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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 22249 | . . . . 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 20247 | . . . 4 ⊢ ((𝑃 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 · 𝐺) ∈ 𝐵) | 
| 10 | 4, 5, 6, 9 | syl3anc 1373 | . . 3 ⊢ (𝜑 → (𝐹 · 𝐺) ∈ 𝐵) | 
| 11 | deg1mul2.d | . . . 4 ⊢ 𝐷 = (deg1‘𝑅) | |
| 12 | 11, 2, 7 | deg1xrcl 26121 | . . 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 26127 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ0) | 
| 17 | 1, 5, 14, 16 | syl3anc 1373 | . . . . 5 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℕ0) | 
| 18 | deg1mul2.gz | . . . . . 6 ⊢ (𝜑 → 𝐺 ≠ 0 ) | |
| 19 | 11, 2, 15, 7 | deg1nn0cl 26127 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝐺 ≠ 0 ) → (𝐷‘𝐺) ∈ ℕ0) | 
| 20 | 1, 6, 18, 19 | syl3anc 1373 | . . . . 5 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℕ0) | 
| 21 | 17, 20 | nn0addcld 12591 | . . . 4 ⊢ (𝜑 → ((𝐷‘𝐹) + (𝐷‘𝐺)) ∈ ℕ0) | 
| 22 | 21 | nn0red 12588 | . . 3 ⊢ (𝜑 → ((𝐷‘𝐹) + (𝐷‘𝐺)) ∈ ℝ) | 
| 23 | 22 | rexrd 11311 | . 2 ⊢ (𝜑 → ((𝐷‘𝐹) + (𝐷‘𝐺)) ∈ ℝ*) | 
| 24 | 17 | nn0red 12588 | . . . 4 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℝ) | 
| 25 | 24 | leidd 11829 | . . 3 ⊢ (𝜑 → (𝐷‘𝐹) ≤ (𝐷‘𝐹)) | 
| 26 | 20 | nn0red 12588 | . . . 4 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℝ) | 
| 27 | 26 | leidd 11829 | . . 3 ⊢ (𝜑 → (𝐷‘𝐺) ≤ (𝐷‘𝐺)) | 
| 28 | 2, 11, 1, 7, 8, 5, 6, 17, 20, 25, 27 | deg1mulle2 26148 | . 2 ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ ((𝐷‘𝐹) + (𝐷‘𝐺))) | 
| 29 | eqid 2737 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
| 30 | 2, 8, 29, 7, 11, 15, 1, 5, 14, 6, 18 | coe1mul4 26139 | . . . 4 ⊢ (𝜑 → ((coe1‘(𝐹 · 𝐺))‘((𝐷‘𝐹) + (𝐷‘𝐺))) = (((coe1‘𝐹)‘(𝐷‘𝐹))(.r‘𝑅)((coe1‘𝐺)‘(𝐷‘𝐺)))) | 
| 31 | eqid 2737 | . . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
| 32 | eqid 2737 | . . . . . . 7 ⊢ (coe1‘𝐺) = (coe1‘𝐺) | |
| 33 | 11, 2, 15, 7, 31, 32 | deg1ldg 26131 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝐺 ≠ 0 ) → ((coe1‘𝐺)‘(𝐷‘𝐺)) ≠ (0g‘𝑅)) | 
| 34 | 1, 6, 18, 33 | syl3anc 1373 | . . . . 5 ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ≠ (0g‘𝑅)) | 
| 35 | deg1mul2.fc | . . . . . . 7 ⊢ (𝜑 → ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝐸) | |
| 36 | eqid 2737 | . . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 37 | 32, 7, 2, 36 | coe1f 22213 | . . . . . . . . 9 ⊢ (𝐺 ∈ 𝐵 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) | 
| 38 | 6, 37 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) | 
| 39 | 38, 20 | ffvelcdmd 7105 | . . . . . . 7 ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ (Base‘𝑅)) | 
| 40 | deg1mul2.e | . . . . . . . 8 ⊢ 𝐸 = (RLReg‘𝑅) | |
| 41 | 40, 36, 29, 31 | rrgeq0i 20699 | . . . . . . 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 584 | . . . . . 6 ⊢ (𝜑 → ((((coe1‘𝐹)‘(𝐷‘𝐹))(.r‘𝑅)((coe1‘𝐺)‘(𝐷‘𝐺))) = (0g‘𝑅) → ((coe1‘𝐺)‘(𝐷‘𝐺)) = (0g‘𝑅))) | 
| 43 | 42 | necon3d 2961 | . . . . 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 3008 | . . 3 ⊢ (𝜑 → ((coe1‘(𝐹 · 𝐺))‘((𝐷‘𝐹) + (𝐷‘𝐺))) ≠ (0g‘𝑅)) | 
| 46 | eqid 2737 | . . . 4 ⊢ (coe1‘(𝐹 · 𝐺)) = (coe1‘(𝐹 · 𝐺)) | |
| 47 | 11, 2, 7, 31, 46 | deg1ge 26137 | . . 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 1373 | . 2 ⊢ (𝜑 → ((𝐷‘𝐹) + (𝐷‘𝐺)) ≤ (𝐷‘(𝐹 · 𝐺))) | 
| 49 | 13, 23, 28, 48 | xrletrid 13197 | 1 ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) = ((𝐷‘𝐹) + (𝐷‘𝐺))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 + caddc 11158 ℝ*cxr 11294 ≤ cle 11296 ℕ0cn0 12526 Basecbs 17247 .rcmulr 17298 0gc0g 17484 Ringcrg 20230 RLRegcrlreg 20691 Poly1cpl1 22178 coe1cco1 22179 deg1cdg1 26093 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-mulg 19086 df-subg 19141 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-subrng 20546 df-subrg 20570 df-rlreg 20694 df-cnfld 21365 df-psr 21929 df-mpl 21931 df-opsr 21933 df-psr1 22181 df-ply1 22183 df-coe1 22184 df-mdeg 26094 df-deg1 26095 | 
| This theorem is referenced by: deg1mul 26154 ply1domn 26163 ply1divmo 26175 fta1glem1 26207 ply1unit 33600 m1pmeq 33608 minplyirredlem 33753 mon1psubm 43211 deg1mhm 43212 | 
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