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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mon1pid | Structured version Visualization version GIF version |
Description: Monicity and degree of the unit polynomial. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
Ref | Expression |
---|---|
mon1pid.p | β’ π = (Poly1βπ ) |
mon1pid.o | β’ 1 = (1rβπ) |
mon1pid.m | β’ π = (Monic1pβπ ) |
mon1pid.d | β’ π· = ( deg1 βπ ) |
Ref | Expression |
---|---|
mon1pid | β’ (π β NzRing β ( 1 β π β§ (π·β 1 ) = 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mon1pid.p | . . . . 5 β’ π = (Poly1βπ ) | |
2 | 1 | ply1nz 25502 | . . . 4 β’ (π β NzRing β π β NzRing) |
3 | nzrring 20747 | . . . 4 β’ (π β NzRing β π β Ring) | |
4 | eqid 2733 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
5 | mon1pid.o | . . . . 5 β’ 1 = (1rβπ) | |
6 | 4, 5 | ringidcl 19994 | . . . 4 β’ (π β Ring β 1 β (Baseβπ)) |
7 | 2, 3, 6 | 3syl 18 | . . 3 β’ (π β NzRing β 1 β (Baseβπ)) |
8 | eqid 2733 | . . . . 5 β’ (0gβπ) = (0gβπ) | |
9 | 5, 8 | nzrnz 20746 | . . . 4 β’ (π β NzRing β 1 β (0gβπ)) |
10 | 2, 9 | syl 17 | . . 3 β’ (π β NzRing β 1 β (0gβπ)) |
11 | nzrring 20747 | . . . . . . . 8 β’ (π β NzRing β π β Ring) | |
12 | eqid 2733 | . . . . . . . . 9 β’ (algScβπ) = (algScβπ) | |
13 | eqid 2733 | . . . . . . . . 9 β’ (1rβπ ) = (1rβπ ) | |
14 | 1, 12, 13, 5 | ply1scl1 21679 | . . . . . . . 8 β’ (π β Ring β ((algScβπ)β(1rβπ )) = 1 ) |
15 | 11, 14 | syl 17 | . . . . . . 7 β’ (π β NzRing β ((algScβπ)β(1rβπ )) = 1 ) |
16 | 15 | fveq2d 6847 | . . . . . 6 β’ (π β NzRing β (coe1β((algScβπ)β(1rβπ ))) = (coe1β 1 )) |
17 | eqid 2733 | . . . . . . . 8 β’ (Baseβπ ) = (Baseβπ ) | |
18 | 17, 13 | ringidcl 19994 | . . . . . . 7 β’ (π β Ring β (1rβπ ) β (Baseβπ )) |
19 | eqid 2733 | . . . . . . . 8 β’ (0gβπ ) = (0gβπ ) | |
20 | 1, 12, 17, 19 | coe1scl 21674 | . . . . . . 7 β’ ((π β Ring β§ (1rβπ ) β (Baseβπ )) β (coe1β((algScβπ)β(1rβπ ))) = (π₯ β β0 β¦ if(π₯ = 0, (1rβπ ), (0gβπ )))) |
21 | 11, 18, 20 | syl2anc2 586 | . . . . . 6 β’ (π β NzRing β (coe1β((algScβπ)β(1rβπ ))) = (π₯ β β0 β¦ if(π₯ = 0, (1rβπ ), (0gβπ )))) |
22 | 16, 21 | eqtr3d 2775 | . . . . 5 β’ (π β NzRing β (coe1β 1 ) = (π₯ β β0 β¦ if(π₯ = 0, (1rβπ ), (0gβπ )))) |
23 | 15 | fveq2d 6847 | . . . . . 6 β’ (π β NzRing β (π·β((algScβπ)β(1rβπ ))) = (π·β 1 )) |
24 | 11, 18 | syl 17 | . . . . . . 7 β’ (π β NzRing β (1rβπ ) β (Baseβπ )) |
25 | 13, 19 | nzrnz 20746 | . . . . . . 7 β’ (π β NzRing β (1rβπ ) β (0gβπ )) |
26 | mon1pid.d | . . . . . . . 8 β’ π· = ( deg1 βπ ) | |
27 | 26, 1, 17, 12, 19 | deg1scl 25494 | . . . . . . 7 β’ ((π β Ring β§ (1rβπ ) β (Baseβπ ) β§ (1rβπ ) β (0gβπ )) β (π·β((algScβπ)β(1rβπ ))) = 0) |
28 | 11, 24, 25, 27 | syl3anc 1372 | . . . . . 6 β’ (π β NzRing β (π·β((algScβπ)β(1rβπ ))) = 0) |
29 | 23, 28 | eqtr3d 2775 | . . . . 5 β’ (π β NzRing β (π·β 1 ) = 0) |
30 | 22, 29 | fveq12d 6850 | . . . 4 β’ (π β NzRing β ((coe1β 1 )β(π·β 1 )) = ((π₯ β β0 β¦ if(π₯ = 0, (1rβπ ), (0gβπ )))β0)) |
31 | 0nn0 12433 | . . . . 5 β’ 0 β β0 | |
32 | iftrue 4493 | . . . . . 6 β’ (π₯ = 0 β if(π₯ = 0, (1rβπ ), (0gβπ )) = (1rβπ )) | |
33 | eqid 2733 | . . . . . 6 β’ (π₯ β β0 β¦ if(π₯ = 0, (1rβπ ), (0gβπ ))) = (π₯ β β0 β¦ if(π₯ = 0, (1rβπ ), (0gβπ ))) | |
34 | fvex 6856 | . . . . . 6 β’ (1rβπ ) β V | |
35 | 32, 33, 34 | fvmpt 6949 | . . . . 5 β’ (0 β β0 β ((π₯ β β0 β¦ if(π₯ = 0, (1rβπ ), (0gβπ )))β0) = (1rβπ )) |
36 | 31, 35 | ax-mp 5 | . . . 4 β’ ((π₯ β β0 β¦ if(π₯ = 0, (1rβπ ), (0gβπ )))β0) = (1rβπ ) |
37 | 30, 36 | eqtrdi 2789 | . . 3 β’ (π β NzRing β ((coe1β 1 )β(π·β 1 )) = (1rβπ )) |
38 | mon1pid.m | . . . 4 β’ π = (Monic1pβπ ) | |
39 | 1, 4, 8, 26, 38, 13 | ismon1p 25523 | . . 3 β’ ( 1 β π β ( 1 β (Baseβπ) β§ 1 β (0gβπ) β§ ((coe1β 1 )β(π·β 1 )) = (1rβπ ))) |
40 | 7, 10, 37, 39 | syl3anbrc 1344 | . 2 β’ (π β NzRing β 1 β π) |
41 | 40, 29 | jca 513 | 1 β’ (π β NzRing β ( 1 β π β§ (π·β 1 ) = 0)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2940 ifcif 4487 β¦ cmpt 5189 βcfv 6497 0cc0 11056 β0cn0 12418 Basecbs 17088 0gc0g 17326 1rcur 19918 Ringcrg 19969 NzRingcnzr 20743 algSccascl 21274 Poly1cpl1 21564 coe1cco1 21565 deg1 cdg1 25432 Monic1pcmn1 25506 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 ax-addf 11135 ax-mulf 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-ofr 7619 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-pm 8771 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 df-sup 9383 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-fzo 13574 df-seq 13913 df-hash 14237 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-starv 17153 df-sca 17154 df-vsca 17155 df-ip 17156 df-tset 17157 df-ple 17158 df-ds 17160 df-unif 17161 df-hom 17162 df-cco 17163 df-0g 17328 df-gsum 17329 df-prds 17334 df-pws 17336 df-mre 17471 df-mrc 17472 df-acs 17474 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-mhm 18606 df-submnd 18607 df-grp 18756 df-minusg 18757 df-sbg 18758 df-mulg 18878 df-subg 18930 df-ghm 19011 df-cntz 19102 df-cmn 19569 df-abl 19570 df-mgp 19902 df-ur 19919 df-ring 19971 df-cring 19972 df-subrg 20234 df-lmod 20338 df-lss 20408 df-nzr 20744 df-cnfld 20813 df-ascl 21277 df-psr 21327 df-mvr 21328 df-mpl 21329 df-opsr 21331 df-psr1 21567 df-vr1 21568 df-ply1 21569 df-coe1 21570 df-mdeg 25433 df-deg1 25434 df-mon1 25511 |
This theorem is referenced by: mon1psubm 41576 deg1mhm 41577 |
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