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| Mirrors > Home > MPE Home > Th. List > deg1sublt | Structured version Visualization version GIF version | ||
| Description: Subtraction of two polynomials limited to the same degree with the same leading coefficient gives a polynomial with a smaller degree. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
| Ref | Expression |
|---|---|
| deg1sublt.d | ⊢ 𝐷 = (deg1‘𝑅) |
| deg1sublt.p | ⊢ 𝑃 = (Poly1‘𝑅) |
| deg1sublt.b | ⊢ 𝐵 = (Base‘𝑃) |
| deg1sublt.m | ⊢ − = (-g‘𝑃) |
| deg1sublt.l | ⊢ (𝜑 → 𝐿 ∈ ℕ0) |
| deg1sublt.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
| deg1sublt.fb | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
| deg1sublt.fd | ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝐿) |
| deg1sublt.gb | ⊢ (𝜑 → 𝐺 ∈ 𝐵) |
| deg1sublt.gd | ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝐿) |
| deg1sublt.a | ⊢ 𝐴 = (coe1‘𝐹) |
| deg1sublt.c | ⊢ 𝐶 = (coe1‘𝐺) |
| deg1sublt.eq | ⊢ (𝜑 → ((coe1‘𝐹)‘𝐿) = ((coe1‘𝐺)‘𝐿)) |
| Ref | Expression |
|---|---|
| deg1sublt | ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) < 𝐿) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | deg1sublt.d | . . . 4 ⊢ 𝐷 = (deg1‘𝑅) | |
| 2 | deg1sublt.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
| 3 | eqid 2737 | . . . 4 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
| 4 | deg1sublt.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
| 5 | eqid 2737 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
| 6 | eqid 2737 | . . . 4 ⊢ (coe1‘(𝐹 − 𝐺)) = (coe1‘(𝐹 − 𝐺)) | |
| 7 | deg1sublt.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
| 8 | 2 | ply1ring 22249 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
| 9 | ringgrp 20235 | . . . . . 6 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Grp) | |
| 10 | 7, 8, 9 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ Grp) |
| 11 | deg1sublt.fb | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
| 12 | deg1sublt.gb | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
| 13 | deg1sublt.m | . . . . . 6 ⊢ − = (-g‘𝑃) | |
| 14 | 4, 13 | grpsubcl 19038 | . . . . 5 ⊢ ((𝑃 ∈ Grp ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 − 𝐺) ∈ 𝐵) |
| 15 | 10, 11, 12, 14 | syl3anc 1373 | . . . 4 ⊢ (𝜑 → (𝐹 − 𝐺) ∈ 𝐵) |
| 16 | deg1sublt.l | . . . 4 ⊢ (𝜑 → 𝐿 ∈ ℕ0) | |
| 17 | eqid 2737 | . . . . . . 7 ⊢ (-g‘𝑅) = (-g‘𝑅) | |
| 18 | 2, 4, 13, 17 | coe1subfv 22269 | . . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝐿 ∈ ℕ0) → ((coe1‘(𝐹 − 𝐺))‘𝐿) = (((coe1‘𝐹)‘𝐿)(-g‘𝑅)((coe1‘𝐺)‘𝐿))) |
| 19 | 7, 11, 12, 16, 18 | syl31anc 1375 | . . . . 5 ⊢ (𝜑 → ((coe1‘(𝐹 − 𝐺))‘𝐿) = (((coe1‘𝐹)‘𝐿)(-g‘𝑅)((coe1‘𝐺)‘𝐿))) |
| 20 | deg1sublt.eq | . . . . . 6 ⊢ (𝜑 → ((coe1‘𝐹)‘𝐿) = ((coe1‘𝐺)‘𝐿)) | |
| 21 | 20 | oveq1d 7446 | . . . . 5 ⊢ (𝜑 → (((coe1‘𝐹)‘𝐿)(-g‘𝑅)((coe1‘𝐺)‘𝐿)) = (((coe1‘𝐺)‘𝐿)(-g‘𝑅)((coe1‘𝐺)‘𝐿))) |
| 22 | ringgrp 20235 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
| 23 | 7, 22 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp) |
| 24 | eqid 2737 | . . . . . . . . 9 ⊢ (coe1‘𝐺) = (coe1‘𝐺) | |
| 25 | eqid 2737 | . . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 26 | 24, 4, 2, 25 | coe1f 22213 | . . . . . . . 8 ⊢ (𝐺 ∈ 𝐵 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
| 27 | 12, 26 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
| 28 | 27, 16 | ffvelcdmd 7105 | . . . . . 6 ⊢ (𝜑 → ((coe1‘𝐺)‘𝐿) ∈ (Base‘𝑅)) |
| 29 | 25, 5, 17 | grpsubid 19042 | . . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ ((coe1‘𝐺)‘𝐿) ∈ (Base‘𝑅)) → (((coe1‘𝐺)‘𝐿)(-g‘𝑅)((coe1‘𝐺)‘𝐿)) = (0g‘𝑅)) |
| 30 | 23, 28, 29 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (((coe1‘𝐺)‘𝐿)(-g‘𝑅)((coe1‘𝐺)‘𝐿)) = (0g‘𝑅)) |
| 31 | 19, 21, 30 | 3eqtrd 2781 | . . . 4 ⊢ (𝜑 → ((coe1‘(𝐹 − 𝐺))‘𝐿) = (0g‘𝑅)) |
| 32 | 1, 2, 3, 4, 5, 6, 7, 15, 16, 31 | deg1ldgn 26132 | . . 3 ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) ≠ 𝐿) |
| 33 | 32 | neneqd 2945 | . 2 ⊢ (𝜑 → ¬ (𝐷‘(𝐹 − 𝐺)) = 𝐿) |
| 34 | 1, 2, 4 | deg1xrcl 26121 | . . . . 5 ⊢ ((𝐹 − 𝐺) ∈ 𝐵 → (𝐷‘(𝐹 − 𝐺)) ∈ ℝ*) |
| 35 | 15, 34 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) ∈ ℝ*) |
| 36 | 1, 2, 4 | deg1xrcl 26121 | . . . . . 6 ⊢ (𝐺 ∈ 𝐵 → (𝐷‘𝐺) ∈ ℝ*) |
| 37 | 12, 36 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℝ*) |
| 38 | 1, 2, 4 | deg1xrcl 26121 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) ∈ ℝ*) |
| 39 | 11, 38 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℝ*) |
| 40 | 37, 39 | ifcld 4572 | . . . 4 ⊢ (𝜑 → if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹)) ∈ ℝ*) |
| 41 | 16 | nn0red 12588 | . . . . 5 ⊢ (𝜑 → 𝐿 ∈ ℝ) |
| 42 | 41 | rexrd 11311 | . . . 4 ⊢ (𝜑 → 𝐿 ∈ ℝ*) |
| 43 | 2, 1, 7, 4, 13, 11, 12 | deg1suble 26146 | . . . 4 ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) ≤ if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹))) |
| 44 | deg1sublt.fd | . . . . 5 ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝐿) | |
| 45 | deg1sublt.gd | . . . . 5 ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝐿) | |
| 46 | xrmaxle 13225 | . . . . . 6 ⊢ (((𝐷‘𝐹) ∈ ℝ* ∧ (𝐷‘𝐺) ∈ ℝ* ∧ 𝐿 ∈ ℝ*) → (if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹)) ≤ 𝐿 ↔ ((𝐷‘𝐹) ≤ 𝐿 ∧ (𝐷‘𝐺) ≤ 𝐿))) | |
| 47 | 39, 37, 42, 46 | syl3anc 1373 | . . . . 5 ⊢ (𝜑 → (if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹)) ≤ 𝐿 ↔ ((𝐷‘𝐹) ≤ 𝐿 ∧ (𝐷‘𝐺) ≤ 𝐿))) |
| 48 | 44, 45, 47 | mpbir2and 713 | . . . 4 ⊢ (𝜑 → if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹)) ≤ 𝐿) |
| 49 | 35, 40, 42, 43, 48 | xrletrd 13204 | . . 3 ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) ≤ 𝐿) |
| 50 | xrleloe 13186 | . . . 4 ⊢ (((𝐷‘(𝐹 − 𝐺)) ∈ ℝ* ∧ 𝐿 ∈ ℝ*) → ((𝐷‘(𝐹 − 𝐺)) ≤ 𝐿 ↔ ((𝐷‘(𝐹 − 𝐺)) < 𝐿 ∨ (𝐷‘(𝐹 − 𝐺)) = 𝐿))) | |
| 51 | 35, 42, 50 | syl2anc 584 | . . 3 ⊢ (𝜑 → ((𝐷‘(𝐹 − 𝐺)) ≤ 𝐿 ↔ ((𝐷‘(𝐹 − 𝐺)) < 𝐿 ∨ (𝐷‘(𝐹 − 𝐺)) = 𝐿))) |
| 52 | 49, 51 | mpbid 232 | . 2 ⊢ (𝜑 → ((𝐷‘(𝐹 − 𝐺)) < 𝐿 ∨ (𝐷‘(𝐹 − 𝐺)) = 𝐿)) |
| 53 | orel2 891 | . 2 ⊢ (¬ (𝐷‘(𝐹 − 𝐺)) = 𝐿 → (((𝐷‘(𝐹 − 𝐺)) < 𝐿 ∨ (𝐷‘(𝐹 − 𝐺)) = 𝐿) → (𝐷‘(𝐹 − 𝐺)) < 𝐿)) | |
| 54 | 33, 52, 53 | sylc 65 | 1 ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) < 𝐿) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ifcif 4525 class class class wbr 5143 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 ℕ0cn0 12526 Basecbs 17247 0gc0g 17484 Grpcgrp 18951 -gcsg 18953 Ringcrg 20230 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-tpos 8251 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-sbg 18956 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-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-subrng 20546 df-subrg 20570 df-rlreg 20694 df-lmod 20860 df-lss 20930 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: ply1divex 26176 deg1submon1p 26192 hbtlem5 43140 |
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