![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2728 | . . . 4 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
4 | deg1sublt.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
5 | eqid 2728 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
6 | eqid 2728 | . . . 4 ⊢ (coe1‘(𝐹 − 𝐺)) = (coe1‘(𝐹 − 𝐺)) | |
7 | deg1sublt.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
8 | 2 | ply1ring 22166 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
9 | ringgrp 20178 | . . . . . 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 18976 | . . . . 5 ⊢ ((𝑃 ∈ Grp ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 − 𝐺) ∈ 𝐵) |
15 | 10, 11, 12, 14 | syl3anc 1369 | . . . 4 ⊢ (𝜑 → (𝐹 − 𝐺) ∈ 𝐵) |
16 | deg1sublt.l | . . . 4 ⊢ (𝜑 → 𝐿 ∈ ℕ0) | |
17 | eqid 2728 | . . . . . . 7 ⊢ (-g‘𝑅) = (-g‘𝑅) | |
18 | 2, 4, 13, 17 | coe1subfv 22185 | . . . . . 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 1371 | . . . . 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 7435 | . . . . 5 ⊢ (𝜑 → (((coe1‘𝐹)‘𝐿)(-g‘𝑅)((coe1‘𝐺)‘𝐿)) = (((coe1‘𝐺)‘𝐿)(-g‘𝑅)((coe1‘𝐺)‘𝐿))) |
22 | ringgrp 20178 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
23 | 7, 22 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp) |
24 | eqid 2728 | . . . . . . . . 9 ⊢ (coe1‘𝐺) = (coe1‘𝐺) | |
25 | eqid 2728 | . . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
26 | 24, 4, 2, 25 | coe1f 22130 | . . . . . . . 8 ⊢ (𝐺 ∈ 𝐵 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
27 | 12, 26 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
28 | 27, 16 | ffvelcdmd 7095 | . . . . . 6 ⊢ (𝜑 → ((coe1‘𝐺)‘𝐿) ∈ (Base‘𝑅)) |
29 | 25, 5, 17 | grpsubid 18980 | . . . . . 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 2772 | . . . 4 ⊢ (𝜑 → ((coe1‘(𝐹 − 𝐺))‘𝐿) = (0g‘𝑅)) |
32 | 1, 2, 3, 4, 5, 6, 7, 15, 16, 31 | deg1ldgn 26042 | . . 3 ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) ≠ 𝐿) |
33 | 32 | neneqd 2942 | . 2 ⊢ (𝜑 → ¬ (𝐷‘(𝐹 − 𝐺)) = 𝐿) |
34 | 1, 2, 4 | deg1xrcl 26031 | . . . . 5 ⊢ ((𝐹 − 𝐺) ∈ 𝐵 → (𝐷‘(𝐹 − 𝐺)) ∈ ℝ*) |
35 | 15, 34 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) ∈ ℝ*) |
36 | 1, 2, 4 | deg1xrcl 26031 | . . . . . 6 ⊢ (𝐺 ∈ 𝐵 → (𝐷‘𝐺) ∈ ℝ*) |
37 | 12, 36 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℝ*) |
38 | 1, 2, 4 | deg1xrcl 26031 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) ∈ ℝ*) |
39 | 11, 38 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℝ*) |
40 | 37, 39 | ifcld 4575 | . . . 4 ⊢ (𝜑 → if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹)) ∈ ℝ*) |
41 | 16 | nn0red 12564 | . . . . 5 ⊢ (𝜑 → 𝐿 ∈ ℝ) |
42 | 41 | rexrd 11295 | . . . 4 ⊢ (𝜑 → 𝐿 ∈ ℝ*) |
43 | 2, 1, 7, 4, 13, 11, 12 | deg1suble 26056 | . . . 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 13195 | . . . . . 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 1369 | . . . . 5 ⊢ (𝜑 → (if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹)) ≤ 𝐿 ↔ ((𝐷‘𝐹) ≤ 𝐿 ∧ (𝐷‘𝐺) ≤ 𝐿))) |
48 | 44, 45, 47 | mpbir2and 712 | . . . 4 ⊢ (𝜑 → if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹)) ≤ 𝐿) |
49 | 35, 40, 42, 43, 48 | xrletrd 13174 | . . 3 ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) ≤ 𝐿) |
50 | xrleloe 13156 | . . . 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 889 | . 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 846 = wceq 1534 ∈ wcel 2099 ifcif 4529 class class class wbr 5148 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 ℝ*cxr 11278 < clt 11279 ≤ cle 11280 ℕ0cn0 12503 Basecbs 17180 0gc0g 17421 Grpcgrp 18890 -gcsg 18892 Ringcrg 20173 Poly1cpl1 22096 coe1cco1 22097 deg1 cdg1 26000 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 ax-addf 11218 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-ofr 7686 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-sup 9466 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 df-fzo 13661 df-seq 14000 df-hash 14323 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-hom 17257 df-cco 17258 df-0g 17423 df-gsum 17424 df-prds 17429 df-pws 17431 df-mre 17566 df-mrc 17567 df-acs 17569 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mhm 18740 df-submnd 18741 df-grp 18893 df-minusg 18894 df-sbg 18895 df-mulg 19024 df-subg 19078 df-ghm 19168 df-cntz 19268 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-cring 20176 df-oppr 20273 df-dvdsr 20296 df-unit 20297 df-invr 20327 df-subrng 20483 df-subrg 20508 df-lmod 20745 df-lss 20816 df-rlreg 21230 df-cnfld 21280 df-psr 21842 df-mpl 21844 df-opsr 21846 df-psr1 22099 df-ply1 22101 df-coe1 22102 df-mdeg 26001 df-deg1 26002 |
This theorem is referenced by: ply1divex 26085 deg1submon1p 26101 hbtlem5 42552 |
Copyright terms: Public domain | W3C validator |