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Mirrors > Home > MPE Home > Th. List > deg1lt0 | Structured version Visualization version GIF version |
Description: A polynomial is zero iff it has negative degree. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
Ref | Expression |
---|---|
deg1z.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
deg1z.p | ⊢ 𝑃 = (Poly1‘𝑅) |
deg1z.z | ⊢ 0 = (0g‘𝑃) |
deg1nn0cl.b | ⊢ 𝐵 = (Base‘𝑃) |
Ref | Expression |
---|---|
deg1lt0 | ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → ((𝐷‘𝐹) < 0 ↔ 𝐹 = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | deg1z.d | . . . . . 6 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
2 | deg1z.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | deg1z.z | . . . . . 6 ⊢ 0 = (0g‘𝑃) | |
4 | deg1nn0cl.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑃) | |
5 | 1, 2, 3, 4 | deg1nn0cl 24375 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ0) |
6 | nn0nlt0 11728 | . . . . 5 ⊢ ((𝐷‘𝐹) ∈ ℕ0 → ¬ (𝐷‘𝐹) < 0) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → ¬ (𝐷‘𝐹) < 0) |
8 | 7 | 3expia 1101 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → (𝐹 ≠ 0 → ¬ (𝐷‘𝐹) < 0)) |
9 | 8 | necon4ad 2980 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → ((𝐷‘𝐹) < 0 → 𝐹 = 0 )) |
10 | 1, 2, 3 | deg1z 24374 | . . . . 5 ⊢ (𝑅 ∈ Ring → (𝐷‘ 0 ) = -∞) |
11 | mnflt0 12330 | . . . . 5 ⊢ -∞ < 0 | |
12 | 10, 11 | syl6eqbr 4962 | . . . 4 ⊢ (𝑅 ∈ Ring → (𝐷‘ 0 ) < 0) |
13 | 12 | adantr 473 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → (𝐷‘ 0 ) < 0) |
14 | fveq2 6493 | . . . 4 ⊢ (𝐹 = 0 → (𝐷‘𝐹) = (𝐷‘ 0 )) | |
15 | 14 | breq1d 4933 | . . 3 ⊢ (𝐹 = 0 → ((𝐷‘𝐹) < 0 ↔ (𝐷‘ 0 ) < 0)) |
16 | 13, 15 | syl5ibrcom 239 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → (𝐹 = 0 → (𝐷‘𝐹) < 0)) |
17 | 9, 16 | impbid 204 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → ((𝐷‘𝐹) < 0 ↔ 𝐹 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2048 ≠ wne 2961 class class class wbr 4923 ‘cfv 6182 0cc0 10327 -∞cmnf 10464 < clt 10466 ℕ0cn0 11700 Basecbs 16329 0gc0g 16559 Ringcrg 19010 Poly1cpl1 20038 deg1 cdg1 24341 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-addf 10406 ax-mulf 10407 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-se 5360 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-of 7221 df-om 7391 df-1st 7494 df-2nd 7495 df-supp 7627 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-oadd 7901 df-er 8081 df-map 8200 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-fsupp 8621 df-sup 8693 df-oi 8761 df-card 9154 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-9 11503 df-n0 11701 df-z 11787 df-dec 11905 df-uz 12052 df-fz 12702 df-fzo 12843 df-seq 13178 df-hash 13499 df-struct 16331 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-ress 16337 df-plusg 16424 df-mulr 16425 df-starv 16426 df-sca 16427 df-vsca 16428 df-tset 16430 df-ple 16431 df-ds 16433 df-unif 16434 df-0g 16561 df-gsum 16562 df-mgm 17700 df-sgrp 17742 df-mnd 17753 df-submnd 17794 df-grp 17884 df-minusg 17885 df-subg 18050 df-cntz 18208 df-cmn 18658 df-abl 18659 df-mgp 18953 df-ur 18965 df-ring 19012 df-cring 19013 df-psr 19840 df-mpl 19842 df-opsr 19844 df-psr1 20041 df-ply1 20043 df-cnfld 20238 df-mdeg 24342 df-deg1 24343 |
This theorem is referenced by: hbtlem5 39069 |
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