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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 2733 | . . . 4 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
4 | deg1sublt.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
5 | eqid 2733 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
6 | eqid 2733 | . . . 4 ⊢ (coe1‘(𝐹 − 𝐺)) = (coe1‘(𝐹 − 𝐺)) | |
7 | deg1sublt.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
8 | 2 | ply1ring 21642 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
9 | ringgrp 19977 | . . . . . 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 18835 | . . . . 5 ⊢ ((𝑃 ∈ Grp ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 − 𝐺) ∈ 𝐵) |
15 | 10, 11, 12, 14 | syl3anc 1372 | . . . 4 ⊢ (𝜑 → (𝐹 − 𝐺) ∈ 𝐵) |
16 | deg1sublt.l | . . . 4 ⊢ (𝜑 → 𝐿 ∈ ℕ0) | |
17 | eqid 2733 | . . . . . . 7 ⊢ (-g‘𝑅) = (-g‘𝑅) | |
18 | 2, 4, 13, 17 | coe1subfv 21660 | . . . . . 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 1374 | . . . . 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 7376 | . . . . 5 ⊢ (𝜑 → (((coe1‘𝐹)‘𝐿)(-g‘𝑅)((coe1‘𝐺)‘𝐿)) = (((coe1‘𝐺)‘𝐿)(-g‘𝑅)((coe1‘𝐺)‘𝐿))) |
22 | ringgrp 19977 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
23 | 7, 22 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp) |
24 | eqid 2733 | . . . . . . . . 9 ⊢ (coe1‘𝐺) = (coe1‘𝐺) | |
25 | eqid 2733 | . . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
26 | 24, 4, 2, 25 | coe1f 21605 | . . . . . . . 8 ⊢ (𝐺 ∈ 𝐵 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
27 | 12, 26 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
28 | 27, 16 | ffvelcdmd 7040 | . . . . . 6 ⊢ (𝜑 → ((coe1‘𝐺)‘𝐿) ∈ (Base‘𝑅)) |
29 | 25, 5, 17 | grpsubid 18839 | . . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ ((coe1‘𝐺)‘𝐿) ∈ (Base‘𝑅)) → (((coe1‘𝐺)‘𝐿)(-g‘𝑅)((coe1‘𝐺)‘𝐿)) = (0g‘𝑅)) |
30 | 23, 28, 29 | syl2anc 585 | . . . . 5 ⊢ (𝜑 → (((coe1‘𝐺)‘𝐿)(-g‘𝑅)((coe1‘𝐺)‘𝐿)) = (0g‘𝑅)) |
31 | 19, 21, 30 | 3eqtrd 2777 | . . . 4 ⊢ (𝜑 → ((coe1‘(𝐹 − 𝐺))‘𝐿) = (0g‘𝑅)) |
32 | 1, 2, 3, 4, 5, 6, 7, 15, 16, 31 | deg1ldgn 25481 | . . 3 ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) ≠ 𝐿) |
33 | 32 | neneqd 2945 | . 2 ⊢ (𝜑 → ¬ (𝐷‘(𝐹 − 𝐺)) = 𝐿) |
34 | 1, 2, 4 | deg1xrcl 25470 | . . . . 5 ⊢ ((𝐹 − 𝐺) ∈ 𝐵 → (𝐷‘(𝐹 − 𝐺)) ∈ ℝ*) |
35 | 15, 34 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) ∈ ℝ*) |
36 | 1, 2, 4 | deg1xrcl 25470 | . . . . . 6 ⊢ (𝐺 ∈ 𝐵 → (𝐷‘𝐺) ∈ ℝ*) |
37 | 12, 36 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℝ*) |
38 | 1, 2, 4 | deg1xrcl 25470 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) ∈ ℝ*) |
39 | 11, 38 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℝ*) |
40 | 37, 39 | ifcld 4536 | . . . 4 ⊢ (𝜑 → if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹)) ∈ ℝ*) |
41 | 16 | nn0red 12482 | . . . . 5 ⊢ (𝜑 → 𝐿 ∈ ℝ) |
42 | 41 | rexrd 11213 | . . . 4 ⊢ (𝜑 → 𝐿 ∈ ℝ*) |
43 | 2, 1, 7, 4, 13, 11, 12 | deg1suble 25495 | . . . 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 13111 | . . . . . 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 1372 | . . . . 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 13090 | . . 3 ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) ≤ 𝐿) |
50 | xrleloe 13072 | . . . 4 ⊢ (((𝐷‘(𝐹 − 𝐺)) ∈ ℝ* ∧ 𝐿 ∈ ℝ*) → ((𝐷‘(𝐹 − 𝐺)) ≤ 𝐿 ↔ ((𝐷‘(𝐹 − 𝐺)) < 𝐿 ∨ (𝐷‘(𝐹 − 𝐺)) = 𝐿))) | |
51 | 35, 42, 50 | syl2anc 585 | . . 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 890 | . 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 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ifcif 4490 class class class wbr 5109 ⟶wf 6496 ‘cfv 6500 (class class class)co 7361 ℝ*cxr 11196 < clt 11197 ≤ cle 11198 ℕ0cn0 12421 Basecbs 17091 0gc0g 17329 Grpcgrp 18756 -gcsg 18758 Ringcrg 19972 Poly1cpl1 21571 coe1cco1 21572 deg1 cdg1 25439 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 ax-addf 11138 ax-mulf 11139 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-ofr 7622 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-tpos 8161 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-pm 8774 df-ixp 8842 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-sup 9386 df-oi 9454 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-fz 13434 df-fzo 13577 df-seq 13916 df-hash 14240 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-starv 17156 df-sca 17157 df-vsca 17158 df-ip 17159 df-tset 17160 df-ple 17161 df-ds 17163 df-unif 17164 df-hom 17165 df-cco 17166 df-0g 17331 df-gsum 17332 df-prds 17337 df-pws 17339 df-mre 17474 df-mrc 17475 df-acs 17477 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-mhm 18609 df-submnd 18610 df-grp 18759 df-minusg 18760 df-sbg 18761 df-mulg 18881 df-subg 18933 df-ghm 19014 df-cntz 19105 df-cmn 19572 df-abl 19573 df-mgp 19905 df-ur 19922 df-ring 19974 df-cring 19975 df-oppr 20057 df-dvdsr 20078 df-unit 20079 df-invr 20109 df-subrg 20262 df-lmod 20367 df-lss 20437 df-rlreg 20776 df-cnfld 20820 df-psr 21334 df-mpl 21336 df-opsr 21338 df-psr1 21574 df-ply1 21576 df-coe1 21577 df-mdeg 25440 df-deg1 25441 |
This theorem is referenced by: ply1divex 25524 deg1submon1p 25540 hbtlem5 41502 |
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