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Mirrors > Home > MPE Home > Th. List > deg1addle2 | Structured version Visualization version GIF version |
Description: If both factors have degree bounded by 𝐿, then the sum of the polynomials also has degree bounded by 𝐿. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
Ref | Expression |
---|---|
deg1addle.y | ⊢ 𝑌 = (Poly1‘𝑅) |
deg1addle.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
deg1addle.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
deg1addle.b | ⊢ 𝐵 = (Base‘𝑌) |
deg1addle.p | ⊢ + = (+g‘𝑌) |
deg1addle.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
deg1addle.g | ⊢ (𝜑 → 𝐺 ∈ 𝐵) |
deg1addle2.l1 | ⊢ (𝜑 → 𝐿 ∈ ℝ*) |
deg1addle2.l2 | ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝐿) |
deg1addle2.l3 | ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝐿) |
Ref | Expression |
---|---|
deg1addle2 | ⊢ (𝜑 → (𝐷‘(𝐹 + 𝐺)) ≤ 𝐿) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | deg1addle.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
2 | deg1addle.y | . . . . . 6 ⊢ 𝑌 = (Poly1‘𝑅) | |
3 | 2 | ply1ring 20419 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑌 ∈ Ring) |
4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ Ring) |
5 | deg1addle.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
6 | deg1addle.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
7 | deg1addle.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑌) | |
8 | deg1addle.p | . . . . 5 ⊢ + = (+g‘𝑌) | |
9 | 7, 8 | ringacl 19331 | . . . 4 ⊢ ((𝑌 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 + 𝐺) ∈ 𝐵) |
10 | 4, 5, 6, 9 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝐹 + 𝐺) ∈ 𝐵) |
11 | deg1addle.d | . . . 4 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
12 | 11, 2, 7 | deg1xrcl 24679 | . . 3 ⊢ ((𝐹 + 𝐺) ∈ 𝐵 → (𝐷‘(𝐹 + 𝐺)) ∈ ℝ*) |
13 | 10, 12 | syl 17 | . 2 ⊢ (𝜑 → (𝐷‘(𝐹 + 𝐺)) ∈ ℝ*) |
14 | 11, 2, 7 | deg1xrcl 24679 | . . . 4 ⊢ (𝐺 ∈ 𝐵 → (𝐷‘𝐺) ∈ ℝ*) |
15 | 6, 14 | syl 17 | . . 3 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℝ*) |
16 | 11, 2, 7 | deg1xrcl 24679 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) ∈ ℝ*) |
17 | 5, 16 | syl 17 | . . 3 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℝ*) |
18 | 15, 17 | ifcld 4515 | . 2 ⊢ (𝜑 → if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹)) ∈ ℝ*) |
19 | deg1addle2.l1 | . 2 ⊢ (𝜑 → 𝐿 ∈ ℝ*) | |
20 | 2, 11, 1, 7, 8, 5, 6 | deg1addle 24698 | . 2 ⊢ (𝜑 → (𝐷‘(𝐹 + 𝐺)) ≤ if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹))) |
21 | deg1addle2.l2 | . . 3 ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝐿) | |
22 | deg1addle2.l3 | . . 3 ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝐿) | |
23 | xrmaxle 12579 | . . . 4 ⊢ (((𝐷‘𝐹) ∈ ℝ* ∧ (𝐷‘𝐺) ∈ ℝ* ∧ 𝐿 ∈ ℝ*) → (if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹)) ≤ 𝐿 ↔ ((𝐷‘𝐹) ≤ 𝐿 ∧ (𝐷‘𝐺) ≤ 𝐿))) | |
24 | 17, 15, 19, 23 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹)) ≤ 𝐿 ↔ ((𝐷‘𝐹) ≤ 𝐿 ∧ (𝐷‘𝐺) ≤ 𝐿))) |
25 | 21, 22, 24 | mpbir2and 711 | . 2 ⊢ (𝜑 → if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹)) ≤ 𝐿) |
26 | 13, 18, 19, 20, 25 | xrletrd 12558 | 1 ⊢ (𝜑 → (𝐷‘(𝐹 + 𝐺)) ≤ 𝐿) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ifcif 4470 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 ℝ*cxr 10677 ≤ cle 10679 Basecbs 16486 +gcplusg 16568 Ringcrg 19300 Poly1cpl1 20348 deg1 cdg1 24651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-addf 10619 ax-mulf 10620 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-ofr 7413 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-sup 8909 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 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 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-sca 16584 df-vsca 16585 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-0g 16718 df-gsum 16719 df-mre 16860 df-mrc 16861 df-acs 16863 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-mhm 17959 df-submnd 17960 df-grp 18109 df-minusg 18110 df-mulg 18228 df-subg 18279 df-ghm 18359 df-cntz 18450 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-cring 19303 df-subrg 19536 df-psr 20139 df-mpl 20141 df-opsr 20143 df-psr1 20351 df-ply1 20353 df-cnfld 20549 df-mdeg 24652 df-deg1 24653 |
This theorem is referenced by: hbtlem2 39730 |
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