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Mirrors > Home > MPE Home > Th. List > mplsubrg | Structured version Visualization version GIF version |
Description: The set of polynomials is closed under multiplication, i.e. it is a subring of the set of power series. (Contributed by Mario Carneiro, 9-Jan-2015.) |
Ref | Expression |
---|---|
mplsubg.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
mplsubg.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mplsubg.u | ⊢ 𝑈 = (Base‘𝑃) |
mplsubg.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
mpllss.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
Ref | Expression |
---|---|
mplsubrg | ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mplsubg.s | . . 3 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
2 | mplsubg.p | . . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
3 | mplsubg.u | . . 3 ⊢ 𝑈 = (Base‘𝑃) | |
4 | mplsubg.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
5 | mpllss.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
6 | ringgrp 20221 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Grp) |
8 | 1, 2, 3, 4, 7 | mplsubg 22011 | . 2 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑆)) |
9 | 1, 4, 5 | psrring 21979 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ Ring) |
10 | eqid 2726 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
11 | eqid 2726 | . . . . 5 ⊢ (1r‘𝑆) = (1r‘𝑆) | |
12 | 10, 11 | ringidcl 20245 | . . . 4 ⊢ (𝑆 ∈ Ring → (1r‘𝑆) ∈ (Base‘𝑆)) |
13 | 9, 12 | syl 17 | . . 3 ⊢ (𝜑 → (1r‘𝑆) ∈ (Base‘𝑆)) |
14 | eqid 2726 | . . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
15 | eqid 2726 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
16 | eqid 2726 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
17 | 1, 4, 5, 14, 15, 16, 11 | psr1 21980 | . . . 4 ⊢ (𝜑 → (1r‘𝑆) = (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑘 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅)))) |
18 | ovex 7457 | . . . . . . . 8 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
19 | 18 | mptrabex 7242 | . . . . . . 7 ⊢ (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑘 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅))) ∈ V |
20 | funmpt 6597 | . . . . . . 7 ⊢ Fun (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑘 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅))) | |
21 | fvex 6914 | . . . . . . 7 ⊢ (0g‘𝑅) ∈ V | |
22 | 19, 20, 21 | 3pm3.2i 1336 | . . . . . 6 ⊢ ((𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑘 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅))) ∈ V ∧ Fun (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑘 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅))) ∧ (0g‘𝑅) ∈ V) |
23 | 22 | a1i 11 | . . . . 5 ⊢ (𝜑 → ((𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑘 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅))) ∈ V ∧ Fun (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑘 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅))) ∧ (0g‘𝑅) ∈ V)) |
24 | snfi 9081 | . . . . . 6 ⊢ {(𝐼 × {0})} ∈ Fin | |
25 | 24 | a1i 11 | . . . . 5 ⊢ (𝜑 → {(𝐼 × {0})} ∈ Fin) |
26 | eldifsni 4799 | . . . . . . . 8 ⊢ (𝑘 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝐼 × {0})}) → 𝑘 ≠ (𝐼 × {0})) | |
27 | 26 | adantl 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝐼 × {0})})) → 𝑘 ≠ (𝐼 × {0})) |
28 | ifnefalse 4545 | . . . . . . 7 ⊢ (𝑘 ≠ (𝐼 × {0}) → if(𝑘 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅)) = (0g‘𝑅)) | |
29 | 27, 28 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝐼 × {0})})) → if(𝑘 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅)) = (0g‘𝑅)) |
30 | 18 | rabex 5339 | . . . . . . 7 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V |
31 | 30 | a1i 11 | . . . . . 6 ⊢ (𝜑 → {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V) |
32 | 29, 31 | suppss2 8215 | . . . . 5 ⊢ (𝜑 → ((𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑘 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅))) supp (0g‘𝑅)) ⊆ {(𝐼 × {0})}) |
33 | suppssfifsupp 9423 | . . . . 5 ⊢ ((((𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑘 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅))) ∈ V ∧ Fun (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑘 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅))) ∧ (0g‘𝑅) ∈ V) ∧ ({(𝐼 × {0})} ∈ Fin ∧ ((𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑘 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅))) supp (0g‘𝑅)) ⊆ {(𝐼 × {0})})) → (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑘 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅))) finSupp (0g‘𝑅)) | |
34 | 23, 25, 32, 33 | syl12anc 835 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑘 = (𝐼 × {0}), (1r‘𝑅), (0g‘𝑅))) finSupp (0g‘𝑅)) |
35 | 17, 34 | eqbrtrd 5175 | . . 3 ⊢ (𝜑 → (1r‘𝑆) finSupp (0g‘𝑅)) |
36 | 2, 1, 10, 15, 3 | mplelbas 22000 | . . 3 ⊢ ((1r‘𝑆) ∈ 𝑈 ↔ ((1r‘𝑆) ∈ (Base‘𝑆) ∧ (1r‘𝑆) finSupp (0g‘𝑅))) |
37 | 13, 35, 36 | sylanbrc 581 | . 2 ⊢ (𝜑 → (1r‘𝑆) ∈ 𝑈) |
38 | 4 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝐼 ∈ 𝑊) |
39 | 5 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑅 ∈ Ring) |
40 | eqid 2726 | . . . 4 ⊢ ( ∘f + “ ((𝑥 supp (0g‘𝑅)) × (𝑦 supp (0g‘𝑅)))) = ( ∘f + “ ((𝑥 supp (0g‘𝑅)) × (𝑦 supp (0g‘𝑅)))) | |
41 | eqid 2726 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
42 | simprl 769 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑥 ∈ 𝑈) | |
43 | simprr 771 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ 𝑈) | |
44 | 1, 2, 3, 38, 39, 14, 15, 40, 41, 42, 43 | mplsubrglem 22013 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑥(.r‘𝑆)𝑦) ∈ 𝑈) |
45 | 44 | ralrimivva 3191 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 (𝑥(.r‘𝑆)𝑦) ∈ 𝑈) |
46 | eqid 2726 | . . . 4 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
47 | 10, 11, 46 | issubrg2 20576 | . . 3 ⊢ (𝑆 ∈ Ring → (𝑈 ∈ (SubRing‘𝑆) ↔ (𝑈 ∈ (SubGrp‘𝑆) ∧ (1r‘𝑆) ∈ 𝑈 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 (𝑥(.r‘𝑆)𝑦) ∈ 𝑈))) |
48 | 9, 47 | syl 17 | . 2 ⊢ (𝜑 → (𝑈 ∈ (SubRing‘𝑆) ↔ (𝑈 ∈ (SubGrp‘𝑆) ∧ (1r‘𝑆) ∈ 𝑈 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 (𝑥(.r‘𝑆)𝑦) ∈ 𝑈))) |
49 | 8, 37, 45, 48 | mpbir3and 1339 | 1 ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∀wral 3051 {crab 3419 Vcvv 3462 ∖ cdif 3944 ⊆ wss 3947 ifcif 4533 {csn 4633 class class class wbr 5153 ↦ cmpt 5236 × cxp 5680 ◡ccnv 5681 “ cima 5685 Fun wfun 6548 ‘cfv 6554 (class class class)co 7424 ∘f cof 7688 supp csupp 8174 ↑m cmap 8855 Fincfn 8974 finSupp cfsupp 9405 0cc0 11158 + caddc 11161 ℕcn 12264 ℕ0cn0 12524 Basecbs 17213 .rcmulr 17267 0gc0g 17454 Grpcgrp 18928 SubGrpcsubg 19114 1rcur 20164 Ringcrg 20216 SubRingcsubrg 20551 mPwSer cmps 21901 mPoly cmpl 21903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-ofr 7691 df-om 7877 df-1st 8003 df-2nd 8004 df-supp 8175 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-map 8857 df-pm 8858 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9406 df-sup 9485 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-uz 12875 df-fz 13539 df-fzo 13682 df-seq 14022 df-hash 14348 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-sca 17282 df-vsca 17283 df-ip 17284 df-tset 17285 df-ple 17286 df-ds 17288 df-hom 17290 df-cco 17291 df-0g 17456 df-gsum 17457 df-prds 17462 df-pws 17464 df-mre 17599 df-mrc 17600 df-acs 17602 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-mhm 18773 df-submnd 18774 df-grp 18931 df-minusg 18932 df-mulg 19062 df-subg 19117 df-ghm 19207 df-cntz 19311 df-cmn 19780 df-abl 19781 df-mgp 20118 df-rng 20136 df-ur 20165 df-ring 20218 df-subrng 20528 df-subrg 20553 df-psr 21906 df-mpl 21908 |
This theorem is referenced by: mpl1 22021 mplring 22028 mplcrng 22030 mplassa 22031 subrgmpl 22039 mplbas2 22049 subrgasclcl 22080 mplind 22083 evlseu 22098 ply1subrg 22187 |
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