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Mirrors > Home > MPE Home > Th. List > deg1xrcl | Structured version Visualization version GIF version |
Description: Closure of univariate polynomial degree in extended reals. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
Ref | Expression |
---|---|
deg1xrf.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
deg1xrf.p | ⊢ 𝑃 = (Poly1‘𝑅) |
deg1xrf.b | ⊢ 𝐵 = (Base‘𝑃) |
Ref | Expression |
---|---|
deg1xrcl | ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) ∈ ℝ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | deg1xrf.d | . . 3 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
2 | deg1xrf.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | deg1xrf.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
4 | 1, 2, 3 | deg1xrf 25581 | . 2 ⊢ 𝐷:𝐵⟶ℝ* |
5 | 4 | ffvelcdmi 7081 | 1 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ‘cfv 6540 ℝ*cxr 11243 Basecbs 17140 Poly1cpl1 21683 deg1 cdg1 25551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7665 df-om 7851 df-1st 7970 df-2nd 7971 df-supp 8142 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-fzo 13624 df-seq 13963 df-hash 14287 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-0g 17383 df-gsum 17384 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-grp 18818 df-minusg 18819 df-cntz 19175 df-cmn 19643 df-abl 19644 df-mgp 19980 df-ur 19997 df-ring 20049 df-cring 20050 df-cnfld 20930 df-psr 21444 df-mpl 21446 df-opsr 21448 df-psr1 21686 df-ply1 21688 df-mdeg 25552 df-deg1 25553 |
This theorem is referenced by: deg1lt 25597 deg1ge 25598 coe1mul3 25599 deg1addle2 25602 deg1add 25603 deg1sublt 25610 deg1mul2 25614 deg1tm 25618 ply1divmo 25635 ply1remlem 25662 ig1peu 25671 plypf1 25708 ply1degltdimlem 32652 ply1degltdim 32653 hbtlem2 41799 idomrootle 41870 |
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