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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 | ⊢ 𝑃 = (Poly1‘𝑅) |
ply1val.2 | ⊢ 𝑆 = (PwSer1‘𝑅) |
ply1bas.u | ⊢ 𝑈 = (Base‘𝑃) |
Ref | Expression |
---|---|
ply1subrg | ⊢ (𝑅 ∈ Ring → 𝑈 ∈ (SubRing‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . 3 ⊢ (1o mPwSer 𝑅) = (1o mPwSer 𝑅) | |
2 | eqid 2738 | . . 3 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
3 | ply1val.1 | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
4 | ply1val.2 | . . . 4 ⊢ 𝑆 = (PwSer1‘𝑅) | |
5 | ply1bas.u | . . . 4 ⊢ 𝑈 = (Base‘𝑃) | |
6 | 3, 4, 5 | ply1bas 20963 | . . 3 ⊢ 𝑈 = (Base‘(1o mPoly 𝑅)) |
7 | 1on 8131 | . . . 4 ⊢ 1o ∈ On | |
8 | 7 | a1i 11 | . . 3 ⊢ (𝑅 ∈ Ring → 1o ∈ On) |
9 | id 22 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Ring) | |
10 | 1, 2, 6, 8, 9 | mplsubrg 20814 | . 2 ⊢ (𝑅 ∈ Ring → 𝑈 ∈ (SubRing‘(1o mPwSer 𝑅))) |
11 | eqidd 2739 | . . 3 ⊢ (𝑅 ∈ Ring → (Base‘(1o mPwSer 𝑅)) = (Base‘(1o mPwSer 𝑅))) | |
12 | 4 | psr1val 20954 | . . . 4 ⊢ 𝑆 = ((1o ordPwSer 𝑅)‘∅) |
13 | 0ss 4282 | . . . . 5 ⊢ ∅ ⊆ (1o × 1o) | |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝑅 ∈ Ring → ∅ ⊆ (1o × 1o)) |
15 | 1, 12, 14 | opsrbas 20854 | . . 3 ⊢ (𝑅 ∈ Ring → (Base‘(1o mPwSer 𝑅)) = (Base‘𝑆)) |
16 | 1, 12, 14 | opsrplusg 20855 | . . . 4 ⊢ (𝑅 ∈ Ring → (+g‘(1o mPwSer 𝑅)) = (+g‘𝑆)) |
17 | 16 | oveqdr 7192 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘(1o mPwSer 𝑅)) ∧ 𝑦 ∈ (Base‘(1o mPwSer 𝑅)))) → (𝑥(+g‘(1o mPwSer 𝑅))𝑦) = (𝑥(+g‘𝑆)𝑦)) |
18 | 1, 12, 14 | opsrmulr 20856 | . . . 4 ⊢ (𝑅 ∈ Ring → (.r‘(1o mPwSer 𝑅)) = (.r‘𝑆)) |
19 | 18 | oveqdr 7192 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘(1o mPwSer 𝑅)) ∧ 𝑦 ∈ (Base‘(1o mPwSer 𝑅)))) → (𝑥(.r‘(1o mPwSer 𝑅))𝑦) = (𝑥(.r‘𝑆)𝑦)) |
20 | 11, 15, 17, 19 | subrgpropd 19682 | . 2 ⊢ (𝑅 ∈ Ring → (SubRing‘(1o mPwSer 𝑅)) = (SubRing‘𝑆)) |
21 | 10, 20 | eleqtrd 2835 | 1 ⊢ (𝑅 ∈ Ring → 𝑈 ∈ (SubRing‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2113 ⊆ wss 3841 ∅c0 4209 × cxp 5517 Oncon0 6166 ‘cfv 6333 (class class class)co 7164 1oc1o 8117 Basecbs 16579 +gcplusg 16661 .rcmulr 16662 Ringcrg 19409 SubRingcsubrg 19643 mPwSer cmps 20710 mPoly cmpl 20712 PwSer1cps1 20943 Poly1cpl1 20945 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-iin 4881 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-isom 6342 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-of 7419 df-ofr 7420 df-om 7594 df-1st 7707 df-2nd 7708 df-supp 7850 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-er 8313 df-map 8432 df-pm 8433 df-ixp 8501 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-fsupp 8900 df-oi 9040 df-card 9434 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-7 11777 df-8 11778 df-9 11779 df-n0 11970 df-z 12056 df-dec 12173 df-uz 12318 df-fz 12975 df-fzo 13118 df-seq 13454 df-hash 13776 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-mulr 16675 df-sca 16677 df-vsca 16678 df-tset 16680 df-ple 16681 df-0g 16811 df-gsum 16812 df-mre 16953 df-mrc 16954 df-acs 16956 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-mhm 18065 df-submnd 18066 df-grp 18215 df-minusg 18216 df-mulg 18336 df-subg 18387 df-ghm 18467 df-cntz 18558 df-cmn 19019 df-abl 19020 df-mgp 19352 df-ur 19364 df-ring 19411 df-subrg 19645 df-psr 20715 df-mpl 20717 df-opsr 20719 df-psr1 20948 df-ply1 20950 |
This theorem is referenced by: ply1crng 20966 ply1assa 20967 ply1ring 21016 |
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