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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 19237 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) |
6 | 1, 2, 5 | psrring 20119 | . 2 ⊢ (𝜑 → 𝑆 ∈ Ring) |
7 | eqid 2818 | . . . . 5 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆) | |
8 | eqid 2818 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
9 | 7, 8 | mgpbas 19174 | . . . 4 ⊢ (Base‘𝑆) = (Base‘(mulGrp‘𝑆)) |
10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → (Base‘𝑆) = (Base‘(mulGrp‘𝑆))) |
11 | eqid 2818 | . . . . 5 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
12 | 7, 11 | mgpplusg 19172 | . . . 4 ⊢ (.r‘𝑆) = (+g‘(mulGrp‘𝑆)) |
13 | 12 | a1i 11 | . . 3 ⊢ (𝜑 → (.r‘𝑆) = (+g‘(mulGrp‘𝑆))) |
14 | 7 | ringmgp 19232 | . . . 4 ⊢ (𝑆 ∈ Ring → (mulGrp‘𝑆) ∈ Mnd) |
15 | 6, 14 | syl 17 | . . 3 ⊢ (𝜑 → (mulGrp‘𝑆) ∈ Mnd) |
16 | 2 | 3ad2ant1 1125 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝐼 ∈ 𝑉) |
17 | 5 | 3ad2ant1 1125 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring) |
18 | eqid 2818 | . . . 4 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
19 | simp2 1129 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆)) | |
20 | simp3 1130 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆)) | |
21 | 3 | 3ad2ant1 1125 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ CRing) |
22 | 1, 16, 17, 18, 11, 8, 19, 20, 21 | psrcom 20117 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(.r‘𝑆)𝑦) = (𝑦(.r‘𝑆)𝑥)) |
23 | 10, 13, 15, 22 | iscmnd 18848 | . 2 ⊢ (𝜑 → (mulGrp‘𝑆) ∈ CMnd) |
24 | 7 | iscrng 19233 | . 2 ⊢ (𝑆 ∈ CRing ↔ (𝑆 ∈ Ring ∧ (mulGrp‘𝑆) ∈ CMnd)) |
25 | 6, 23, 24 | sylanbrc 583 | 1 ⊢ (𝜑 → 𝑆 ∈ CRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 {crab 3139 ◡ccnv 5547 “ cima 5551 ‘cfv 6348 (class class class)co 7145 ↑m cmap 8395 Fincfn 8497 ℕcn 11626 ℕ0cn0 11885 Basecbs 16471 +gcplusg 16553 .rcmulr 16554 Mndcmnd 17899 CMndccmn 18835 mulGrpcmgp 19168 Ringcrg 19226 CRingccrg 19227 mPwSer cmps 20059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-ofr 7399 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-seq 13358 df-hash 13679 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-tset 16572 df-0g 16703 df-gsum 16704 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-submnd 17945 df-grp 18044 df-minusg 18045 df-mulg 18163 df-ghm 18294 df-cntz 18385 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-psr 20064 |
This theorem is referenced by: mplcrng 20162 opsrcrng 20196 |
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