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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 | ⊢ 𝑃 = (Poly1‘𝑅) |
Ref | Expression |
---|---|
ply1nz | ⊢ (𝑅 ∈ NzRing → 𝑃 ∈ NzRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nzrring 20288 | . . 3 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
2 | ply1domn.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | 2 | ply1ring 21762 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝑅 ∈ NzRing → 𝑃 ∈ Ring) |
5 | eqid 2733 | . . . . . 6 ⊢ (algSc‘𝑃) = (algSc‘𝑃) | |
6 | eqid 2733 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | eqid 2733 | . . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
8 | 2, 5, 6, 7 | ply1sclf 21799 | . . . . 5 ⊢ (𝑅 ∈ Ring → (algSc‘𝑃):(Base‘𝑅)⟶(Base‘𝑃)) |
9 | 1, 8 | syl 17 | . . . 4 ⊢ (𝑅 ∈ NzRing → (algSc‘𝑃):(Base‘𝑅)⟶(Base‘𝑃)) |
10 | eqid 2733 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
11 | 6, 10 | ringidcl 20077 | . . . . 5 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅)) |
12 | 1, 11 | syl 17 | . . . 4 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ∈ (Base‘𝑅)) |
13 | 9, 12 | ffvelcdmd 7085 | . . 3 ⊢ (𝑅 ∈ NzRing → ((algSc‘𝑃)‘(1r‘𝑅)) ∈ (Base‘𝑃)) |
14 | eqid 2733 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
15 | 10, 14 | nzrnz 20287 | . . . 4 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ≠ (0g‘𝑅)) |
16 | eqid 2733 | . . . . 5 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
17 | 2, 5, 14, 16, 6 | ply1scln0 21806 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ∈ (Base‘𝑅) ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → ((algSc‘𝑃)‘(1r‘𝑅)) ≠ (0g‘𝑃)) |
18 | 1, 12, 15, 17 | syl3anc 1372 | . . 3 ⊢ (𝑅 ∈ NzRing → ((algSc‘𝑃)‘(1r‘𝑅)) ≠ (0g‘𝑃)) |
19 | eldifsn 4790 | . . 3 ⊢ (((algSc‘𝑃)‘(1r‘𝑅)) ∈ ((Base‘𝑃) ∖ {(0g‘𝑃)}) ↔ (((algSc‘𝑃)‘(1r‘𝑅)) ∈ (Base‘𝑃) ∧ ((algSc‘𝑃)‘(1r‘𝑅)) ≠ (0g‘𝑃))) | |
20 | 13, 18, 19 | sylanbrc 584 | . 2 ⊢ (𝑅 ∈ NzRing → ((algSc‘𝑃)‘(1r‘𝑅)) ∈ ((Base‘𝑃) ∖ {(0g‘𝑃)})) |
21 | 16, 7 | ringelnzr 20293 | . 2 ⊢ ((𝑃 ∈ Ring ∧ ((algSc‘𝑃)‘(1r‘𝑅)) ∈ ((Base‘𝑃) ∖ {(0g‘𝑃)})) → 𝑃 ∈ NzRing) |
22 | 4, 20, 21 | syl2anc 585 | 1 ⊢ (𝑅 ∈ NzRing → 𝑃 ∈ NzRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∖ cdif 3945 {csn 4628 ⟶wf 6537 ‘cfv 6541 Basecbs 17141 0gc0g 17382 1rcur 19999 Ringcrg 20050 NzRingcnzr 20284 algSccascl 21399 Poly1cpl1 21693 |
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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 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 |
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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-ofr 7668 df-om 7853 df-1st 7972 df-2nd 7973 df-supp 8144 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-sup 9434 df-oi 9502 df-card 9931 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 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-sca 17210 df-vsca 17211 df-ip 17212 df-tset 17213 df-ple 17214 df-ds 17216 df-hom 17218 df-cco 17219 df-0g 17384 df-gsum 17385 df-prds 17390 df-pws 17392 df-mre 17527 df-mrc 17528 df-acs 17530 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-mhm 18668 df-submnd 18669 df-grp 18819 df-minusg 18820 df-sbg 18821 df-mulg 18946 df-subg 18998 df-ghm 19085 df-cntz 19176 df-cmn 19645 df-abl 19646 df-mgp 19983 df-ur 20000 df-ring 20052 df-nzr 20285 df-subrg 20354 df-lmod 20466 df-lss 20536 df-ascl 21402 df-psr 21454 df-mvr 21455 df-mpl 21456 df-opsr 21458 df-psr1 21696 df-vr1 21697 df-ply1 21698 df-coe1 21699 |
This theorem is referenced by: ply1nzb 25632 ply1domn 25633 algextdeglem1 32761 mon1pid 41933 mon1psubm 41934 |
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