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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 20418 | . . . . 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 19330 | . . . 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 24678 | . . 3 ⊢ ((𝐹 + 𝐺) ∈ 𝐵 → (𝐷‘(𝐹 + 𝐺)) ∈ ℝ*) |
13 | 10, 12 | syl 17 | . 2 ⊢ (𝜑 → (𝐷‘(𝐹 + 𝐺)) ∈ ℝ*) |
14 | 11, 2, 7 | deg1xrcl 24678 | . . . 4 ⊢ (𝐺 ∈ 𝐵 → (𝐷‘𝐺) ∈ ℝ*) |
15 | 6, 14 | syl 17 | . . 3 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℝ*) |
16 | 11, 2, 7 | deg1xrcl 24678 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) ∈ ℝ*) |
17 | 5, 16 | syl 17 | . . 3 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℝ*) |
18 | 15, 17 | ifcld 4514 | . 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 24697 | . 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 1537 ∈ wcel 2114 ifcif 4469 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ℝ*cxr 10676 ≤ cle 10678 Basecbs 16485 +gcplusg 16567 Ringcrg 19299 Poly1cpl1 20347 deg1 cdg1 24650 |
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-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-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-subrg 19535 df-psr 20138 df-mpl 20140 df-opsr 20142 df-psr1 20350 df-ply1 20352 df-cnfld 20548 df-mdeg 24651 df-deg1 24652 |
This theorem is referenced by: hbtlem2 39731 |
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