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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 6676 | . . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0) ∧ 𝑃 = (ℂ × {(𝑃‘0)})) → (𝑃‘𝐴) = ((ℂ × {(𝑃‘0)})‘𝐴)) |
3 | simplr 769 | . . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0) ∧ 𝑃 = (ℂ × {(𝑃‘0)})) → (𝑃‘𝐴) = 0) | |
4 | fvex 6687 | . . . . . . . . . . . . . 14 ⊢ (𝑃‘0) ∈ V | |
5 | 4 | fvconst2 6976 | . . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → ((ℂ × {(𝑃‘0)})‘𝐴) = (𝑃‘0)) |
6 | 5 | ad2antrr 726 | . . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0) ∧ 𝑃 = (ℂ × {(𝑃‘0)})) → ((ℂ × {(𝑃‘0)})‘𝐴) = (𝑃‘0)) |
7 | 2, 3, 6 | 3eqtr3rd 2782 | . . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0) ∧ 𝑃 = (ℂ × {(𝑃‘0)})) → (𝑃‘0) = 0) |
8 | 7 | sneqd 4528 | . . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0) ∧ 𝑃 = (ℂ × {(𝑃‘0)})) → {(𝑃‘0)} = {0}) |
9 | 8 | xpeq2d 5555 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0) ∧ 𝑃 = (ℂ × {(𝑃‘0)})) → (ℂ × {(𝑃‘0)}) = (ℂ × {0})) |
10 | 1, 9 | eqtrd 2773 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0) ∧ 𝑃 = (ℂ × {(𝑃‘0)})) → 𝑃 = (ℂ × {0})) |
11 | df-0p 24422 | . . . . . . . 8 ⊢ 0𝑝 = (ℂ × {0}) | |
12 | 10, 11 | eqtr4di 2791 | . . . . . . 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 2947 | . . . . 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 714 | . . 3 ⊢ (((𝑃 ∈ (Poly‘𝑆) ∧ 𝑃 ≠ 0𝑝) ∧ (𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0)) → ¬ 𝑃 = (ℂ × {(𝑃‘0)})) |
17 | 0dgrb 24995 | . . . 4 ⊢ (𝑃 ∈ (Poly‘𝑆) → ((deg‘𝑃) = 0 ↔ 𝑃 = (ℂ × {(𝑃‘0)}))) | |
18 | 17 | ad2antrr 726 | . . 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 24982 | . . . 4 ⊢ (𝑃 ∈ (Poly‘𝑆) → (deg‘𝑃) ∈ ℕ0) | |
21 | 20 | ad2antrr 726 | . . 3 ⊢ (((𝑃 ∈ (Poly‘𝑆) ∧ 𝑃 ≠ 0𝑝) ∧ (𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0)) → (deg‘𝑃) ∈ ℕ0) |
22 | elnn0 11978 | . . 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 890 | . 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 846 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 {csn 4516 × cxp 5523 ‘cfv 6339 ℂcc 10613 0cc0 10615 ℕcn 11716 ℕ0cn0 11976 0𝑝c0p 24421 Polycply 24933 degcdgr 24936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-inf2 9177 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-of 7425 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-map 8439 df-pm 8440 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-sup 8979 df-inf 8980 df-oi 9047 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-n0 11977 df-z 12063 df-uz 12325 df-rp 12473 df-fz 12982 df-fzo 13125 df-fl 13253 df-seq 13461 df-exp 13522 df-hash 13783 df-cj 14548 df-re 14549 df-im 14550 df-sqrt 14684 df-abs 14685 df-clim 14935 df-rlim 14936 df-sum 15136 df-0p 24422 df-ply 24937 df-coe 24939 df-dgr 24940 |
This theorem is referenced by: dgraalem 40542 dgraaub 40545 etransclem47 43364 |
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