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Mirrors > Home > MPE Home > Th. List > mplring | Structured version Visualization version GIF version |
Description: The polynomial ring is a ring. (Contributed by Mario Carneiro, 9-Jan-2015.) |
Ref | Expression |
---|---|
mplgrp.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
Ref | Expression |
---|---|
mplring | ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝑃 ∈ Ring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . 3 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
2 | mplgrp.p | . . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
3 | eqid 2736 | . . 3 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
4 | simpl 483 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝐼 ∈ 𝑉) | |
5 | simpr 485 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Ring) | |
6 | 1, 2, 3, 4, 5 | mplsubrg 21409 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (Base‘𝑃) ∈ (SubRing‘(𝐼 mPwSer 𝑅))) |
7 | 2, 1, 3 | mplval2 21400 | . . 3 ⊢ 𝑃 = ((𝐼 mPwSer 𝑅) ↾s (Base‘𝑃)) |
8 | 7 | subrgring 20223 | . 2 ⊢ ((Base‘𝑃) ∈ (SubRing‘(𝐼 mPwSer 𝑅)) → 𝑃 ∈ Ring) |
9 | 6, 8 | syl 17 | 1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝑃 ∈ Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ‘cfv 6496 (class class class)co 7356 Basecbs 17082 Ringcrg 19962 SubRingcsubrg 20216 mPwSer cmps 21304 mPoly cmpl 21306 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7616 df-ofr 7617 df-om 7802 df-1st 7920 df-2nd 7921 df-supp 8092 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-er 8647 df-map 8766 df-pm 8767 df-ixp 8835 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-fsupp 9305 df-sup 9377 df-oi 9445 df-card 9874 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-5 12218 df-6 12219 df-7 12220 df-8 12221 df-9 12222 df-n0 12413 df-z 12499 df-dec 12618 df-uz 12763 df-fz 13424 df-fzo 13567 df-seq 13906 df-hash 14230 df-struct 17018 df-sets 17035 df-slot 17053 df-ndx 17065 df-base 17083 df-ress 17112 df-plusg 17145 df-mulr 17146 df-sca 17148 df-vsca 17149 df-ip 17150 df-tset 17151 df-ple 17152 df-ds 17154 df-hom 17156 df-cco 17157 df-0g 17322 df-gsum 17323 df-prds 17328 df-pws 17330 df-mre 17465 df-mrc 17466 df-acs 17468 df-mgm 18496 df-sgrp 18545 df-mnd 18556 df-mhm 18600 df-submnd 18601 df-grp 18750 df-minusg 18751 df-mulg 18871 df-subg 18923 df-ghm 19004 df-cntz 19095 df-cmn 19562 df-abl 19563 df-mgp 19895 df-ur 19912 df-ring 19964 df-subrg 20218 df-psr 21309 df-mpl 21311 |
This theorem is referenced by: mplcoe1 21436 mplcoe3 21437 mplcoe5lem 21438 mplcoe5 21439 mplbas2 21441 mplasclf 21471 subrgasclcl 21473 evlslem2 21487 evlslem1 21490 evlseu 21491 evlsgsumadd 21499 mpfconst 21509 mpfsubrg 21511 mpfind 21515 mhpmulcl 21537 mhppwdeg 21538 ply1coe 21665 mdegaddle 25437 mdegmullem 25441 mplringd 40712 mplascl0 40721 selvcllem5 40738 |
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