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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 2737 | . . 3 β’ (1o mPwSer π ) = (1o mPwSer π ) | |
2 | eqid 2737 | . . 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 8391 | . . . 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 2738 | . . 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 4354 | . . . . 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 7377 | . . 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 7377 | . . 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 2840 | 1 β’ (π β Ring β π β (SubRingβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 β wss 3908 β c0 4280 Γ cxp 5628 Oncon0 6313 βcfv 6491 (class class class)co 7349 1oc1o 8372 Basecbs 17017 +gcplusg 17067 .rcmulr 17068 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 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 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7662 ax-cnex 11040 ax-resscn 11041 ax-1cn 11042 ax-icn 11043 ax-addcl 11044 ax-addrcl 11045 ax-mulcl 11046 ax-mulrcl 11047 ax-mulcom 11048 ax-addass 11049 ax-mulass 11050 ax-distr 11051 ax-i2m1 11052 ax-1ne0 11053 ax-1rid 11054 ax-rnegex 11055 ax-rrecex 11056 ax-cnre 11057 ax-pre-lttri 11058 ax-pre-lttrn 11059 ax-pre-ltadd 11060 ax-pre-mulgt0 11061 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5528 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5585 df-se 5586 df-we 5587 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-rn 5641 df-res 5642 df-ima 5643 df-pred 6249 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-isom 6500 df-riota 7305 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7607 df-ofr 7608 df-om 7793 df-1st 7911 df-2nd 7912 df-supp 8060 df-frecs 8179 df-wrecs 8210 df-recs 8284 df-rdg 8323 df-1o 8379 df-er 8581 df-map 8700 df-pm 8701 df-ixp 8769 df-en 8817 df-dom 8818 df-sdom 8819 df-fin 8820 df-fsupp 9239 df-oi 9379 df-card 9808 df-pnf 11124 df-mnf 11125 df-xr 11126 df-ltxr 11127 df-le 11128 df-sub 11320 df-neg 11321 df-nn 12087 df-2 12149 df-3 12150 df-4 12151 df-5 12152 df-6 12153 df-7 12154 df-8 12155 df-9 12156 df-n0 12347 df-z 12433 df-dec 12551 df-uz 12696 df-fz 13353 df-fzo 13496 df-seq 13835 df-hash 14158 df-struct 16953 df-sets 16970 df-slot 16988 df-ndx 17000 df-base 17018 df-ress 17047 df-plusg 17080 df-mulr 17081 df-sca 17083 df-vsca 17084 df-tset 17086 df-ple 17087 df-0g 17257 df-gsum 17258 df-mre 17400 df-mrc 17401 df-acs 17403 df-mgm 18431 df-sgrp 18480 df-mnd 18491 df-mhm 18535 df-submnd 18536 df-grp 18685 df-minusg 18686 df-mulg 18806 df-subg 18857 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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