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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 2610 | . . . 4 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
4 | deg1sublt.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
5 | eqid 2610 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
6 | eqid 2610 | . . . 4 ⊢ (coe1‘(𝐹 − 𝐺)) = (coe1‘(𝐹 − 𝐺)) | |
7 | deg1sublt.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
8 | 2 | ply1ring 19439 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
9 | ringgrp 18375 | . . . . . 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 17318 | . . . . 5 ⊢ ((𝑃 ∈ Grp ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 − 𝐺) ∈ 𝐵) |
15 | 10, 11, 12, 14 | syl3anc 1318 | . . . 4 ⊢ (𝜑 → (𝐹 − 𝐺) ∈ 𝐵) |
16 | deg1sublt.l | . . . 4 ⊢ (𝜑 → 𝐿 ∈ ℕ0) | |
17 | eqid 2610 | . . . . . . 7 ⊢ (-g‘𝑅) = (-g‘𝑅) | |
18 | 2, 4, 13, 17 | coe1subfv 19457 | . . . . . 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 1321 | . . . . 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 6564 | . . . . 5 ⊢ (𝜑 → (((coe1‘𝐹)‘𝐿)(-g‘𝑅)((coe1‘𝐺)‘𝐿)) = (((coe1‘𝐺)‘𝐿)(-g‘𝑅)((coe1‘𝐺)‘𝐿))) |
22 | ringgrp 18375 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
23 | 7, 22 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp) |
24 | eqid 2610 | . . . . . . . . 9 ⊢ (coe1‘𝐺) = (coe1‘𝐺) | |
25 | eqid 2610 | . . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
26 | 24, 4, 2, 25 | coe1f 19402 | . . . . . . . 8 ⊢ (𝐺 ∈ 𝐵 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
27 | 12, 26 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
28 | 27, 16 | ffvelrnd 6268 | . . . . . 6 ⊢ (𝜑 → ((coe1‘𝐺)‘𝐿) ∈ (Base‘𝑅)) |
29 | 25, 5, 17 | grpsubid 17322 | . . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ ((coe1‘𝐺)‘𝐿) ∈ (Base‘𝑅)) → (((coe1‘𝐺)‘𝐿)(-g‘𝑅)((coe1‘𝐺)‘𝐿)) = (0g‘𝑅)) |
30 | 23, 28, 29 | syl2anc 691 | . . . . 5 ⊢ (𝜑 → (((coe1‘𝐺)‘𝐿)(-g‘𝑅)((coe1‘𝐺)‘𝐿)) = (0g‘𝑅)) |
31 | 19, 21, 30 | 3eqtrd 2648 | . . . 4 ⊢ (𝜑 → ((coe1‘(𝐹 − 𝐺))‘𝐿) = (0g‘𝑅)) |
32 | 1, 2, 3, 4, 5, 6, 7, 15, 16, 31 | deg1ldgn 23657 | . . 3 ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) ≠ 𝐿) |
33 | 32 | neneqd 2787 | . 2 ⊢ (𝜑 → ¬ (𝐷‘(𝐹 − 𝐺)) = 𝐿) |
34 | 1, 2, 4 | deg1xrcl 23646 | . . . . 5 ⊢ ((𝐹 − 𝐺) ∈ 𝐵 → (𝐷‘(𝐹 − 𝐺)) ∈ ℝ*) |
35 | 15, 34 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) ∈ ℝ*) |
36 | 1, 2, 4 | deg1xrcl 23646 | . . . . . 6 ⊢ (𝐺 ∈ 𝐵 → (𝐷‘𝐺) ∈ ℝ*) |
37 | 12, 36 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℝ*) |
38 | 1, 2, 4 | deg1xrcl 23646 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) ∈ ℝ*) |
39 | 11, 38 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℝ*) |
40 | 37, 39 | ifcld 4081 | . . . 4 ⊢ (𝜑 → if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹)) ∈ ℝ*) |
41 | 16 | nn0red 11229 | . . . . 5 ⊢ (𝜑 → 𝐿 ∈ ℝ) |
42 | 41 | rexrd 9968 | . . . 4 ⊢ (𝜑 → 𝐿 ∈ ℝ*) |
43 | 2, 1, 7, 4, 13, 11, 12 | deg1suble 23671 | . . . 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 11888 | . . . . . 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 1318 | . . . . 5 ⊢ (𝜑 → (if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹)) ≤ 𝐿 ↔ ((𝐷‘𝐹) ≤ 𝐿 ∧ (𝐷‘𝐺) ≤ 𝐿))) |
48 | 44, 45, 47 | mpbir2and 959 | . . . 4 ⊢ (𝜑 → if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹)) ≤ 𝐿) |
49 | 35, 40, 42, 43, 48 | xrletrd 11869 | . . 3 ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) ≤ 𝐿) |
50 | xrleloe 11853 | . . . 4 ⊢ (((𝐷‘(𝐹 − 𝐺)) ∈ ℝ* ∧ 𝐿 ∈ ℝ*) → ((𝐷‘(𝐹 − 𝐺)) ≤ 𝐿 ↔ ((𝐷‘(𝐹 − 𝐺)) < 𝐿 ∨ (𝐷‘(𝐹 − 𝐺)) = 𝐿))) | |
51 | 35, 42, 50 | syl2anc 691 | . . 3 ⊢ (𝜑 → ((𝐷‘(𝐹 − 𝐺)) ≤ 𝐿 ↔ ((𝐷‘(𝐹 − 𝐺)) < 𝐿 ∨ (𝐷‘(𝐹 − 𝐺)) = 𝐿))) |
52 | 49, 51 | mpbid 221 | . 2 ⊢ (𝜑 → ((𝐷‘(𝐹 − 𝐺)) < 𝐿 ∨ (𝐷‘(𝐹 − 𝐺)) = 𝐿)) |
53 | orel2 397 | . 2 ⊢ (¬ (𝐷‘(𝐹 − 𝐺)) = 𝐿 → (((𝐷‘(𝐹 − 𝐺)) < 𝐿 ∨ (𝐷‘(𝐹 − 𝐺)) = 𝐿) → (𝐷‘(𝐹 − 𝐺)) < 𝐿)) | |
54 | 33, 52, 53 | sylc 63 | 1 ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) < 𝐿) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ifcif 4036 class class class wbr 4583 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 ℕ0cn0 11169 Basecbs 15695 0gc0g 15923 Grpcgrp 17245 -gcsg 17247 Ringcrg 18370 Poly1cpl1 19368 coe1cco1 19369 deg1 cdg1 23618 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-ofr 6796 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-0g 15925 df-gsum 15926 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-ghm 17481 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-subrg 18601 df-lmod 18688 df-lss 18754 df-rlreg 19104 df-psr 19177 df-mpl 19179 df-opsr 19181 df-psr1 19371 df-ply1 19373 df-coe1 19374 df-cnfld 19568 df-mdeg 23619 df-deg1 23620 |
This theorem is referenced by: ply1divex 23700 deg1submon1p 23716 hbtlem5 36717 |
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