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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 2724 | . . . 4 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
4 | deg1sublt.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
5 | eqid 2724 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
6 | eqid 2724 | . . . 4 ⊢ (coe1‘(𝐹 − 𝐺)) = (coe1‘(𝐹 − 𝐺)) | |
7 | deg1sublt.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
8 | 2 | ply1ring 22110 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
9 | ringgrp 20139 | . . . . . 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 18944 | . . . . 5 ⊢ ((𝑃 ∈ Grp ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 − 𝐺) ∈ 𝐵) |
15 | 10, 11, 12, 14 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → (𝐹 − 𝐺) ∈ 𝐵) |
16 | deg1sublt.l | . . . 4 ⊢ (𝜑 → 𝐿 ∈ ℕ0) | |
17 | eqid 2724 | . . . . . . 7 ⊢ (-g‘𝑅) = (-g‘𝑅) | |
18 | 2, 4, 13, 17 | coe1subfv 22129 | . . . . . 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 1370 | . . . . 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 7417 | . . . . 5 ⊢ (𝜑 → (((coe1‘𝐹)‘𝐿)(-g‘𝑅)((coe1‘𝐺)‘𝐿)) = (((coe1‘𝐺)‘𝐿)(-g‘𝑅)((coe1‘𝐺)‘𝐿))) |
22 | ringgrp 20139 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
23 | 7, 22 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp) |
24 | eqid 2724 | . . . . . . . . 9 ⊢ (coe1‘𝐺) = (coe1‘𝐺) | |
25 | eqid 2724 | . . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
26 | 24, 4, 2, 25 | coe1f 22074 | . . . . . . . 8 ⊢ (𝐺 ∈ 𝐵 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
27 | 12, 26 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
28 | 27, 16 | ffvelcdmd 7078 | . . . . . 6 ⊢ (𝜑 → ((coe1‘𝐺)‘𝐿) ∈ (Base‘𝑅)) |
29 | 25, 5, 17 | grpsubid 18948 | . . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ ((coe1‘𝐺)‘𝐿) ∈ (Base‘𝑅)) → (((coe1‘𝐺)‘𝐿)(-g‘𝑅)((coe1‘𝐺)‘𝐿)) = (0g‘𝑅)) |
30 | 23, 28, 29 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → (((coe1‘𝐺)‘𝐿)(-g‘𝑅)((coe1‘𝐺)‘𝐿)) = (0g‘𝑅)) |
31 | 19, 21, 30 | 3eqtrd 2768 | . . . 4 ⊢ (𝜑 → ((coe1‘(𝐹 − 𝐺))‘𝐿) = (0g‘𝑅)) |
32 | 1, 2, 3, 4, 5, 6, 7, 15, 16, 31 | deg1ldgn 25973 | . . 3 ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) ≠ 𝐿) |
33 | 32 | neneqd 2937 | . 2 ⊢ (𝜑 → ¬ (𝐷‘(𝐹 − 𝐺)) = 𝐿) |
34 | 1, 2, 4 | deg1xrcl 25962 | . . . . 5 ⊢ ((𝐹 − 𝐺) ∈ 𝐵 → (𝐷‘(𝐹 − 𝐺)) ∈ ℝ*) |
35 | 15, 34 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) ∈ ℝ*) |
36 | 1, 2, 4 | deg1xrcl 25962 | . . . . . 6 ⊢ (𝐺 ∈ 𝐵 → (𝐷‘𝐺) ∈ ℝ*) |
37 | 12, 36 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℝ*) |
38 | 1, 2, 4 | deg1xrcl 25962 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) ∈ ℝ*) |
39 | 11, 38 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℝ*) |
40 | 37, 39 | ifcld 4567 | . . . 4 ⊢ (𝜑 → if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹)) ∈ ℝ*) |
41 | 16 | nn0red 12532 | . . . . 5 ⊢ (𝜑 → 𝐿 ∈ ℝ) |
42 | 41 | rexrd 11263 | . . . 4 ⊢ (𝜑 → 𝐿 ∈ ℝ*) |
43 | 2, 1, 7, 4, 13, 11, 12 | deg1suble 25987 | . . . 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 13163 | . . . . . 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 1368 | . . . . 5 ⊢ (𝜑 → (if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹)) ≤ 𝐿 ↔ ((𝐷‘𝐹) ≤ 𝐿 ∧ (𝐷‘𝐺) ≤ 𝐿))) |
48 | 44, 45, 47 | mpbir2and 710 | . . . 4 ⊢ (𝜑 → if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹)) ≤ 𝐿) |
49 | 35, 40, 42, 43, 48 | xrletrd 13142 | . . 3 ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) ≤ 𝐿) |
50 | xrleloe 13124 | . . . 4 ⊢ (((𝐷‘(𝐹 − 𝐺)) ∈ ℝ* ∧ 𝐿 ∈ ℝ*) → ((𝐷‘(𝐹 − 𝐺)) ≤ 𝐿 ↔ ((𝐷‘(𝐹 − 𝐺)) < 𝐿 ∨ (𝐷‘(𝐹 − 𝐺)) = 𝐿))) | |
51 | 35, 42, 50 | syl2anc 583 | . . 3 ⊢ (𝜑 → ((𝐷‘(𝐹 − 𝐺)) ≤ 𝐿 ↔ ((𝐷‘(𝐹 − 𝐺)) < 𝐿 ∨ (𝐷‘(𝐹 − 𝐺)) = 𝐿))) |
52 | 49, 51 | mpbid 231 | . 2 ⊢ (𝜑 → ((𝐷‘(𝐹 − 𝐺)) < 𝐿 ∨ (𝐷‘(𝐹 − 𝐺)) = 𝐿)) |
53 | orel2 887 | . 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 205 ∧ wa 395 ∨ wo 844 = wceq 1533 ∈ wcel 2098 ifcif 4521 class class class wbr 5139 ⟶wf 6530 ‘cfv 6534 (class class class)co 7402 ℝ*cxr 11246 < clt 11247 ≤ cle 11248 ℕ0cn0 12471 Basecbs 17149 0gc0g 17390 Grpcgrp 18859 -gcsg 18861 Ringcrg 20134 Poly1cpl1 22040 coe1cco1 22041 deg1 cdg1 25931 |
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 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187 |
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 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-ofr 7665 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-sup 9434 df-oi 9502 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13486 df-fzo 13629 df-seq 13968 df-hash 14292 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-0g 17392 df-gsum 17393 df-prds 17398 df-pws 17400 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-mhm 18709 df-submnd 18710 df-grp 18862 df-minusg 18863 df-sbg 18864 df-mulg 18992 df-subg 19046 df-ghm 19135 df-cntz 19229 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-cring 20137 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-subrng 20442 df-subrg 20467 df-lmod 20704 df-lss 20775 df-rlreg 21189 df-cnfld 21235 df-psr 21792 df-mpl 21794 df-opsr 21796 df-psr1 22043 df-ply1 22045 df-coe1 22046 df-mdeg 25932 df-deg1 25933 |
This theorem is referenced by: ply1divex 26016 deg1submon1p 26032 hbtlem5 42422 |
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