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Mirrors > Home > MPE Home > Th. List > 0dgrb | Structured version Visualization version Unicode version |
Description: A function has degree zero iff it is a constant function. (Contributed by Mario Carneiro, 23-Jul-2014.) |
Ref | Expression |
---|---|
0dgrb | Poly deg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . . . . . 8 coeff coeff | |
2 | eqid 2622 | . . . . . . . 8 deg deg | |
3 | 1, 2 | coeid 23994 | . . . . . . 7 Poly degcoeff |
4 | 3 | adantr 481 | . . . . . 6 Poly deg degcoeff |
5 | simplr 792 | . . . . . . . . . 10 Poly deg deg | |
6 | 5 | oveq2d 6666 | . . . . . . . . 9 Poly deg deg |
7 | 6 | sumeq1d 14431 | . . . . . . . 8 Poly deg degcoeff coeff |
8 | 0z 11388 | . . . . . . . . . 10 | |
9 | exp0 12864 | . . . . . . . . . . . . . 14 | |
10 | 9 | adantl 482 | . . . . . . . . . . . . 13 Poly deg |
11 | 10 | oveq2d 6666 | . . . . . . . . . . . 12 Poly deg coeff coeff |
12 | 1 | coef3 23988 | . . . . . . . . . . . . . . 15 Poly coeff |
13 | 0nn0 11307 | . . . . . . . . . . . . . . 15 | |
14 | ffvelrn 6357 | . . . . . . . . . . . . . . 15 coeff coeff | |
15 | 12, 13, 14 | sylancl 694 | . . . . . . . . . . . . . 14 Poly coeff |
16 | 15 | ad2antrr 762 | . . . . . . . . . . . . 13 Poly deg coeff |
17 | 16 | mulid1d 10057 | . . . . . . . . . . . 12 Poly deg coeff coeff |
18 | 11, 17 | eqtrd 2656 | . . . . . . . . . . 11 Poly deg coeff coeff |
19 | 18, 16 | eqeltrd 2701 | . . . . . . . . . 10 Poly deg coeff |
20 | fveq2 6191 | . . . . . . . . . . . 12 coeff coeff | |
21 | oveq2 6658 | . . . . . . . . . . . 12 | |
22 | 20, 21 | oveq12d 6668 | . . . . . . . . . . 11 coeff coeff |
23 | 22 | fsum1 14476 | . . . . . . . . . 10 coeff coeff coeff |
24 | 8, 19, 23 | sylancr 695 | . . . . . . . . 9 Poly deg coeff coeff |
25 | 24, 18 | eqtrd 2656 | . . . . . . . 8 Poly deg coeff coeff |
26 | 7, 25 | eqtrd 2656 | . . . . . . 7 Poly deg degcoeff coeff |
27 | 26 | mpteq2dva 4744 | . . . . . 6 Poly deg degcoeff coeff |
28 | 4, 27 | eqtrd 2656 | . . . . 5 Poly deg coeff |
29 | fconstmpt 5163 | . . . . 5 coeff coeff | |
30 | 28, 29 | syl6eqr 2674 | . . . 4 Poly deg coeff |
31 | 30 | fveq1d 6193 | . . . . . . 7 Poly deg coeff |
32 | 0cn 10032 | . . . . . . . 8 | |
33 | fvex 6201 | . . . . . . . . 9 coeff | |
34 | 33 | fvconst2 6469 | . . . . . . . 8 coeff coeff |
35 | 32, 34 | ax-mp 5 | . . . . . . 7 coeff coeff |
36 | 31, 35 | syl6eq 2672 | . . . . . 6 Poly deg coeff |
37 | 36 | sneqd 4189 | . . . . 5 Poly deg coeff |
38 | 37 | xpeq2d 5139 | . . . 4 Poly deg coeff |
39 | 30, 38 | eqtr4d 2659 | . . 3 Poly deg |
40 | 39 | ex 450 | . 2 Poly deg |
41 | plyf 23954 | . . . . 5 Poly | |
42 | ffvelrn 6357 | . . . . 5 | |
43 | 41, 32, 42 | sylancl 694 | . . . 4 Poly |
44 | 0dgr 24001 | . . . 4 deg | |
45 | 43, 44 | syl 17 | . . 3 Poly deg |
46 | fveq2 6191 | . . . 4 deg deg | |
47 | 46 | eqeq1d 2624 | . . 3 deg deg |
48 | 45, 47 | syl5ibrcom 237 | . 2 Poly deg |
49 | 40, 48 | impbid 202 | 1 Poly deg |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 csn 4177 cmpt 4729 cxp 5112 wf 5884 cfv 5888 (class class class)co 6650 cc 9934 cc0 9936 c1 9937 cmul 9941 cn0 11292 cz 11377 cfz 12326 cexp 12860 csu 14416 Polycply 23940 coeffccoe 23942 degcdgr 23943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-rlim 14220 df-sum 14417 df-0p 23437 df-ply 23944 df-coe 23946 df-dgr 23947 |
This theorem is referenced by: dgrnznn 24003 dgreq0 24021 dgrcolem2 24030 dgrco 24031 plyrem 24060 fta1 24063 aaliou2 24095 |
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