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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 2798 | . 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 24745 | . . . . 5 ⊢ (𝐹 ∈ 𝑂 → 𝐹 ∈ (Base‘𝑃)) |
10 | 7, 9 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (Base‘𝑃)) |
11 | eqid 2798 | . . . . . 6 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
12 | 2, 11, 8 | mon1pn0 24747 | . . . . 5 ⊢ (𝐹 ∈ 𝑂 → 𝐹 ≠ (0g‘𝑃)) |
13 | 7, 12 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹 ≠ (0g‘𝑃)) |
14 | 1, 2, 11, 3 | deg1nn0cl 24689 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ (Base‘𝑃) ∧ 𝐹 ≠ (0g‘𝑃)) → (𝐷‘𝐹) ∈ ℕ0) |
15 | 6, 10, 13, 14 | syl3anc 1368 | . . 3 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℕ0) |
16 | 5, 15 | eqeltrrd 2891 | . 2 ⊢ (𝜑 → 𝑋 ∈ ℕ0) |
17 | 16 | nn0red 11944 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ) |
18 | 17 | leidd 11195 | . . 3 ⊢ (𝜑 → 𝑋 ≤ 𝑋) |
19 | 5, 18 | eqbrtrd 5052 | . 2 ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝑋) |
20 | deg1submon1p.g1 | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑂) | |
21 | 2, 3, 8 | mon1pcl 24745 | . . 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 5052 | . 2 ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝑋) |
25 | eqid 2798 | . 2 ⊢ (coe1‘𝐹) = (coe1‘𝐹) | |
26 | eqid 2798 | . 2 ⊢ (coe1‘𝐺) = (coe1‘𝐺) | |
27 | 5 | fveq2d 6649 | . . . 4 ⊢ (𝜑 → ((coe1‘𝐹)‘(𝐷‘𝐹)) = ((coe1‘𝐹)‘𝑋)) |
28 | eqid 2798 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
29 | 1, 28, 8 | mon1pldg 24750 | . . . . 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 2835 | . . 3 ⊢ (𝜑 → ((coe1‘𝐹)‘𝑋) = (1r‘𝑅)) |
32 | 1, 28, 8 | mon1pldg 24750 | . . . 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 6649 | . . 3 ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) = ((coe1‘𝐺)‘𝑋)) |
35 | 31, 33, 34 | 3eqtr2d 2839 | . 2 ⊢ (𝜑 → ((coe1‘𝐹)‘𝑋) = ((coe1‘𝐺)‘𝑋)) |
36 | 1, 2, 3, 4, 16, 6, 10, 19, 22, 24, 25, 26, 35 | deg1sublt 24711 | 1 ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) < 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 < clt 10664 ≤ cle 10665 ℕ0cn0 11885 Basecbs 16475 0gc0g 16705 -gcsg 18097 1rcur 19244 Ringcrg 19290 Poly1cpl1 20806 coe1cco1 20807 deg1 cdg1 24655 Monic1pcmn1 24726 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-ofr 7390 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-sup 8890 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-0g 16707 df-gsum 16708 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-ghm 18348 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-subrg 19526 df-lmod 19629 df-lss 19697 df-rlreg 20049 df-cnfld 20092 df-psr 20594 df-mpl 20596 df-opsr 20598 df-psr1 20809 df-ply1 20811 df-coe1 20812 df-mdeg 24656 df-deg1 24657 df-mon1 24731 |
This theorem is referenced by: ig1peu 24772 |
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