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Mirrors > Home > MPE Home > Th. List > ply1nz | Structured version Visualization version GIF version |
Description: Univariate polynomials over a nonzero ring are a nonzero ring. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
Ref | Expression |
---|---|
ply1domn.p | β’ π = (Poly1βπ ) |
Ref | Expression |
---|---|
ply1nz | β’ (π β NzRing β π β NzRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nzrring 20408 | . . 3 β’ (π β NzRing β π β Ring) | |
2 | ply1domn.p | . . . 4 β’ π = (Poly1βπ ) | |
3 | 2 | ply1ring 22089 | . . 3 β’ (π β Ring β π β Ring) |
4 | 1, 3 | syl 17 | . 2 β’ (π β NzRing β π β Ring) |
5 | eqid 2724 | . . . . . 6 β’ (algScβπ) = (algScβπ) | |
6 | eqid 2724 | . . . . . 6 β’ (Baseβπ ) = (Baseβπ ) | |
7 | eqid 2724 | . . . . . 6 β’ (Baseβπ) = (Baseβπ) | |
8 | 2, 5, 6, 7 | ply1sclf 22126 | . . . . 5 β’ (π β Ring β (algScβπ):(Baseβπ )βΆ(Baseβπ)) |
9 | 1, 8 | syl 17 | . . . 4 β’ (π β NzRing β (algScβπ):(Baseβπ )βΆ(Baseβπ)) |
10 | eqid 2724 | . . . . . 6 β’ (1rβπ ) = (1rβπ ) | |
11 | 6, 10 | ringidcl 20155 | . . . . 5 β’ (π β Ring β (1rβπ ) β (Baseβπ )) |
12 | 1, 11 | syl 17 | . . . 4 β’ (π β NzRing β (1rβπ ) β (Baseβπ )) |
13 | 9, 12 | ffvelcdmd 7077 | . . 3 β’ (π β NzRing β ((algScβπ)β(1rβπ )) β (Baseβπ)) |
14 | eqid 2724 | . . . . 5 β’ (0gβπ ) = (0gβπ ) | |
15 | 10, 14 | nzrnz 20407 | . . . 4 β’ (π β NzRing β (1rβπ ) β (0gβπ )) |
16 | eqid 2724 | . . . . 5 β’ (0gβπ) = (0gβπ) | |
17 | 2, 5, 14, 16, 6 | ply1scln0 22133 | . . . 4 β’ ((π β Ring β§ (1rβπ ) β (Baseβπ ) β§ (1rβπ ) β (0gβπ )) β ((algScβπ)β(1rβπ )) β (0gβπ)) |
18 | 1, 12, 15, 17 | syl3anc 1368 | . . 3 β’ (π β NzRing β ((algScβπ)β(1rβπ )) β (0gβπ)) |
19 | eldifsn 4782 | . . 3 β’ (((algScβπ)β(1rβπ )) β ((Baseβπ) β {(0gβπ)}) β (((algScβπ)β(1rβπ )) β (Baseβπ) β§ ((algScβπ)β(1rβπ )) β (0gβπ))) | |
20 | 13, 18, 19 | sylanbrc 582 | . 2 β’ (π β NzRing β ((algScβπ)β(1rβπ )) β ((Baseβπ) β {(0gβπ)})) |
21 | 16, 7 | ringelnzr 20413 | . 2 β’ ((π β Ring β§ ((algScβπ)β(1rβπ )) β ((Baseβπ) β {(0gβπ)})) β π β NzRing) |
22 | 4, 20, 21 | syl2anc 583 | 1 β’ (π β NzRing β π β NzRing) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wne 2932 β cdif 3937 {csn 4620 βΆwf 6529 βcfv 6533 Basecbs 17143 0gc0g 17384 1rcur 20076 Ringcrg 20128 NzRingcnzr 20404 algSccascl 21715 Poly1cpl1 22019 |
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 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
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 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-ofr 7664 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-fzo 13625 df-seq 13964 df-hash 14288 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-hom 17220 df-cco 17221 df-0g 17386 df-gsum 17387 df-prds 17392 df-pws 17394 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mhm 18703 df-submnd 18704 df-grp 18856 df-minusg 18857 df-sbg 18858 df-mulg 18986 df-subg 19040 df-ghm 19129 df-cntz 19223 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-nzr 20405 df-subrng 20436 df-subrg 20461 df-lmod 20698 df-lss 20769 df-ascl 21718 df-psr 21771 df-mvr 21772 df-mpl 21773 df-opsr 21775 df-psr1 22022 df-vr1 22023 df-ply1 22024 df-coe1 22025 |
This theorem is referenced by: ply1nzb 25980 ply1domn 25981 algextdeglem4 33256 mon1pid 42436 mon1psubm 42437 |
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