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Mirrors > Home > MPE Home > Th. List > deg1submon1p | Structured version Visualization version GIF version |
Description: The difference of two monic polynomials of the same degree is a polynomial of lesser degree. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
Ref | Expression |
---|---|
deg1submon1p.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
deg1submon1p.o | ⊢ 𝑂 = (Monic1p‘𝑅) |
deg1submon1p.p | ⊢ 𝑃 = (Poly1‘𝑅) |
deg1submon1p.m | ⊢ − = (-g‘𝑃) |
deg1submon1p.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
deg1submon1p.f1 | ⊢ (𝜑 → 𝐹 ∈ 𝑂) |
deg1submon1p.f2 | ⊢ (𝜑 → (𝐷‘𝐹) = 𝑋) |
deg1submon1p.g1 | ⊢ (𝜑 → 𝐺 ∈ 𝑂) |
deg1submon1p.g2 | ⊢ (𝜑 → (𝐷‘𝐺) = 𝑋) |
Ref | Expression |
---|---|
deg1submon1p | ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) < 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | deg1submon1p.d | . 2 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
2 | deg1submon1p.p | . 2 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | eqid 2823 | . 2 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
4 | deg1submon1p.m | . 2 ⊢ − = (-g‘𝑃) | |
5 | deg1submon1p.f2 | . . 3 ⊢ (𝜑 → (𝐷‘𝐹) = 𝑋) | |
6 | deg1submon1p.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
7 | deg1submon1p.f1 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝑂) | |
8 | deg1submon1p.o | . . . . . 6 ⊢ 𝑂 = (Monic1p‘𝑅) | |
9 | 2, 3, 8 | mon1pcl 24740 | . . . . 5 ⊢ (𝐹 ∈ 𝑂 → 𝐹 ∈ (Base‘𝑃)) |
10 | 7, 9 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (Base‘𝑃)) |
11 | eqid 2823 | . . . . . 6 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
12 | 2, 11, 8 | mon1pn0 24742 | . . . . 5 ⊢ (𝐹 ∈ 𝑂 → 𝐹 ≠ (0g‘𝑃)) |
13 | 7, 12 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹 ≠ (0g‘𝑃)) |
14 | 1, 2, 11, 3 | deg1nn0cl 24684 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ (Base‘𝑃) ∧ 𝐹 ≠ (0g‘𝑃)) → (𝐷‘𝐹) ∈ ℕ0) |
15 | 6, 10, 13, 14 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℕ0) |
16 | 5, 15 | eqeltrrd 2916 | . 2 ⊢ (𝜑 → 𝑋 ∈ ℕ0) |
17 | 16 | nn0red 11959 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ) |
18 | 17 | leidd 11208 | . . 3 ⊢ (𝜑 → 𝑋 ≤ 𝑋) |
19 | 5, 18 | eqbrtrd 5090 | . 2 ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝑋) |
20 | deg1submon1p.g1 | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑂) | |
21 | 2, 3, 8 | mon1pcl 24740 | . . 3 ⊢ (𝐺 ∈ 𝑂 → 𝐺 ∈ (Base‘𝑃)) |
22 | 20, 21 | syl 17 | . 2 ⊢ (𝜑 → 𝐺 ∈ (Base‘𝑃)) |
23 | deg1submon1p.g2 | . . 3 ⊢ (𝜑 → (𝐷‘𝐺) = 𝑋) | |
24 | 23, 18 | eqbrtrd 5090 | . 2 ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝑋) |
25 | eqid 2823 | . 2 ⊢ (coe1‘𝐹) = (coe1‘𝐹) | |
26 | eqid 2823 | . 2 ⊢ (coe1‘𝐺) = (coe1‘𝐺) | |
27 | 5 | fveq2d 6676 | . . . 4 ⊢ (𝜑 → ((coe1‘𝐹)‘(𝐷‘𝐹)) = ((coe1‘𝐹)‘𝑋)) |
28 | eqid 2823 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
29 | 1, 28, 8 | mon1pldg 24745 | . . . . 5 ⊢ (𝐹 ∈ 𝑂 → ((coe1‘𝐹)‘(𝐷‘𝐹)) = (1r‘𝑅)) |
30 | 7, 29 | syl 17 | . . . 4 ⊢ (𝜑 → ((coe1‘𝐹)‘(𝐷‘𝐹)) = (1r‘𝑅)) |
31 | 27, 30 | eqtr3d 2860 | . . 3 ⊢ (𝜑 → ((coe1‘𝐹)‘𝑋) = (1r‘𝑅)) |
32 | 1, 28, 8 | mon1pldg 24745 | . . . 4 ⊢ (𝐺 ∈ 𝑂 → ((coe1‘𝐺)‘(𝐷‘𝐺)) = (1r‘𝑅)) |
33 | 20, 32 | syl 17 | . . 3 ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) = (1r‘𝑅)) |
34 | 23 | fveq2d 6676 | . . 3 ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) = ((coe1‘𝐺)‘𝑋)) |
35 | 31, 33, 34 | 3eqtr2d 2864 | . 2 ⊢ (𝜑 → ((coe1‘𝐹)‘𝑋) = ((coe1‘𝐺)‘𝑋)) |
36 | 1, 2, 3, 4, 16, 6, 10, 19, 22, 24, 25, 26, 35 | deg1sublt 24706 | 1 ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) < 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 < clt 10677 ≤ cle 10678 ℕ0cn0 11900 Basecbs 16485 0gc0g 16715 -gcsg 18107 1rcur 19253 Ringcrg 19299 Poly1cpl1 20347 coe1cco1 20348 deg1 cdg1 24650 Monic1pcmn1 24721 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-ofr 7412 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-0g 16717 df-gsum 16718 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-ghm 18358 df-cntz 18449 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-subrg 19535 df-lmod 19638 df-lss 19706 df-rlreg 20058 df-psr 20138 df-mpl 20140 df-opsr 20142 df-psr1 20350 df-ply1 20352 df-coe1 20353 df-cnfld 20548 df-mdeg 24651 df-deg1 24652 df-mon1 24726 |
This theorem is referenced by: ig1peu 24767 |
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