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Mirrors > Home > MPE Home > Th. List > dgrnznn | Structured version Visualization version GIF version |
Description: A nonzero polynomial with a root has positive degree. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
Ref | Expression |
---|---|
dgrnznn | ⊢ (((𝑃 ∈ (Poly‘𝑆) ∧ 𝑃 ≠ 0𝑝) ∧ (𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0)) → (deg‘𝑃) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 488 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0) ∧ 𝑃 = (ℂ × {(𝑃‘0)})) → 𝑃 = (ℂ × {(𝑃‘0)})) | |
2 | 1 | fveq1d 6647 | . . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0) ∧ 𝑃 = (ℂ × {(𝑃‘0)})) → (𝑃‘𝐴) = ((ℂ × {(𝑃‘0)})‘𝐴)) |
3 | simplr 768 | . . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0) ∧ 𝑃 = (ℂ × {(𝑃‘0)})) → (𝑃‘𝐴) = 0) | |
4 | fvex 6658 | . . . . . . . . . . . . . 14 ⊢ (𝑃‘0) ∈ V | |
5 | 4 | fvconst2 6943 | . . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → ((ℂ × {(𝑃‘0)})‘𝐴) = (𝑃‘0)) |
6 | 5 | ad2antrr 725 | . . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0) ∧ 𝑃 = (ℂ × {(𝑃‘0)})) → ((ℂ × {(𝑃‘0)})‘𝐴) = (𝑃‘0)) |
7 | 2, 3, 6 | 3eqtr3rd 2842 | . . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0) ∧ 𝑃 = (ℂ × {(𝑃‘0)})) → (𝑃‘0) = 0) |
8 | 7 | sneqd 4537 | . . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0) ∧ 𝑃 = (ℂ × {(𝑃‘0)})) → {(𝑃‘0)} = {0}) |
9 | 8 | xpeq2d 5549 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0) ∧ 𝑃 = (ℂ × {(𝑃‘0)})) → (ℂ × {(𝑃‘0)}) = (ℂ × {0})) |
10 | 1, 9 | eqtrd 2833 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0) ∧ 𝑃 = (ℂ × {(𝑃‘0)})) → 𝑃 = (ℂ × {0})) |
11 | df-0p 24274 | . . . . . . . 8 ⊢ 0𝑝 = (ℂ × {0}) | |
12 | 10, 11 | eqtr4di 2851 | . . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0) ∧ 𝑃 = (ℂ × {(𝑃‘0)})) → 𝑃 = 0𝑝) |
13 | 12 | ex 416 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0) → (𝑃 = (ℂ × {(𝑃‘0)}) → 𝑃 = 0𝑝)) |
14 | 13 | necon3ad 3000 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0) → (𝑃 ≠ 0𝑝 → ¬ 𝑃 = (ℂ × {(𝑃‘0)}))) |
15 | 14 | impcom 411 | . . . 4 ⊢ ((𝑃 ≠ 0𝑝 ∧ (𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0)) → ¬ 𝑃 = (ℂ × {(𝑃‘0)})) |
16 | 15 | adantll 713 | . . 3 ⊢ (((𝑃 ∈ (Poly‘𝑆) ∧ 𝑃 ≠ 0𝑝) ∧ (𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0)) → ¬ 𝑃 = (ℂ × {(𝑃‘0)})) |
17 | 0dgrb 24843 | . . . 4 ⊢ (𝑃 ∈ (Poly‘𝑆) → ((deg‘𝑃) = 0 ↔ 𝑃 = (ℂ × {(𝑃‘0)}))) | |
18 | 17 | ad2antrr 725 | . . 3 ⊢ (((𝑃 ∈ (Poly‘𝑆) ∧ 𝑃 ≠ 0𝑝) ∧ (𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0)) → ((deg‘𝑃) = 0 ↔ 𝑃 = (ℂ × {(𝑃‘0)}))) |
19 | 16, 18 | mtbird 328 | . 2 ⊢ (((𝑃 ∈ (Poly‘𝑆) ∧ 𝑃 ≠ 0𝑝) ∧ (𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0)) → ¬ (deg‘𝑃) = 0) |
20 | dgrcl 24830 | . . . 4 ⊢ (𝑃 ∈ (Poly‘𝑆) → (deg‘𝑃) ∈ ℕ0) | |
21 | 20 | ad2antrr 725 | . . 3 ⊢ (((𝑃 ∈ (Poly‘𝑆) ∧ 𝑃 ≠ 0𝑝) ∧ (𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0)) → (deg‘𝑃) ∈ ℕ0) |
22 | elnn0 11887 | . . 3 ⊢ ((deg‘𝑃) ∈ ℕ0 ↔ ((deg‘𝑃) ∈ ℕ ∨ (deg‘𝑃) = 0)) | |
23 | 21, 22 | sylib 221 | . 2 ⊢ (((𝑃 ∈ (Poly‘𝑆) ∧ 𝑃 ≠ 0𝑝) ∧ (𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0)) → ((deg‘𝑃) ∈ ℕ ∨ (deg‘𝑃) = 0)) |
24 | orel2 888 | . 2 ⊢ (¬ (deg‘𝑃) = 0 → (((deg‘𝑃) ∈ ℕ ∨ (deg‘𝑃) = 0) → (deg‘𝑃) ∈ ℕ)) | |
25 | 19, 23, 24 | sylc 65 | 1 ⊢ (((𝑃 ∈ (Poly‘𝑆) ∧ 𝑃 ≠ 0𝑝) ∧ (𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0)) → (deg‘𝑃) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 {csn 4525 × cxp 5517 ‘cfv 6324 ℂcc 10524 0cc0 10526 ℕcn 11625 ℕ0cn0 11885 0𝑝c0p 24273 Polycply 24781 degcdgr 24784 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-rlim 14838 df-sum 15035 df-0p 24274 df-ply 24785 df-coe 24787 df-dgr 24788 |
This theorem is referenced by: dgraalem 40089 dgraaub 40092 etransclem47 42923 |
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