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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 22090 | . . . . 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 20173 | . . . 4 ⊢ ((𝑌 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 + 𝐺) ∈ 𝐵) |
10 | 4, 5, 6, 9 | syl3anc 1370 | . . 3 ⊢ (𝜑 → (𝐹 + 𝐺) ∈ 𝐵) |
11 | deg1addle.d | . . . 4 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
12 | 11, 2, 7 | deg1xrcl 25938 | . . 3 ⊢ ((𝐹 + 𝐺) ∈ 𝐵 → (𝐷‘(𝐹 + 𝐺)) ∈ ℝ*) |
13 | 10, 12 | syl 17 | . 2 ⊢ (𝜑 → (𝐷‘(𝐹 + 𝐺)) ∈ ℝ*) |
14 | 11, 2, 7 | deg1xrcl 25938 | . . . 4 ⊢ (𝐺 ∈ 𝐵 → (𝐷‘𝐺) ∈ ℝ*) |
15 | 6, 14 | syl 17 | . . 3 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℝ*) |
16 | 11, 2, 7 | deg1xrcl 25938 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) ∈ ℝ*) |
17 | 5, 16 | syl 17 | . . 3 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℝ*) |
18 | 15, 17 | ifcld 4574 | . 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 25957 | . 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 13169 | . . . 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 1370 | . . 3 ⊢ (𝜑 → (if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹)) ≤ 𝐿 ↔ ((𝐷‘𝐹) ≤ 𝐿 ∧ (𝐷‘𝐺) ≤ 𝐿))) |
25 | 21, 22, 24 | mpbir2and 710 | . 2 ⊢ (𝜑 → if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹)) ≤ 𝐿) |
26 | 13, 18, 19, 20, 25 | xrletrd 13148 | 1 ⊢ (𝜑 → (𝐷‘(𝐹 + 𝐺)) ≤ 𝐿) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ifcif 4528 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 ℝ*cxr 11254 ≤ cle 11256 Basecbs 17151 +gcplusg 17204 Ringcrg 20134 Poly1cpl1 22020 deg1 cdg1 25907 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-addf 11195 ax-mulf 11196 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-ofr 7675 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-sup 9443 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-fz 13492 df-fzo 13635 df-seq 13974 df-hash 14298 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-0g 17394 df-gsum 17395 df-prds 17400 df-pws 17402 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mhm 18711 df-submnd 18712 df-grp 18864 df-minusg 18865 df-mulg 18994 df-subg 19046 df-ghm 19135 df-cntz 19229 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-cring 20137 df-subrng 20442 df-subrg 20467 df-cnfld 21234 df-psr 21772 df-mpl 21774 df-opsr 21776 df-psr1 22023 df-ply1 22025 df-mdeg 25908 df-deg1 25909 |
This theorem is referenced by: ply1degltlss 33109 hbtlem2 42331 |
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