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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.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
2 | deg1mul2.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | 2 | ply1ring 20408 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ Ring) |
5 | deg1mul2.fb | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
6 | deg1mul2.gb | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
7 | deg1mul2.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑃) | |
8 | deg1mul2.t | . . . . 5 ⊢ · = (.r‘𝑃) | |
9 | 7, 8 | ringcl 19303 | . . . 4 ⊢ ((𝑃 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 · 𝐺) ∈ 𝐵) |
10 | 4, 5, 6, 9 | syl3anc 1366 | . . 3 ⊢ (𝜑 → (𝐹 · 𝐺) ∈ 𝐵) |
11 | deg1mul2.d | . . . 4 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
12 | 11, 2, 7 | deg1xrcl 24668 | . . 3 ⊢ ((𝐹 · 𝐺) ∈ 𝐵 → (𝐷‘(𝐹 · 𝐺)) ∈ ℝ*) |
13 | 10, 12 | syl 17 | . 2 ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) ∈ ℝ*) |
14 | deg1mul2.fz | . . . . . 6 ⊢ (𝜑 → 𝐹 ≠ 0 ) | |
15 | deg1mul2.z | . . . . . . 7 ⊢ 0 = (0g‘𝑃) | |
16 | 11, 2, 15, 7 | deg1nn0cl 24674 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ0) |
17 | 1, 5, 14, 16 | syl3anc 1366 | . . . . 5 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℕ0) |
18 | deg1mul2.gz | . . . . . 6 ⊢ (𝜑 → 𝐺 ≠ 0 ) | |
19 | 11, 2, 15, 7 | deg1nn0cl 24674 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝐺 ≠ 0 ) → (𝐷‘𝐺) ∈ ℕ0) |
20 | 1, 6, 18, 19 | syl3anc 1366 | . . . . 5 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℕ0) |
21 | 17, 20 | nn0addcld 11951 | . . . 4 ⊢ (𝜑 → ((𝐷‘𝐹) + (𝐷‘𝐺)) ∈ ℕ0) |
22 | 21 | nn0red 11948 | . . 3 ⊢ (𝜑 → ((𝐷‘𝐹) + (𝐷‘𝐺)) ∈ ℝ) |
23 | 22 | rexrd 10683 | . 2 ⊢ (𝜑 → ((𝐷‘𝐹) + (𝐷‘𝐺)) ∈ ℝ*) |
24 | 17 | nn0red 11948 | . . . 4 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℝ) |
25 | 24 | leidd 11198 | . . 3 ⊢ (𝜑 → (𝐷‘𝐹) ≤ (𝐷‘𝐹)) |
26 | 20 | nn0red 11948 | . . . 4 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℝ) |
27 | 26 | leidd 11198 | . . 3 ⊢ (𝜑 → (𝐷‘𝐺) ≤ (𝐷‘𝐺)) |
28 | 2, 11, 1, 7, 8, 5, 6, 17, 20, 25, 27 | deg1mulle2 24695 | . 2 ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ ((𝐷‘𝐹) + (𝐷‘𝐺))) |
29 | eqid 2819 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
30 | 2, 8, 29, 7, 11, 15, 1, 5, 14, 6, 18 | coe1mul4 24686 | . . . 4 ⊢ (𝜑 → ((coe1‘(𝐹 · 𝐺))‘((𝐷‘𝐹) + (𝐷‘𝐺))) = (((coe1‘𝐹)‘(𝐷‘𝐹))(.r‘𝑅)((coe1‘𝐺)‘(𝐷‘𝐺)))) |
31 | eqid 2819 | . . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
32 | eqid 2819 | . . . . . . 7 ⊢ (coe1‘𝐺) = (coe1‘𝐺) | |
33 | 11, 2, 15, 7, 31, 32 | deg1ldg 24678 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝐺 ≠ 0 ) → ((coe1‘𝐺)‘(𝐷‘𝐺)) ≠ (0g‘𝑅)) |
34 | 1, 6, 18, 33 | syl3anc 1366 | . . . . 5 ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ≠ (0g‘𝑅)) |
35 | deg1mul2.fc | . . . . . . 7 ⊢ (𝜑 → ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝐸) | |
36 | eqid 2819 | . . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
37 | 32, 7, 2, 36 | coe1f 20371 | . . . . . . . . 9 ⊢ (𝐺 ∈ 𝐵 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
38 | 6, 37 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
39 | 38, 20 | ffvelrnd 6845 | . . . . . . 7 ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ (Base‘𝑅)) |
40 | deg1mul2.e | . . . . . . . 8 ⊢ 𝐸 = (RLReg‘𝑅) | |
41 | 40, 36, 29, 31 | rrgeq0i 20054 | . . . . . . 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))) |
42 | 35, 39, 41 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → ((((coe1‘𝐹)‘(𝐷‘𝐹))(.r‘𝑅)((coe1‘𝐺)‘(𝐷‘𝐺))) = (0g‘𝑅) → ((coe1‘𝐺)‘(𝐷‘𝐺)) = (0g‘𝑅))) |
43 | 42 | necon3d 3035 | . . . . 5 ⊢ (𝜑 → (((coe1‘𝐺)‘(𝐷‘𝐺)) ≠ (0g‘𝑅) → (((coe1‘𝐹)‘(𝐷‘𝐹))(.r‘𝑅)((coe1‘𝐺)‘(𝐷‘𝐺))) ≠ (0g‘𝑅))) |
44 | 34, 43 | mpd 15 | . . . 4 ⊢ (𝜑 → (((coe1‘𝐹)‘(𝐷‘𝐹))(.r‘𝑅)((coe1‘𝐺)‘(𝐷‘𝐺))) ≠ (0g‘𝑅)) |
45 | 30, 44 | eqnetrd 3081 | . . 3 ⊢ (𝜑 → ((coe1‘(𝐹 · 𝐺))‘((𝐷‘𝐹) + (𝐷‘𝐺))) ≠ (0g‘𝑅)) |
46 | eqid 2819 | . . . 4 ⊢ (coe1‘(𝐹 · 𝐺)) = (coe1‘(𝐹 · 𝐺)) | |
47 | 11, 2, 7, 31, 46 | deg1ge 24684 | . . 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))) |
48 | 10, 21, 45, 47 | syl3anc 1366 | . 2 ⊢ (𝜑 → ((𝐷‘𝐹) + (𝐷‘𝐺)) ≤ (𝐷‘(𝐹 · 𝐺))) |
49 | 13, 23, 28, 48 | xrletrid 12540 | 1 ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) = ((𝐷‘𝐹) + (𝐷‘𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 class class class wbr 5057 ⟶wf 6344 ‘cfv 6348 (class class class)co 7148 + caddc 10532 ℝ*cxr 10666 ≤ cle 10668 ℕ0cn0 11889 Basecbs 16475 .rcmulr 16558 0gc0g 16705 Ringcrg 19289 RLRegcrlreg 20044 Poly1cpl1 20337 coe1cco1 20338 deg1 cdg1 24640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 ax-addf 10608 ax-mulf 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-of 7401 df-ofr 7402 df-om 7573 df-1st 7681 df-2nd 7682 df-supp 7823 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-er 8281 df-map 8400 df-pm 8401 df-ixp 8454 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-fsupp 8826 df-sup 8898 df-oi 8966 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-fz 12885 df-fzo 13026 df-seq 13362 df-hash 13683 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-0g 16707 df-gsum 16708 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-grp 18098 df-minusg 18099 df-mulg 18217 df-subg 18268 df-ghm 18348 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19232 df-ur 19244 df-ring 19291 df-cring 19292 df-subrg 19525 df-rlreg 20048 df-psr 20128 df-mpl 20130 df-opsr 20132 df-psr1 20340 df-ply1 20342 df-coe1 20343 df-cnfld 20538 df-mdeg 24641 df-deg1 24642 |
This theorem is referenced by: ply1domn 24709 ply1divmo 24721 fta1glem1 24751 mon1psubm 39797 deg1mhm 39798 |
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