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Mirrors > Home > MPE Home > Th. List > dgr0 | Structured version Visualization version GIF version |
Description: The degree of the zero polynomial is zero. Note: this differs from some other definitions of the degree of the zero polynomial, such as -1, -β or undefined. But it is convenient for us to define it this way, so that we have dgrcl 25754, dgreq0 25786 and coeid 25759 without having to special-case zero, although plydivalg 25819 is a little more complicated as a result. (Contributed by Mario Carneiro, 22-Jul-2014.) |
Ref | Expression |
---|---|
dgr0 | β’ (degβ0π) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-0p 25194 | . . 3 β’ 0π = (β Γ {0}) | |
2 | 1 | fveq2i 6894 | . 2 β’ (degβ0π) = (degβ(β Γ {0})) |
3 | 0cn 11208 | . . 3 β’ 0 β β | |
4 | 0dgr 25766 | . . 3 β’ (0 β β β (degβ(β Γ {0})) = 0) | |
5 | 3, 4 | ax-mp 5 | . 2 β’ (degβ(β Γ {0})) = 0 |
6 | 2, 5 | eqtri 2760 | 1 β’ (degβ0π) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 β wcel 2106 {csn 4628 Γ cxp 5674 βcfv 6543 βcc 11110 0cc0 11112 0πc0p 25193 degcdgr 25708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 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-op 4635 df-uni 4909 df-int 4951 df-iun 4999 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 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-n0 12475 df-z 12561 df-uz 12825 df-rp 12977 df-fz 13487 df-fzo 13630 df-fl 13759 df-seq 13969 df-exp 14030 df-hash 14293 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434 df-rlim 15435 df-sum 15635 df-0p 25194 df-ply 25709 df-coe 25711 df-dgr 25712 |
This theorem is referenced by: dgreq0 25786 dgrlt 25787 dgradd2 25789 dgrmulc 25792 dgrcolem1 25794 dgrcolem2 25795 plyrem 25825 facth 25826 fta1lem 25827 vieta1lem1 25830 vieta1lem2 25831 vieta1 25832 aalioulem2 25853 ftalem2 26585 ftalem4 26587 ftalem5 26588 basellem4 26595 dgrsub2 41965 mncn0 41969 aaitgo 41992 |
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