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Mathbox for Stefan O'Rear |
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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 | β’ (π β ( Monic βπ) β π β 0π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnccoe 42432 | . 2 β’ (π β ( Monic βπ) β ((coeffβπ)β(degβπ)) = 1) | |
2 | coe0 26134 | . . . . . . 7 β’ (coeffβ0π) = (β0 Γ {0}) | |
3 | 2 | fveq1i 6883 | . . . . . 6 β’ ((coeffβ0π)β(degβ0π)) = ((β0 Γ {0})β(degβ0π)) |
4 | dgr0 26141 | . . . . . . . 8 β’ (degβ0π) = 0 | |
5 | 0nn0 12486 | . . . . . . . 8 β’ 0 β β0 | |
6 | 4, 5 | eqeltri 2821 | . . . . . . 7 β’ (degβ0π) β β0 |
7 | c0ex 11207 | . . . . . . . 8 β’ 0 β V | |
8 | 7 | fvconst2 7198 | . . . . . . 7 β’ ((degβ0π) β β0 β ((β0 Γ {0})β(degβ0π)) = 0) |
9 | 6, 8 | ax-mp 5 | . . . . . 6 β’ ((β0 Γ {0})β(degβ0π)) = 0 |
10 | 3, 9 | eqtri 2752 | . . . . 5 β’ ((coeffβ0π)β(degβ0π)) = 0 |
11 | 0ne1 12282 | . . . . 5 β’ 0 β 1 | |
12 | 10, 11 | eqnetri 3003 | . . . 4 β’ ((coeffβ0π)β(degβ0π)) β 1 |
13 | fveq2 6882 | . . . . . 6 β’ (π = 0π β (coeffβπ) = (coeffβ0π)) | |
14 | fveq2 6882 | . . . . . 6 β’ (π = 0π β (degβπ) = (degβ0π)) | |
15 | 13, 14 | fveq12d 6889 | . . . . 5 β’ (π = 0π β ((coeffβπ)β(degβπ)) = ((coeffβ0π)β(degβ0π))) |
16 | 15 | neeq1d 2992 | . . . 4 β’ (π = 0π β (((coeffβπ)β(degβπ)) β 1 β ((coeffβ0π)β(degβ0π)) β 1)) |
17 | 12, 16 | mpbiri 258 | . . 3 β’ (π = 0π β ((coeffβπ)β(degβπ)) β 1) |
18 | 17 | necon2i 2967 | . 2 β’ (((coeffβπ)β(degβπ)) = 1 β π β 0π) |
19 | 1, 18 | syl 17 | 1 β’ (π β ( Monic βπ) β π β 0π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wne 2932 {csn 4621 Γ cxp 5665 βcfv 6534 0cc0 11107 1c1 11108 β0cn0 12471 0πc0p 25542 coeffccoe 26064 degcdgr 26065 Monic cmnc 42425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-inf2 9633 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-pm 8820 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-z 12558 df-uz 12822 df-rp 12976 df-fz 13486 df-fzo 13629 df-fl 13758 df-seq 13968 df-exp 14029 df-hash 14292 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434 df-rlim 15435 df-sum 15635 df-0p 25543 df-ply 26066 df-coe 26068 df-dgr 26069 df-mnc 42427 |
This theorem is referenced by: (None) |
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