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Mirrors > Home > MPE Home > Th. List > deg1mulle2 | Structured version Visualization version GIF version |
Description: Produce a bound on the product of two univariate polynomials given bounds on the factors. (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) |
deg1mulle2.b | ⊢ 𝐵 = (Base‘𝑌) |
deg1mulle2.t | ⊢ · = (.r‘𝑌) |
deg1mulle2.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
deg1mulle2.g | ⊢ (𝜑 → 𝐺 ∈ 𝐵) |
deg1mulle2.j1 | ⊢ (𝜑 → 𝐽 ∈ ℕ0) |
deg1mulle2.k1 | ⊢ (𝜑 → 𝐾 ∈ ℕ0) |
deg1mulle2.j2 | ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝐽) |
deg1mulle2.k2 | ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝐾) |
Ref | Expression |
---|---|
deg1mulle2 | ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ (𝐽 + 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . 2 ⊢ (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅) | |
2 | deg1addle.d | . . 3 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
3 | 2 | deg1fval 23644 | . 2 ⊢ 𝐷 = (1𝑜 mDeg 𝑅) |
4 | 1on 7454 | . . 3 ⊢ 1𝑜 ∈ On | |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → 1𝑜 ∈ On) |
6 | deg1addle.r | . 2 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
7 | deg1addle.y | . . 3 ⊢ 𝑌 = (Poly1‘𝑅) | |
8 | eqid 2610 | . . 3 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
9 | deg1mulle2.b | . . 3 ⊢ 𝐵 = (Base‘𝑌) | |
10 | 7, 8, 9 | ply1bas 19386 | . 2 ⊢ 𝐵 = (Base‘(1𝑜 mPoly 𝑅)) |
11 | deg1mulle2.t | . . 3 ⊢ · = (.r‘𝑌) | |
12 | 7, 1, 11 | ply1mulr 19418 | . 2 ⊢ · = (.r‘(1𝑜 mPoly 𝑅)) |
13 | deg1mulle2.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
14 | deg1mulle2.g | . 2 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
15 | deg1mulle2.j1 | . 2 ⊢ (𝜑 → 𝐽 ∈ ℕ0) | |
16 | deg1mulle2.k1 | . 2 ⊢ (𝜑 → 𝐾 ∈ ℕ0) | |
17 | deg1mulle2.j2 | . 2 ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝐽) | |
18 | deg1mulle2.k2 | . 2 ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝐾) | |
19 | 1, 3, 5, 6, 10, 12, 13, 14, 15, 16, 17, 18 | mdegmulle2 23643 | 1 ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ (𝐽 + 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 Oncon0 5640 ‘cfv 5804 (class class class)co 6549 1𝑜c1o 7440 + caddc 9818 ≤ cle 9954 ℕ0cn0 11169 Basecbs 15695 .rcmulr 15769 Ringcrg 18370 mPoly cmpl 19174 PwSer1cps1 19366 Poly1cpl1 19368 deg1 cdg1 23618 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-ofr 6796 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-0g 15925 df-gsum 15926 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-mulg 17364 df-subg 17414 df-ghm 17481 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-subrg 18601 df-psr 19177 df-mpl 19179 df-opsr 19181 df-psr1 19371 df-ply1 19373 df-cnfld 19568 df-mdeg 23619 df-deg1 23620 |
This theorem is referenced by: deg1mul2 23678 ply1divex 23700 hbtlem4 36715 |
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