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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 20418 | . . . . 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 19313 | . . . 4 ⊢ ((𝑃 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 · 𝐺) ∈ 𝐵) |
10 | 4, 5, 6, 9 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝐹 · 𝐺) ∈ 𝐵) |
11 | deg1mul2.d | . . . 4 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
12 | 11, 2, 7 | deg1xrcl 24678 | . . 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 24684 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ0) |
17 | 1, 5, 14, 16 | syl3anc 1367 | . . . . 5 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℕ0) |
18 | deg1mul2.gz | . . . . . 6 ⊢ (𝜑 → 𝐺 ≠ 0 ) | |
19 | 11, 2, 15, 7 | deg1nn0cl 24684 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝐺 ≠ 0 ) → (𝐷‘𝐺) ∈ ℕ0) |
20 | 1, 6, 18, 19 | syl3anc 1367 | . . . . 5 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℕ0) |
21 | 17, 20 | nn0addcld 11962 | . . . 4 ⊢ (𝜑 → ((𝐷‘𝐹) + (𝐷‘𝐺)) ∈ ℕ0) |
22 | 21 | nn0red 11959 | . . 3 ⊢ (𝜑 → ((𝐷‘𝐹) + (𝐷‘𝐺)) ∈ ℝ) |
23 | 22 | rexrd 10693 | . 2 ⊢ (𝜑 → ((𝐷‘𝐹) + (𝐷‘𝐺)) ∈ ℝ*) |
24 | 17 | nn0red 11959 | . . . 4 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℝ) |
25 | 24 | leidd 11208 | . . 3 ⊢ (𝜑 → (𝐷‘𝐹) ≤ (𝐷‘𝐹)) |
26 | 20 | nn0red 11959 | . . . 4 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℝ) |
27 | 26 | leidd 11208 | . . 3 ⊢ (𝜑 → (𝐷‘𝐺) ≤ (𝐷‘𝐺)) |
28 | 2, 11, 1, 7, 8, 5, 6, 17, 20, 25, 27 | deg1mulle2 24705 | . 2 ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ ((𝐷‘𝐹) + (𝐷‘𝐺))) |
29 | eqid 2823 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
30 | 2, 8, 29, 7, 11, 15, 1, 5, 14, 6, 18 | coe1mul4 24696 | . . . 4 ⊢ (𝜑 → ((coe1‘(𝐹 · 𝐺))‘((𝐷‘𝐹) + (𝐷‘𝐺))) = (((coe1‘𝐹)‘(𝐷‘𝐹))(.r‘𝑅)((coe1‘𝐺)‘(𝐷‘𝐺)))) |
31 | eqid 2823 | . . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
32 | eqid 2823 | . . . . . . 7 ⊢ (coe1‘𝐺) = (coe1‘𝐺) | |
33 | 11, 2, 15, 7, 31, 32 | deg1ldg 24688 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝐺 ≠ 0 ) → ((coe1‘𝐺)‘(𝐷‘𝐺)) ≠ (0g‘𝑅)) |
34 | 1, 6, 18, 33 | syl3anc 1367 | . . . . 5 ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ≠ (0g‘𝑅)) |
35 | deg1mul2.fc | . . . . . . 7 ⊢ (𝜑 → ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝐸) | |
36 | eqid 2823 | . . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
37 | 32, 7, 2, 36 | coe1f 20381 | . . . . . . . . 9 ⊢ (𝐺 ∈ 𝐵 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
38 | 6, 37 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
39 | 38, 20 | ffvelrnd 6854 | . . . . . . 7 ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ (Base‘𝑅)) |
40 | deg1mul2.e | . . . . . . . 8 ⊢ 𝐸 = (RLReg‘𝑅) | |
41 | 40, 36, 29, 31 | rrgeq0i 20064 | . . . . . . 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 3039 | . . . . 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 3085 | . . 3 ⊢ (𝜑 → ((coe1‘(𝐹 · 𝐺))‘((𝐷‘𝐹) + (𝐷‘𝐺))) ≠ (0g‘𝑅)) |
46 | eqid 2823 | . . . 4 ⊢ (coe1‘(𝐹 · 𝐺)) = (coe1‘(𝐹 · 𝐺)) | |
47 | 11, 2, 7, 31, 46 | deg1ge 24694 | . . 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 1367 | . 2 ⊢ (𝜑 → ((𝐷‘𝐹) + (𝐷‘𝐺)) ≤ (𝐷‘(𝐹 · 𝐺))) |
49 | 13, 23, 28, 48 | xrletrid 12551 | 1 ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) = ((𝐷‘𝐹) + (𝐷‘𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 class class class wbr 5068 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 + caddc 10542 ℝ*cxr 10676 ≤ cle 10678 ℕ0cn0 11900 Basecbs 16485 .rcmulr 16568 0gc0g 16715 Ringcrg 19299 RLRegcrlreg 20054 Poly1cpl1 20347 coe1cco1 20348 deg1 cdg1 24650 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-ofr 7412 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 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 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-0g 16717 df-gsum 16718 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-grp 18108 df-minusg 18109 df-mulg 18227 df-subg 18278 df-ghm 18358 df-cntz 18449 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-subrg 19535 df-rlreg 20058 df-psr 20138 df-mpl 20140 df-opsr 20142 df-psr1 20350 df-ply1 20352 df-coe1 20353 df-cnfld 20548 df-mdeg 24651 df-deg1 24652 |
This theorem is referenced by: ply1domn 24719 ply1divmo 24731 fta1glem1 24761 mon1psubm 39813 deg1mhm 39814 |
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