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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 20420 | . . . 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 2821 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
7 | 1, 3, 4, 5, 6 | ply1moncl 20439 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ ℕ0) → (𝐹 ↑ 𝑋) ∈ (Base‘𝑃)) |
8 | eqid 2821 | . . . . 5 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃) | |
9 | eqid 2821 | . . . . 5 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃) | |
10 | eqid 2821 | . . . . 5 ⊢ (1r‘(Scalar‘𝑃)) = (1r‘(Scalar‘𝑃)) | |
11 | 6, 8, 9, 10 | lmodvs1 19662 | . . . 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 6674 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ ℕ0) → (𝐷‘((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝐹 ↑ 𝑋))) = (𝐷‘(𝐹 ↑ 𝑋))) |
14 | simpl 485 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ ℕ0) → 𝑅 ∈ Ring) | |
15 | 1 | ply1sca 20421 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃)) |
16 | 15 | fveq2d 6674 | . . . . 5 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) = (1r‘(Scalar‘𝑃))) |
17 | eqid 2821 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
18 | eqid 2821 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
19 | 17, 18 | ringidcl 19318 | . . . . 5 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅)) |
20 | 16, 19 | eqeltrrd 2914 | . . . 4 ⊢ (𝑅 ∈ Ring → (1r‘(Scalar‘𝑃)) ∈ (Base‘𝑅)) |
21 | 20 | adantr 483 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ ℕ0) → (1r‘(Scalar‘𝑃)) ∈ (Base‘𝑅)) |
22 | simpr 487 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ ℕ0) → 𝐹 ∈ ℕ0) | |
23 | deg1pw.d | . . . 4 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
24 | 23, 17, 1, 3, 9, 4, 5 | deg1tmle 24711 | . . 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 1367 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ ℕ0) → (𝐷‘((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝐹 ↑ 𝑋))) ≤ 𝐹) |
26 | 13, 25 | eqbrtrrd 5090 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐹 ↑ 𝑋)) ≤ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ≤ cle 10676 ℕ0cn0 11898 Basecbs 16483 Scalarcsca 16568 ·𝑠 cvsca 16569 .gcmg 18224 mulGrpcmgp 19239 1rcur 19251 Ringcrg 19297 LModclmod 19634 var1cv1 20344 Poly1cpl1 20345 deg1 cdg1 24648 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-ofr 7410 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-sup 8906 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-0g 16715 df-gsum 16716 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-mulg 18225 df-subg 18276 df-ghm 18356 df-cntz 18447 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-subrg 19533 df-lmod 19636 df-lss 19704 df-psr 20136 df-mvr 20137 df-mpl 20138 df-opsr 20140 df-psr1 20348 df-vr1 20349 df-ply1 20350 df-coe1 20351 df-cnfld 20546 df-mdeg 24649 df-deg1 24650 |
This theorem is referenced by: hbtlem4 39746 |
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