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Mirrors > Home > MPE Home > Th. List > ply1subrg | Structured version Visualization version GIF version |
Description: Univariate polynomials form a subring of the set of univariate power series. (Contributed by Mario Carneiro, 9-Feb-2015.) |
Ref | Expression |
---|---|
ply1val.1 | β’ π = (Poly1βπ ) |
ply1val.2 | β’ π = (PwSer1βπ ) |
ply1bas.u | β’ π = (Baseβπ) |
Ref | Expression |
---|---|
ply1subrg | β’ (π β Ring β π β (SubRingβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . 3 β’ (1o mPwSer π ) = (1o mPwSer π ) | |
2 | eqid 2738 | . . 3 β’ (1o mPoly π ) = (1o mPoly π ) | |
3 | ply1val.1 | . . . 4 β’ π = (Poly1βπ ) | |
4 | ply1val.2 | . . . 4 β’ π = (PwSer1βπ ) | |
5 | ply1bas.u | . . . 4 β’ π = (Baseβπ) | |
6 | 3, 4, 5 | ply1bas 21488 | . . 3 β’ π = (Baseβ(1o mPoly π )) |
7 | 1on 8392 | . . . 4 β’ 1o β On | |
8 | 7 | a1i 11 | . . 3 β’ (π β Ring β 1o β On) |
9 | id 22 | . . 3 β’ (π β Ring β π β Ring) | |
10 | 1, 2, 6, 8, 9 | mplsubrg 21333 | . 2 β’ (π β Ring β π β (SubRingβ(1o mPwSer π ))) |
11 | eqidd 2739 | . . 3 β’ (π β Ring β (Baseβ(1o mPwSer π )) = (Baseβ(1o mPwSer π ))) | |
12 | 4 | psr1val 21479 | . . . 4 β’ π = ((1o ordPwSer π )ββ ) |
13 | 0ss 4355 | . . . . 5 β’ β β (1o Γ 1o) | |
14 | 13 | a1i 11 | . . . 4 β’ (π β Ring β β β (1o Γ 1o)) |
15 | 1, 12, 14 | opsrbas 21374 | . . 3 β’ (π β Ring β (Baseβ(1o mPwSer π )) = (Baseβπ)) |
16 | 1, 12, 14 | opsrplusg 21376 | . . . 4 β’ (π β Ring β (+gβ(1o mPwSer π )) = (+gβπ)) |
17 | 16 | oveqdr 7378 | . . 3 β’ ((π β Ring β§ (π₯ β (Baseβ(1o mPwSer π )) β§ π¦ β (Baseβ(1o mPwSer π )))) β (π₯(+gβ(1o mPwSer π ))π¦) = (π₯(+gβπ)π¦)) |
18 | 1, 12, 14 | opsrmulr 21378 | . . . 4 β’ (π β Ring β (.rβ(1o mPwSer π )) = (.rβπ)) |
19 | 18 | oveqdr 7378 | . . 3 β’ ((π β Ring β§ (π₯ β (Baseβ(1o mPwSer π )) β§ π¦ β (Baseβ(1o mPwSer π )))) β (π₯(.rβ(1o mPwSer π ))π¦) = (π₯(.rβπ)π¦)) |
20 | 11, 15, 17, 19 | subrgpropd 20180 | . 2 β’ (π β Ring β (SubRingβ(1o mPwSer π )) = (SubRingβπ)) |
21 | 10, 20 | eleqtrd 2841 | 1 β’ (π β Ring β π β (SubRingβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wss 3909 β c0 4281 Γ cxp 5629 Oncon0 6314 βcfv 6492 (class class class)co 7350 1oc1o 8373 Basecbs 17018 +gcplusg 17068 .rcmulr 17069 Ringcrg 19888 SubRingcsubrg 20141 mPwSer cmps 21229 mPoly cmpl 21231 PwSer1cps1 21468 Poly1cpl1 21470 |
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 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-of 7608 df-ofr 7609 df-om 7794 df-1st 7912 df-2nd 7913 df-supp 8061 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-er 8582 df-map 8701 df-pm 8702 df-ixp 8770 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-fsupp 9240 df-oi 9380 df-card 9809 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-5 12153 df-6 12154 df-7 12155 df-8 12156 df-9 12157 df-n0 12348 df-z 12434 df-dec 12552 df-uz 12697 df-fz 13354 df-fzo 13497 df-seq 13836 df-hash 14159 df-struct 16954 df-sets 16971 df-slot 16989 df-ndx 17001 df-base 17019 df-ress 17048 df-plusg 17081 df-mulr 17082 df-sca 17084 df-vsca 17085 df-tset 17087 df-ple 17088 df-0g 17258 df-gsum 17259 df-mre 17401 df-mrc 17402 df-acs 17404 df-mgm 18432 df-sgrp 18481 df-mnd 18492 df-mhm 18536 df-submnd 18537 df-grp 18686 df-minusg 18687 df-mulg 18807 df-subg 18858 df-ghm 18938 df-cntz 19029 df-cmn 19493 df-abl 19494 df-mgp 19826 df-ur 19843 df-ring 19890 df-subrg 20143 df-psr 21234 df-mpl 21236 df-opsr 21238 df-psr1 21473 df-ply1 21475 |
This theorem is referenced by: ply1crng 21491 ply1assa 21492 ply1ring 21541 |
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