![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 20877 | . . . . 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 19307 | . . . 4 ⊢ ((𝑃 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 · 𝐺) ∈ 𝐵) |
10 | 4, 5, 6, 9 | syl3anc 1368 | . . 3 ⊢ (𝜑 → (𝐹 · 𝐺) ∈ 𝐵) |
11 | deg1mul2.d | . . . 4 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
12 | 11, 2, 7 | deg1xrcl 24683 | . . 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 24689 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ0) |
17 | 1, 5, 14, 16 | syl3anc 1368 | . . . . 5 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℕ0) |
18 | deg1mul2.gz | . . . . . 6 ⊢ (𝜑 → 𝐺 ≠ 0 ) | |
19 | 11, 2, 15, 7 | deg1nn0cl 24689 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝐺 ≠ 0 ) → (𝐷‘𝐺) ∈ ℕ0) |
20 | 1, 6, 18, 19 | syl3anc 1368 | . . . . 5 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℕ0) |
21 | 17, 20 | nn0addcld 11947 | . . . 4 ⊢ (𝜑 → ((𝐷‘𝐹) + (𝐷‘𝐺)) ∈ ℕ0) |
22 | 21 | nn0red 11944 | . . 3 ⊢ (𝜑 → ((𝐷‘𝐹) + (𝐷‘𝐺)) ∈ ℝ) |
23 | 22 | rexrd 10680 | . 2 ⊢ (𝜑 → ((𝐷‘𝐹) + (𝐷‘𝐺)) ∈ ℝ*) |
24 | 17 | nn0red 11944 | . . . 4 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℝ) |
25 | 24 | leidd 11195 | . . 3 ⊢ (𝜑 → (𝐷‘𝐹) ≤ (𝐷‘𝐹)) |
26 | 20 | nn0red 11944 | . . . 4 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℝ) |
27 | 26 | leidd 11195 | . . 3 ⊢ (𝜑 → (𝐷‘𝐺) ≤ (𝐷‘𝐺)) |
28 | 2, 11, 1, 7, 8, 5, 6, 17, 20, 25, 27 | deg1mulle2 24710 | . 2 ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ ((𝐷‘𝐹) + (𝐷‘𝐺))) |
29 | eqid 2798 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
30 | 2, 8, 29, 7, 11, 15, 1, 5, 14, 6, 18 | coe1mul4 24701 | . . . 4 ⊢ (𝜑 → ((coe1‘(𝐹 · 𝐺))‘((𝐷‘𝐹) + (𝐷‘𝐺))) = (((coe1‘𝐹)‘(𝐷‘𝐹))(.r‘𝑅)((coe1‘𝐺)‘(𝐷‘𝐺)))) |
31 | eqid 2798 | . . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
32 | eqid 2798 | . . . . . . 7 ⊢ (coe1‘𝐺) = (coe1‘𝐺) | |
33 | 11, 2, 15, 7, 31, 32 | deg1ldg 24693 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝐺 ≠ 0 ) → ((coe1‘𝐺)‘(𝐷‘𝐺)) ≠ (0g‘𝑅)) |
34 | 1, 6, 18, 33 | syl3anc 1368 | . . . . 5 ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ≠ (0g‘𝑅)) |
35 | deg1mul2.fc | . . . . . . 7 ⊢ (𝜑 → ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝐸) | |
36 | eqid 2798 | . . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
37 | 32, 7, 2, 36 | coe1f 20840 | . . . . . . . . 9 ⊢ (𝐺 ∈ 𝐵 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
38 | 6, 37 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
39 | 38, 20 | ffvelrnd 6829 | . . . . . . 7 ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ (Base‘𝑅)) |
40 | deg1mul2.e | . . . . . . . 8 ⊢ 𝐸 = (RLReg‘𝑅) | |
41 | 40, 36, 29, 31 | rrgeq0i 20055 | . . . . . . 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 587 | . . . . . 6 ⊢ (𝜑 → ((((coe1‘𝐹)‘(𝐷‘𝐹))(.r‘𝑅)((coe1‘𝐺)‘(𝐷‘𝐺))) = (0g‘𝑅) → ((coe1‘𝐺)‘(𝐷‘𝐺)) = (0g‘𝑅))) |
43 | 42 | necon3d 3008 | . . . . 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 3054 | . . 3 ⊢ (𝜑 → ((coe1‘(𝐹 · 𝐺))‘((𝐷‘𝐹) + (𝐷‘𝐺))) ≠ (0g‘𝑅)) |
46 | eqid 2798 | . . . 4 ⊢ (coe1‘(𝐹 · 𝐺)) = (coe1‘(𝐹 · 𝐺)) | |
47 | 11, 2, 7, 31, 46 | deg1ge 24699 | . . 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 1368 | . 2 ⊢ (𝜑 → ((𝐷‘𝐹) + (𝐷‘𝐺)) ≤ (𝐷‘(𝐹 · 𝐺))) |
49 | 13, 23, 28, 48 | xrletrid 12536 | 1 ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) = ((𝐷‘𝐹) + (𝐷‘𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 class class class wbr 5030 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 + caddc 10529 ℝ*cxr 10663 ≤ cle 10665 ℕ0cn0 11885 Basecbs 16475 .rcmulr 16558 0gc0g 16705 Ringcrg 19290 RLRegcrlreg 20045 Poly1cpl1 20806 coe1cco1 20807 deg1 cdg1 24655 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-ofr 7390 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-sup 8890 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 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 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-subrg 19526 df-rlreg 20049 df-cnfld 20092 df-psr 20594 df-mpl 20596 df-opsr 20598 df-psr1 20809 df-ply1 20811 df-coe1 20812 df-mdeg 24656 df-deg1 24657 |
This theorem is referenced by: ply1domn 24724 ply1divmo 24736 fta1glem1 24766 mon1psubm 40150 deg1mhm 40151 |
Copyright terms: Public domain | W3C validator |