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Mirrors > Home > MPE Home > Th. List > Mathboxes > mncn0 | Structured version Visualization version GIF version |
Description: A monic polynomial is not zero. (Contributed by Stefan O'Rear, 5-Dec-2014.) |
Ref | Expression |
---|---|
mncn0 | ⊢ (𝑃 ∈ ( Monic ‘𝑆) → 𝑃 ≠ 0𝑝) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnccoe 40952 | . 2 ⊢ (𝑃 ∈ ( Monic ‘𝑆) → ((coeff‘𝑃)‘(deg‘𝑃)) = 1) | |
2 | coe0 25407 | . . . . . . 7 ⊢ (coeff‘0𝑝) = (ℕ0 × {0}) | |
3 | 2 | fveq1i 6770 | . . . . . 6 ⊢ ((coeff‘0𝑝)‘(deg‘0𝑝)) = ((ℕ0 × {0})‘(deg‘0𝑝)) |
4 | dgr0 25413 | . . . . . . . 8 ⊢ (deg‘0𝑝) = 0 | |
5 | 0nn0 12240 | . . . . . . . 8 ⊢ 0 ∈ ℕ0 | |
6 | 4, 5 | eqeltri 2837 | . . . . . . 7 ⊢ (deg‘0𝑝) ∈ ℕ0 |
7 | c0ex 10962 | . . . . . . . 8 ⊢ 0 ∈ V | |
8 | 7 | fvconst2 7074 | . . . . . . 7 ⊢ ((deg‘0𝑝) ∈ ℕ0 → ((ℕ0 × {0})‘(deg‘0𝑝)) = 0) |
9 | 6, 8 | ax-mp 5 | . . . . . 6 ⊢ ((ℕ0 × {0})‘(deg‘0𝑝)) = 0 |
10 | 3, 9 | eqtri 2768 | . . . . 5 ⊢ ((coeff‘0𝑝)‘(deg‘0𝑝)) = 0 |
11 | 0ne1 12036 | . . . . 5 ⊢ 0 ≠ 1 | |
12 | 10, 11 | eqnetri 3016 | . . . 4 ⊢ ((coeff‘0𝑝)‘(deg‘0𝑝)) ≠ 1 |
13 | fveq2 6769 | . . . . . 6 ⊢ (𝑃 = 0𝑝 → (coeff‘𝑃) = (coeff‘0𝑝)) | |
14 | fveq2 6769 | . . . . . 6 ⊢ (𝑃 = 0𝑝 → (deg‘𝑃) = (deg‘0𝑝)) | |
15 | 13, 14 | fveq12d 6776 | . . . . 5 ⊢ (𝑃 = 0𝑝 → ((coeff‘𝑃)‘(deg‘𝑃)) = ((coeff‘0𝑝)‘(deg‘0𝑝))) |
16 | 15 | neeq1d 3005 | . . . 4 ⊢ (𝑃 = 0𝑝 → (((coeff‘𝑃)‘(deg‘𝑃)) ≠ 1 ↔ ((coeff‘0𝑝)‘(deg‘0𝑝)) ≠ 1)) |
17 | 12, 16 | mpbiri 257 | . . 3 ⊢ (𝑃 = 0𝑝 → ((coeff‘𝑃)‘(deg‘𝑃)) ≠ 1) |
18 | 17 | necon2i 2980 | . 2 ⊢ (((coeff‘𝑃)‘(deg‘𝑃)) = 1 → 𝑃 ≠ 0𝑝) |
19 | 1, 18 | syl 17 | 1 ⊢ (𝑃 ∈ ( Monic ‘𝑆) → 𝑃 ≠ 0𝑝) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 {csn 4567 × cxp 5587 ‘cfv 6431 0cc0 10864 1c1 10865 ℕ0cn0 12225 0𝑝c0p 24823 coeffccoe 25337 degcdgr 25338 Monic cmnc 40945 |
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 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-inf2 9369 ax-cnex 10920 ax-resscn 10921 ax-1cn 10922 ax-icn 10923 ax-addcl 10924 ax-addrcl 10925 ax-mulcl 10926 ax-mulrcl 10927 ax-mulcom 10928 ax-addass 10929 ax-mulass 10930 ax-distr 10931 ax-i2m1 10932 ax-1ne0 10933 ax-1rid 10934 ax-rnegex 10935 ax-rrecex 10936 ax-cnre 10937 ax-pre-lttri 10938 ax-pre-lttrn 10939 ax-pre-ltadd 10940 ax-pre-mulgt0 10941 ax-pre-sup 10942 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-isom 6440 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-of 7525 df-om 7702 df-1st 7818 df-2nd 7819 df-frecs 8082 df-wrecs 8113 df-recs 8187 df-rdg 8226 df-1o 8282 df-er 8473 df-map 8592 df-pm 8593 df-en 8709 df-dom 8710 df-sdom 8711 df-fin 8712 df-sup 9171 df-inf 9172 df-oi 9239 df-card 9690 df-pnf 11004 df-mnf 11005 df-xr 11006 df-ltxr 11007 df-le 11008 df-sub 11199 df-neg 11200 df-div 11625 df-nn 11966 df-2 12028 df-3 12029 df-n0 12226 df-z 12312 df-uz 12574 df-rp 12722 df-fz 13231 df-fzo 13374 df-fl 13502 df-seq 13712 df-exp 13773 df-hash 14035 df-cj 14800 df-re 14801 df-im 14802 df-sqrt 14936 df-abs 14937 df-clim 15187 df-rlim 15188 df-sum 15388 df-0p 24824 df-ply 25339 df-coe 25341 df-dgr 25342 df-mnc 40947 |
This theorem is referenced by: (None) |
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