![]() |
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 22089 | . . . . 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 20145 | . . . 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 25940 | . . 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 25946 | . . . . . 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 25946 | . . . . . 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 12533 | . . . 4 ⊢ (𝜑 → ((𝐷‘𝐹) + (𝐷‘𝐺)) ∈ ℕ0) |
22 | 21 | nn0red 12530 | . . 3 ⊢ (𝜑 → ((𝐷‘𝐹) + (𝐷‘𝐺)) ∈ ℝ) |
23 | 22 | rexrd 11261 | . 2 ⊢ (𝜑 → ((𝐷‘𝐹) + (𝐷‘𝐺)) ∈ ℝ*) |
24 | 17 | nn0red 12530 | . . . 4 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℝ) |
25 | 24 | leidd 11777 | . . 3 ⊢ (𝜑 → (𝐷‘𝐹) ≤ (𝐷‘𝐹)) |
26 | 20 | nn0red 12530 | . . . 4 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℝ) |
27 | 26 | leidd 11777 | . . 3 ⊢ (𝜑 → (𝐷‘𝐺) ≤ (𝐷‘𝐺)) |
28 | 2, 11, 1, 7, 8, 5, 6, 17, 20, 25, 27 | deg1mulle2 25967 | . 2 ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ ((𝐷‘𝐹) + (𝐷‘𝐺))) |
29 | eqid 2724 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
30 | 2, 8, 29, 7, 11, 15, 1, 5, 14, 6, 18 | coe1mul4 25958 | . . . 4 ⊢ (𝜑 → ((coe1‘(𝐹 · 𝐺))‘((𝐷‘𝐹) + (𝐷‘𝐺))) = (((coe1‘𝐹)‘(𝐷‘𝐹))(.r‘𝑅)((coe1‘𝐺)‘(𝐷‘𝐺)))) |
31 | eqid 2724 | . . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
32 | eqid 2724 | . . . . . . 7 ⊢ (coe1‘𝐺) = (coe1‘𝐺) | |
33 | 11, 2, 15, 7, 31, 32 | deg1ldg 25950 | . . . . . 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 2724 | . . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
37 | 32, 7, 2, 36 | coe1f 22053 | . . . . . . . . 9 ⊢ (𝐺 ∈ 𝐵 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
38 | 6, 37 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
39 | 38, 20 | ffvelcdmd 7077 | . . . . . . 7 ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ (Base‘𝑅)) |
40 | deg1mul2.e | . . . . . . . 8 ⊢ 𝐸 = (RLReg‘𝑅) | |
41 | 40, 36, 29, 31 | rrgeq0i 21189 | . . . . . . 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 583 | . . . . . 6 ⊢ (𝜑 → ((((coe1‘𝐹)‘(𝐷‘𝐹))(.r‘𝑅)((coe1‘𝐺)‘(𝐷‘𝐺))) = (0g‘𝑅) → ((coe1‘𝐺)‘(𝐷‘𝐺)) = (0g‘𝑅))) |
43 | 42 | necon3d 2953 | . . . . 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 3000 | . . 3 ⊢ (𝜑 → ((coe1‘(𝐹 · 𝐺))‘((𝐷‘𝐹) + (𝐷‘𝐺))) ≠ (0g‘𝑅)) |
46 | eqid 2724 | . . . 4 ⊢ (coe1‘(𝐹 · 𝐺)) = (coe1‘(𝐹 · 𝐺)) | |
47 | 11, 2, 7, 31, 46 | deg1ge 25956 | . . 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 13131 | 1 ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) = ((𝐷‘𝐹) + (𝐷‘𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 class class class wbr 5138 ⟶wf 6529 ‘cfv 6533 (class class class)co 7401 + caddc 11109 ℝ*cxr 11244 ≤ cle 11246 ℕ0cn0 12469 Basecbs 17143 .rcmulr 17197 0gc0g 17384 Ringcrg 20128 RLRegcrlreg 21179 Poly1cpl1 22019 coe1cco1 22020 deg1 cdg1 25909 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-ofr 7664 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-fzo 13625 df-seq 13964 df-hash 14288 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-0g 17386 df-gsum 17387 df-prds 17392 df-pws 17394 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mhm 18703 df-submnd 18704 df-grp 18856 df-minusg 18857 df-mulg 18986 df-subg 19040 df-ghm 19129 df-cntz 19223 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-cring 20131 df-subrng 20436 df-subrg 20461 df-rlreg 21183 df-cnfld 21229 df-psr 21771 df-mpl 21773 df-opsr 21775 df-psr1 22022 df-ply1 22024 df-coe1 22025 df-mdeg 25910 df-deg1 25911 |
This theorem is referenced by: ply1domn 25981 ply1divmo 25993 fta1glem1 26023 minplyirredlem 33249 mon1psubm 42437 deg1mhm 42438 |
Copyright terms: Public domain | W3C validator |