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Mirrors > Home > MPE Home > Th. List > deg1mul2 | Structured version Visualization version GIF version |
Description: Degree of multiplication of two nonzero polynomials when the first leads with a nonzero-divisor coefficient. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
Ref | Expression |
---|---|
deg1mul2.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
deg1mul2.p | ⊢ 𝑃 = (Poly1‘𝑅) |
deg1mul2.e | ⊢ 𝐸 = (RLReg‘𝑅) |
deg1mul2.b | ⊢ 𝐵 = (Base‘𝑃) |
deg1mul2.t | ⊢ · = (.r‘𝑃) |
deg1mul2.z | ⊢ 0 = (0g‘𝑃) |
deg1mul2.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
deg1mul2.fb | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
deg1mul2.fz | ⊢ (𝜑 → 𝐹 ≠ 0 ) |
deg1mul2.fc | ⊢ (𝜑 → ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝐸) |
deg1mul2.gb | ⊢ (𝜑 → 𝐺 ∈ 𝐵) |
deg1mul2.gz | ⊢ (𝜑 → 𝐺 ≠ 0 ) |
Ref | Expression |
---|---|
deg1mul2 | ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) = ((𝐷‘𝐹) + (𝐷‘𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | deg1mul2.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | deg1mul2.d | . . 3 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
3 | deg1mul2.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
4 | deg1mul2.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
5 | deg1mul2.t | . . 3 ⊢ · = (.r‘𝑃) | |
6 | deg1mul2.fb | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
7 | deg1mul2.gb | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
8 | deg1mul2.fz | . . . 4 ⊢ (𝜑 → 𝐹 ≠ 0 ) | |
9 | deg1mul2.z | . . . . 5 ⊢ 0 = (0g‘𝑃) | |
10 | 2, 1, 9, 4 | deg1nn0cl 24254 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ0) |
11 | 3, 6, 8, 10 | syl3anc 1494 | . . 3 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℕ0) |
12 | deg1mul2.gz | . . . 4 ⊢ (𝜑 → 𝐺 ≠ 0 ) | |
13 | 2, 1, 9, 4 | deg1nn0cl 24254 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝐺 ≠ 0 ) → (𝐷‘𝐺) ∈ ℕ0) |
14 | 3, 7, 12, 13 | syl3anc 1494 | . . 3 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℕ0) |
15 | 11 | nn0red 11686 | . . . 4 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℝ) |
16 | 15 | leidd 10925 | . . 3 ⊢ (𝜑 → (𝐷‘𝐹) ≤ (𝐷‘𝐹)) |
17 | 14 | nn0red 11686 | . . . 4 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℝ) |
18 | 17 | leidd 10925 | . . 3 ⊢ (𝜑 → (𝐷‘𝐺) ≤ (𝐷‘𝐺)) |
19 | 1, 2, 3, 4, 5, 6, 7, 11, 14, 16, 18 | deg1mulle2 24275 | . 2 ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ ((𝐷‘𝐹) + (𝐷‘𝐺))) |
20 | 1 | ply1ring 19985 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
21 | 3, 20 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ Ring) |
22 | 4, 5 | ringcl 18922 | . . . 4 ⊢ ((𝑃 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 · 𝐺) ∈ 𝐵) |
23 | 21, 6, 7, 22 | syl3anc 1494 | . . 3 ⊢ (𝜑 → (𝐹 · 𝐺) ∈ 𝐵) |
24 | 11, 14 | nn0addcld 11689 | . . 3 ⊢ (𝜑 → ((𝐷‘𝐹) + (𝐷‘𝐺)) ∈ ℕ0) |
25 | eqid 2825 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
26 | 1, 5, 25, 4, 2, 9, 3, 6, 8, 7, 12 | coe1mul4 24266 | . . . 4 ⊢ (𝜑 → ((coe1‘(𝐹 · 𝐺))‘((𝐷‘𝐹) + (𝐷‘𝐺))) = (((coe1‘𝐹)‘(𝐷‘𝐹))(.r‘𝑅)((coe1‘𝐺)‘(𝐷‘𝐺)))) |
27 | eqid 2825 | . . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
28 | eqid 2825 | . . . . . . 7 ⊢ (coe1‘𝐺) = (coe1‘𝐺) | |
29 | 2, 1, 9, 4, 27, 28 | deg1ldg 24258 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝐺 ≠ 0 ) → ((coe1‘𝐺)‘(𝐷‘𝐺)) ≠ (0g‘𝑅)) |
30 | 3, 7, 12, 29 | syl3anc 1494 | . . . . 5 ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ≠ (0g‘𝑅)) |
31 | deg1mul2.fc | . . . . . . 7 ⊢ (𝜑 → ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝐸) | |
32 | eqid 2825 | . . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
33 | 28, 4, 1, 32 | coe1f 19948 | . . . . . . . . 9 ⊢ (𝐺 ∈ 𝐵 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
34 | 7, 33 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
35 | 34, 14 | ffvelrnd 6614 | . . . . . . 7 ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ (Base‘𝑅)) |
36 | deg1mul2.e | . . . . . . . 8 ⊢ 𝐸 = (RLReg‘𝑅) | |
37 | 36, 32, 25, 27 | rrgeq0i 19657 | . . . . . . 7 ⊢ ((((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝐸 ∧ ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ (Base‘𝑅)) → ((((coe1‘𝐹)‘(𝐷‘𝐹))(.r‘𝑅)((coe1‘𝐺)‘(𝐷‘𝐺))) = (0g‘𝑅) → ((coe1‘𝐺)‘(𝐷‘𝐺)) = (0g‘𝑅))) |
38 | 31, 35, 37 | syl2anc 579 | . . . . . 6 ⊢ (𝜑 → ((((coe1‘𝐹)‘(𝐷‘𝐹))(.r‘𝑅)((coe1‘𝐺)‘(𝐷‘𝐺))) = (0g‘𝑅) → ((coe1‘𝐺)‘(𝐷‘𝐺)) = (0g‘𝑅))) |
39 | 38 | necon3d 3020 | . . . . 5 ⊢ (𝜑 → (((coe1‘𝐺)‘(𝐷‘𝐺)) ≠ (0g‘𝑅) → (((coe1‘𝐹)‘(𝐷‘𝐹))(.r‘𝑅)((coe1‘𝐺)‘(𝐷‘𝐺))) ≠ (0g‘𝑅))) |
40 | 30, 39 | mpd 15 | . . . 4 ⊢ (𝜑 → (((coe1‘𝐹)‘(𝐷‘𝐹))(.r‘𝑅)((coe1‘𝐺)‘(𝐷‘𝐺))) ≠ (0g‘𝑅)) |
41 | 26, 40 | eqnetrd 3066 | . . 3 ⊢ (𝜑 → ((coe1‘(𝐹 · 𝐺))‘((𝐷‘𝐹) + (𝐷‘𝐺))) ≠ (0g‘𝑅)) |
42 | eqid 2825 | . . . 4 ⊢ (coe1‘(𝐹 · 𝐺)) = (coe1‘(𝐹 · 𝐺)) | |
43 | 2, 1, 4, 27, 42 | deg1ge 24264 | . . 3 ⊢ (((𝐹 · 𝐺) ∈ 𝐵 ∧ ((𝐷‘𝐹) + (𝐷‘𝐺)) ∈ ℕ0 ∧ ((coe1‘(𝐹 · 𝐺))‘((𝐷‘𝐹) + (𝐷‘𝐺))) ≠ (0g‘𝑅)) → ((𝐷‘𝐹) + (𝐷‘𝐺)) ≤ (𝐷‘(𝐹 · 𝐺))) |
44 | 23, 24, 41, 43 | syl3anc 1494 | . 2 ⊢ (𝜑 → ((𝐷‘𝐹) + (𝐷‘𝐺)) ≤ (𝐷‘(𝐹 · 𝐺))) |
45 | 2, 1, 4 | deg1xrcl 24248 | . . . 4 ⊢ ((𝐹 · 𝐺) ∈ 𝐵 → (𝐷‘(𝐹 · 𝐺)) ∈ ℝ*) |
46 | 23, 45 | syl 17 | . . 3 ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) ∈ ℝ*) |
47 | 24 | nn0red 11686 | . . . 4 ⊢ (𝜑 → ((𝐷‘𝐹) + (𝐷‘𝐺)) ∈ ℝ) |
48 | 47 | rexrd 10413 | . . 3 ⊢ (𝜑 → ((𝐷‘𝐹) + (𝐷‘𝐺)) ∈ ℝ*) |
49 | xrletri3 12280 | . . 3 ⊢ (((𝐷‘(𝐹 · 𝐺)) ∈ ℝ* ∧ ((𝐷‘𝐹) + (𝐷‘𝐺)) ∈ ℝ*) → ((𝐷‘(𝐹 · 𝐺)) = ((𝐷‘𝐹) + (𝐷‘𝐺)) ↔ ((𝐷‘(𝐹 · 𝐺)) ≤ ((𝐷‘𝐹) + (𝐷‘𝐺)) ∧ ((𝐷‘𝐹) + (𝐷‘𝐺)) ≤ (𝐷‘(𝐹 · 𝐺))))) | |
50 | 46, 48, 49 | syl2anc 579 | . 2 ⊢ (𝜑 → ((𝐷‘(𝐹 · 𝐺)) = ((𝐷‘𝐹) + (𝐷‘𝐺)) ↔ ((𝐷‘(𝐹 · 𝐺)) ≤ ((𝐷‘𝐹) + (𝐷‘𝐺)) ∧ ((𝐷‘𝐹) + (𝐷‘𝐺)) ≤ (𝐷‘(𝐹 · 𝐺))))) |
51 | 19, 44, 50 | mpbir2and 704 | 1 ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) = ((𝐷‘𝐹) + (𝐷‘𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 class class class wbr 4875 ⟶wf 6123 ‘cfv 6127 (class class class)co 6910 + caddc 10262 ℝ*cxr 10397 ≤ cle 10399 ℕ0cn0 11625 Basecbs 16229 .rcmulr 16313 0gc0g 16460 Ringcrg 18908 RLRegcrlreg 19647 Poly1cpl1 19914 coe1cco1 19915 deg1 cdg1 24220 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 ax-addf 10338 ax-mulf 10339 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-ofr 7163 df-om 7332 df-1st 7433 df-2nd 7434 df-supp 7565 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-2o 7832 df-oadd 7835 df-er 8014 df-map 8129 df-pm 8130 df-ixp 8182 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-fsupp 8551 df-sup 8623 df-oi 8691 df-card 9085 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-fz 12627 df-fzo 12768 df-seq 13103 df-hash 13418 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-starv 16327 df-sca 16328 df-vsca 16329 df-tset 16331 df-ple 16332 df-ds 16334 df-unif 16335 df-0g 16462 df-gsum 16463 df-mre 16606 df-mrc 16607 df-acs 16609 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-mhm 17695 df-submnd 17696 df-grp 17786 df-minusg 17787 df-mulg 17902 df-subg 17949 df-ghm 18016 df-cntz 18107 df-cmn 18555 df-abl 18556 df-mgp 18851 df-ur 18863 df-ring 18910 df-cring 18911 df-subrg 19141 df-rlreg 19651 df-psr 19724 df-mpl 19726 df-opsr 19728 df-psr1 19917 df-ply1 19919 df-coe1 19920 df-cnfld 20114 df-mdeg 24221 df-deg1 24222 |
This theorem is referenced by: ply1domn 24289 ply1divmo 24301 fta1glem1 24331 mon1psubm 38622 deg1mhm 38623 |
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