Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > deg1tm | Structured version Visualization version GIF version |
Description: Exact degree of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.) |
Ref | Expression |
---|---|
deg1tm.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
deg1tm.k | ⊢ 𝐾 = (Base‘𝑅) |
deg1tm.p | ⊢ 𝑃 = (Poly1‘𝑅) |
deg1tm.x | ⊢ 𝑋 = (var1‘𝑅) |
deg1tm.m | ⊢ · = ( ·𝑠 ‘𝑃) |
deg1tm.n | ⊢ 𝑁 = (mulGrp‘𝑃) |
deg1tm.e | ⊢ ↑ = (.g‘𝑁) |
deg1tm.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
deg1tm | ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝐶 ≠ 0 ) ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐶 · (𝐹 ↑ 𝑋))) = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | deg1tm.k | . . . . 5 ⊢ 𝐾 = (Base‘𝑅) | |
2 | deg1tm.p | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | deg1tm.x | . . . . 5 ⊢ 𝑋 = (var1‘𝑅) | |
4 | deg1tm.m | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑃) | |
5 | deg1tm.n | . . . . 5 ⊢ 𝑁 = (mulGrp‘𝑃) | |
6 | deg1tm.e | . . . . 5 ⊢ ↑ = (.g‘𝑁) | |
7 | eqid 2824 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ply1tmcl 20443 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐹 ∈ ℕ0) → (𝐶 · (𝐹 ↑ 𝑋)) ∈ (Base‘𝑃)) |
9 | 8 | 3adant2r 1175 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝐶 ≠ 0 ) ∧ 𝐹 ∈ ℕ0) → (𝐶 · (𝐹 ↑ 𝑋)) ∈ (Base‘𝑃)) |
10 | deg1tm.d | . . . 4 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
11 | 10, 2, 7 | deg1xrcl 24679 | . . 3 ⊢ ((𝐶 · (𝐹 ↑ 𝑋)) ∈ (Base‘𝑃) → (𝐷‘(𝐶 · (𝐹 ↑ 𝑋))) ∈ ℝ*) |
12 | 9, 11 | syl 17 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝐶 ≠ 0 ) ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐶 · (𝐹 ↑ 𝑋))) ∈ ℝ*) |
13 | simp3 1134 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝐶 ≠ 0 ) ∧ 𝐹 ∈ ℕ0) → 𝐹 ∈ ℕ0) | |
14 | 13 | nn0red 11959 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝐶 ≠ 0 ) ∧ 𝐹 ∈ ℕ0) → 𝐹 ∈ ℝ) |
15 | 14 | rexrd 10694 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝐶 ≠ 0 ) ∧ 𝐹 ∈ ℕ0) → 𝐹 ∈ ℝ*) |
16 | 10, 1, 2, 3, 4, 5, 6 | deg1tmle 24714 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐶 · (𝐹 ↑ 𝑋))) ≤ 𝐹) |
17 | 16 | 3adant2r 1175 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝐶 ≠ 0 ) ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐶 · (𝐹 ↑ 𝑋))) ≤ 𝐹) |
18 | deg1tm.z | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
19 | 18, 1, 2, 3, 4, 5, 6 | coe1tmfv1 20445 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐹 ∈ ℕ0) → ((coe1‘(𝐶 · (𝐹 ↑ 𝑋)))‘𝐹) = 𝐶) |
20 | 19 | 3adant2r 1175 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝐶 ≠ 0 ) ∧ 𝐹 ∈ ℕ0) → ((coe1‘(𝐶 · (𝐹 ↑ 𝑋)))‘𝐹) = 𝐶) |
21 | simp2r 1196 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝐶 ≠ 0 ) ∧ 𝐹 ∈ ℕ0) → 𝐶 ≠ 0 ) | |
22 | 20, 21 | eqnetrd 3086 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝐶 ≠ 0 ) ∧ 𝐹 ∈ ℕ0) → ((coe1‘(𝐶 · (𝐹 ↑ 𝑋)))‘𝐹) ≠ 0 ) |
23 | eqid 2824 | . . . 4 ⊢ (coe1‘(𝐶 · (𝐹 ↑ 𝑋))) = (coe1‘(𝐶 · (𝐹 ↑ 𝑋))) | |
24 | 10, 2, 7, 18, 23 | deg1ge 24695 | . . 3 ⊢ (((𝐶 · (𝐹 ↑ 𝑋)) ∈ (Base‘𝑃) ∧ 𝐹 ∈ ℕ0 ∧ ((coe1‘(𝐶 · (𝐹 ↑ 𝑋)))‘𝐹) ≠ 0 ) → 𝐹 ≤ (𝐷‘(𝐶 · (𝐹 ↑ 𝑋)))) |
25 | 9, 13, 22, 24 | syl3anc 1367 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝐶 ≠ 0 ) ∧ 𝐹 ∈ ℕ0) → 𝐹 ≤ (𝐷‘(𝐶 · (𝐹 ↑ 𝑋)))) |
26 | 12, 15, 17, 25 | xrletrid 12551 | 1 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝐶 ≠ 0 ) ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐶 · (𝐹 ↑ 𝑋))) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 ℝ*cxr 10677 ≤ cle 10679 ℕ0cn0 11900 Basecbs 16486 ·𝑠 cvsca 16572 0gc0g 16716 .gcmg 18227 mulGrpcmgp 19242 Ringcrg 19300 var1cv1 20347 Poly1cpl1 20348 coe1cco1 20349 deg1 cdg1 24651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-addf 10619 ax-mulf 10620 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-ofr 7413 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-sup 8909 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-sca 16584 df-vsca 16585 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-0g 16718 df-gsum 16719 df-mre 16860 df-mrc 16861 df-acs 16863 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-mhm 17959 df-submnd 17960 df-grp 18109 df-minusg 18110 df-sbg 18111 df-mulg 18228 df-subg 18279 df-ghm 18359 df-cntz 18450 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-cring 19303 df-subrg 19536 df-lmod 19639 df-lss 19707 df-psr 20139 df-mvr 20140 df-mpl 20141 df-opsr 20143 df-psr1 20351 df-vr1 20352 df-ply1 20353 df-coe1 20354 df-cnfld 20549 df-mdeg 24652 df-deg1 24653 |
This theorem is referenced by: deg1pw 24717 fta1blem 24765 |
Copyright terms: Public domain | W3C validator |