | 6, 8, 9, 10 | lmodvs1 20297 | . . . 4 ⊢ ((𝑃 ∈ LMod ∧ (𝐹 ↑ 𝑋) ∈ (Base‘𝑃)) → ((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝐹 ↑ 𝑋)) = (𝐹 ↑ 𝑋)) |
12 | 2, 7, 11 | syl2an2r 683 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ ℕ0) → ((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝐹 ↑ 𝑋)) = (𝐹 ↑ 𝑋)) |
13 | 12 | fveq2d 6843 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ ℕ0) → (𝐷‘((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝐹 ↑ 𝑋))) = (𝐷‘(𝐹 ↑ 𝑋))) |
14 | simpl 483 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ ℕ0) → 𝑅 ∈ Ring) | |
15 | 1 | ply1sca 21570 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃)) |
16 | 15 | fveq2d 6843 | . . . . 5 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) = (1r‘(Scalar‘𝑃))) |
17 | eqid 2736 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
18 | eqid 2736 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
19 | 17, 18 | ringidcl 19937 | . . . . 5 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅)) |
20 | 16, 19 | eqeltrrd 2839 | . . . 4 ⊢ (𝑅 ∈ Ring → (1r‘(Scalar‘𝑃)) ∈ (Base‘𝑅)) |
21 | 20 | adantr 481 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ ℕ0) → (1r‘(Scalar‘𝑃)) ∈ (Base‘𝑅)) |
22 | simpr 485 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ ℕ0) → 𝐹 ∈ ℕ0) | |
23 | deg1pw.d | . . . 4 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
24 | 23, 17, 1, 3, 9, 4, 5 | deg1tmle 25428 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (1r‘(Scalar‘𝑃)) ∈ (Base‘𝑅) ∧ 𝐹 ∈ ℕ0) → (𝐷‘((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝐹 ↑ 𝑋))) ≤ 𝐹) |
25 | 14, 21, 22, 24 | syl3anc 1371 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ ℕ0) → (𝐷‘((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝐹 ↑ 𝑋))) ≤ 𝐹) |
26 | 13, 25 | eqbrtrrd 5127 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐹 ↑ 𝑋)) ≤ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 class class class wbr 5103 ‘cfv 6493 (class class class)co 7351 ≤ cle 11148 ℕ0cn0 12371 Basecbs 17037 Scalarcsca 17090 ·𝑠 cvsca 17091 .gcmg 18825 mulGrpcmgp 19849 1rcur 19866 Ringcrg 19912 LModclmod 20269 var1cv1 21493 Poly1cpl1 21494 deg1 cdg1 25362 |
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 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088 ax-mulf 11089 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-of 7609 df-ofr 7610 df-om 7795 df-1st 7913 df-2nd 7914 df-supp 8085 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-map 8725 df-pm 8726 df-ixp 8794 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-fsupp 9264 df-sup 9336 df-oi 9404 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-dec 12577 df-uz 12722 df-fz 13379 df-fzo 13522 df-seq 13861 df-hash 14185 df-struct 16973 df-sets 16990 df-slot 17008 df-ndx 17020 df-base 17038 df-ress 17067 df-plusg 17100 df-mulr 17101 df-starv 17102 df-sca 17103 df-vsca 17104 df-tset 17106 df-ple 17107 df-ds 17109 df-unif 17110 df-0g 17277 df-gsum 17278 df-mre 17420 df-mrc 17421 df-acs 17423 df-mgm 18451 df-sgrp 18500 df-mnd 18511 df-mhm 18555 df-submnd 18556 df-grp 18705 df-minusg 18706 df-sbg 18707 df-mulg 18826 df-subg 18878 df-ghm 18959 df-cntz 19050 df-cmn 19517 df-abl 19518 df-mgp 19850 df-ur 19867 df-ring 19914 df-cring 19915 df-subrg 20167 df-lmod 20271 df-lss 20340 df-cnfld 20744 df-psr 21258 df-mvr 21259 df-mpl 21260 df-opsr 21262 df-psr1 21497 df-vr1 21498 df-ply1 21499 df-coe1 21500 df-mdeg 25363 df-deg1 25364 |
This theorem is referenced by: hbtlem4 41356 |
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