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Mirrors > Home > MPE Home > Th. List > deg1suble | Structured version Visualization version GIF version |
Description: The degree of a difference of polynomials is bounded by the maximum of degrees. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
Ref | Expression |
---|---|
deg1addle.y | ⊢ 𝑌 = (Poly1‘𝑅) |
deg1addle.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
deg1addle.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
deg1suble.b | ⊢ 𝐵 = (Base‘𝑌) |
deg1suble.m | ⊢ − = (-g‘𝑌) |
deg1suble.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
deg1suble.g | ⊢ (𝜑 → 𝐺 ∈ 𝐵) |
Ref | Expression |
---|---|
deg1suble | ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) ≤ if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | deg1addle.y | . . 3 ⊢ 𝑌 = (Poly1‘𝑅) | |
2 | deg1addle.d | . . 3 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
3 | deg1addle.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
4 | deg1suble.b | . . 3 ⊢ 𝐵 = (Base‘𝑌) | |
5 | eqid 2821 | . . 3 ⊢ (+g‘𝑌) = (+g‘𝑌) | |
6 | deg1suble.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
7 | 1 | ply1ring 20410 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑌 ∈ Ring) |
8 | ringgrp 19296 | . . . . 5 ⊢ (𝑌 ∈ Ring → 𝑌 ∈ Grp) | |
9 | 3, 7, 8 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ Grp) |
10 | deg1suble.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
11 | eqid 2821 | . . . . 5 ⊢ (invg‘𝑌) = (invg‘𝑌) | |
12 | 4, 11 | grpinvcl 18145 | . . . 4 ⊢ ((𝑌 ∈ Grp ∧ 𝐺 ∈ 𝐵) → ((invg‘𝑌)‘𝐺) ∈ 𝐵) |
13 | 9, 10, 12 | syl2anc 586 | . . 3 ⊢ (𝜑 → ((invg‘𝑌)‘𝐺) ∈ 𝐵) |
14 | 1, 2, 3, 4, 5, 6, 13 | deg1addle 24689 | . 2 ⊢ (𝜑 → (𝐷‘(𝐹(+g‘𝑌)((invg‘𝑌)‘𝐺))) ≤ if((𝐷‘𝐹) ≤ (𝐷‘((invg‘𝑌)‘𝐺)), (𝐷‘((invg‘𝑌)‘𝐺)), (𝐷‘𝐹))) |
15 | deg1suble.m | . . . . 5 ⊢ − = (-g‘𝑌) | |
16 | 4, 5, 11, 15 | grpsubval 18143 | . . . 4 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 − 𝐺) = (𝐹(+g‘𝑌)((invg‘𝑌)‘𝐺))) |
17 | 6, 10, 16 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝐹 − 𝐺) = (𝐹(+g‘𝑌)((invg‘𝑌)‘𝐺))) |
18 | 17 | fveq2d 6669 | . 2 ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) = (𝐷‘(𝐹(+g‘𝑌)((invg‘𝑌)‘𝐺)))) |
19 | 1, 2, 3, 4, 11, 10 | deg1invg 24694 | . . . . 5 ⊢ (𝜑 → (𝐷‘((invg‘𝑌)‘𝐺)) = (𝐷‘𝐺)) |
20 | 19 | eqcomd 2827 | . . . 4 ⊢ (𝜑 → (𝐷‘𝐺) = (𝐷‘((invg‘𝑌)‘𝐺))) |
21 | 20 | breq2d 5071 | . . 3 ⊢ (𝜑 → ((𝐷‘𝐹) ≤ (𝐷‘𝐺) ↔ (𝐷‘𝐹) ≤ (𝐷‘((invg‘𝑌)‘𝐺)))) |
22 | 21, 20 | ifbieq1d 4490 | . 2 ⊢ (𝜑 → if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹)) = if((𝐷‘𝐹) ≤ (𝐷‘((invg‘𝑌)‘𝐺)), (𝐷‘((invg‘𝑌)‘𝐺)), (𝐷‘𝐹))) |
23 | 14, 18, 22 | 3brtr4d 5091 | 1 ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) ≤ if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ifcif 4467 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 ≤ cle 10670 Basecbs 16477 +gcplusg 16559 Grpcgrp 18097 invgcminusg 18098 -gcsg 18099 Ringcrg 19291 Poly1cpl1 20339 deg1 cdg1 24642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-ofr 7404 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-0g 16709 df-gsum 16710 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-ghm 18350 df-cntz 18441 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-subrg 19527 df-lmod 19630 df-lss 19698 df-rlreg 20050 df-psr 20130 df-mpl 20132 df-opsr 20134 df-psr1 20342 df-ply1 20344 df-cnfld 20540 df-mdeg 24643 df-deg1 24644 |
This theorem is referenced by: deg1sublt 24698 ply1divmo 24723 |
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