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Theorem psrcrng 22015

Description: The ring of power series is commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)

**Hypotheses**
Ref	Expression
psrcnrg.s	⊢ 𝑆 = (𝐼 mPwSer 𝑅)
psrcnrg.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
psrcnrg.r	⊢ (𝜑 → 𝑅 ∈ CRing)

**Assertion**
Ref	Expression
psrcrng	⊢ (𝜑 → 𝑆 ∈ CRing)

Proof of Theorem psrcrng Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.

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 20272	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5	3, 4	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
6	1, 2, 5	psrring 22013	. 2 ⊢ (𝜑 → 𝑆 ∈ Ring)
7		eqid 2740	. . . . 5 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
8		eqid 2740	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
9	7, 8	mgpbas 20167	. . . 4 ⊢ (Base‘𝑆) = (Base‘(mulGrp‘𝑆))
10	9	a1i 11	. . 3 ⊢ (𝜑 → (Base‘𝑆) = (Base‘(mulGrp‘𝑆)))
11		eqid 2740	. . . . 5 ⊢ (._r‘𝑆) = (._r‘𝑆)
12	7, 11	mgpplusg 20165	. . . 4 ⊢ (._r‘𝑆) = (+_g‘(mulGrp‘𝑆))
13	12	a1i 11	. . 3 ⊢ (𝜑 → (._r‘𝑆) = (+_g‘(mulGrp‘𝑆)))
14	7	ringmgp 20266	. . . 4 ⊢ (𝑆 ∈ Ring → (mulGrp‘𝑆) ∈ Mnd)
15	6, 14	syl 17	. . 3 ⊢ (𝜑 → (mulGrp‘𝑆) ∈ Mnd)
16	2	3ad2ant1 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝐼 ∈ 𝑉)
17	5	3ad2ant1 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring)
18		eqid 2740	. . . 4 ⊢ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
19		simp2 1137	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
20		simp3 1138	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆))
21	3	3ad2ant1 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ CRing)
22	1, 16, 17, 18, 11, 8, 19, 20, 21	psrcom 22011	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(._r‘𝑆)𝑦) = (𝑦(._r‘𝑆)𝑥))
23	10, 13, 15, 22	iscmnd 19836	. 2 ⊢ (𝜑 → (mulGrp‘𝑆) ∈ CMnd)
24	7	iscrng 20267	. 2 ⊢ (𝑆 ∈ CRing ↔ (𝑆 ∈ Ring ∧ (mulGrp‘𝑆) ∈ CMnd))
25	6, 23, 24	sylanbrc 582	1 ⊢ (𝜑 → 𝑆 ∈ CRing)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 {crab 3443 ^◡ccnv 5699 “ cima 5703 ‘cfv 6573 (class class class)co 7448 ↑_m cmap 8884 Fincfn 9003 ℕcn 12293 ℕ₀cn0 12553 Basecbs 17258 +_gcplusg 17311 ._rcmulr 17312 Mndcmnd 18772 CMndccmn 19822 mulGrpcmgp 20161 Ringcrg 20260 CRingccrg 20261 mPwSer cmps 21947

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-ofr 7715 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-sup 9511 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-fz 13568 df-fzo 13712 df-seq 14053 df-hash 14380 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-hom 17335 df-cco 17336 df-0g 17501 df-gsum 17502 df-prds 17507 df-pws 17509 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-mulg 19108 df-ghm 19253 df-cntz 19357 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-cring 20263 df-psr 21952

This theorem is referenced by: mplcrng 22064 opsrcrng 22106 psd1 22194

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