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Mirrors > Home > MPE Home > Th. List > psrcrng | Structured version Visualization version GIF version |
Description: The ring of power series is commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.) |
Ref | Expression |
---|---|
psrcnrg.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
psrcnrg.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
psrcnrg.r | ⊢ (𝜑 → 𝑅 ∈ CRing) |
Ref | Expression |
---|---|
psrcrng | ⊢ (𝜑 → 𝑆 ∈ CRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psrcnrg.s | . . 3 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
2 | psrcnrg.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
3 | psrcnrg.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
4 | crngring 19421 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) |
6 | 1, 2, 5 | psrring 20783 | . 2 ⊢ (𝜑 → 𝑆 ∈ Ring) |
7 | eqid 2738 | . . . . 5 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆) | |
8 | eqid 2738 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
9 | 7, 8 | mgpbas 19357 | . . . 4 ⊢ (Base‘𝑆) = (Base‘(mulGrp‘𝑆)) |
10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → (Base‘𝑆) = (Base‘(mulGrp‘𝑆))) |
11 | eqid 2738 | . . . . 5 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
12 | 7, 11 | mgpplusg 19355 | . . . 4 ⊢ (.r‘𝑆) = (+g‘(mulGrp‘𝑆)) |
13 | 12 | a1i 11 | . . 3 ⊢ (𝜑 → (.r‘𝑆) = (+g‘(mulGrp‘𝑆))) |
14 | 7 | ringmgp 19415 | . . . 4 ⊢ (𝑆 ∈ Ring → (mulGrp‘𝑆) ∈ Mnd) |
15 | 6, 14 | syl 17 | . . 3 ⊢ (𝜑 → (mulGrp‘𝑆) ∈ Mnd) |
16 | 2 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝐼 ∈ 𝑉) |
17 | 5 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring) |
18 | eqid 2738 | . . . 4 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
19 | simp2 1138 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆)) | |
20 | simp3 1139 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆)) | |
21 | 3 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ CRing) |
22 | 1, 16, 17, 18, 11, 8, 19, 20, 21 | psrcom 20781 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(.r‘𝑆)𝑦) = (𝑦(.r‘𝑆)𝑥)) |
23 | 10, 13, 15, 22 | iscmnd 19030 | . 2 ⊢ (𝜑 → (mulGrp‘𝑆) ∈ CMnd) |
24 | 7 | iscrng 19416 | . 2 ⊢ (𝑆 ∈ CRing ↔ (𝑆 ∈ Ring ∧ (mulGrp‘𝑆) ∈ CMnd)) |
25 | 6, 23, 24 | sylanbrc 586 | 1 ⊢ (𝜑 → 𝑆 ∈ CRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2113 {crab 3057 ◡ccnv 5518 “ cima 5522 ‘cfv 6333 (class class class)co 7164 ↑m cmap 8430 Fincfn 8548 ℕcn 11709 ℕ0cn0 11969 Basecbs 16579 +gcplusg 16661 .rcmulr 16662 Mndcmnd 18020 CMndccmn 19017 mulGrpcmgp 19351 Ringcrg 19409 CRingccrg 19410 mPwSer cmps 20710 |
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-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-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-ghm 18467 df-cntz 18558 df-cmn 19019 df-abl 19020 df-mgp 19352 df-ur 19364 df-ring 19411 df-cring 19412 df-psr 20715 |
This theorem is referenced by: mplcrng 20829 opsrcrng 20863 |
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