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Theorem psrcrng 21092

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 19710	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5	3, 4	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
6	1, 2, 5	psrring 21090	. 2 ⊢ (𝜑 → 𝑆 ∈ Ring)
7		eqid 2738	. . . . 5 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
8		eqid 2738	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
9	7, 8	mgpbas 19641	. . . 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 19639	. . . 4 ⊢ (._r‘𝑆) = (+_g‘(mulGrp‘𝑆))
13	12	a1i 11	. . 3 ⊢ (𝜑 → (._r‘𝑆) = (+_g‘(mulGrp‘𝑆)))
14	7	ringmgp 19704	. . . 4 ⊢ (𝑆 ∈ Ring → (mulGrp‘𝑆) ∈ Mnd)
15	6, 14	syl 17	. . 3 ⊢ (𝜑 → (mulGrp‘𝑆) ∈ Mnd)
16	2	3ad2ant1 1131	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝐼 ∈ 𝑉)
17	5	3ad2ant1 1131	. . . 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 1135	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
20		simp3 1136	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆))
21	3	3ad2ant1 1131	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ CRing)
22	1, 16, 17, 18, 11, 8, 19, 20, 21	psrcom 21088	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(._r‘𝑆)𝑦) = (𝑦(._r‘𝑆)𝑥))
23	10, 13, 15, 22	iscmnd 19314	. 2 ⊢ (𝜑 → (mulGrp‘𝑆) ∈ CMnd)
24	7	iscrng 19705	. 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 1085 = wceq 1539 ∈ wcel 2108 {crab 3067 ^◡ccnv 5579 “ cima 5583 ‘cfv 6418 (class class class)co 7255 ↑_m cmap 8573 Fincfn 8691 ℕcn 11903 ℕ₀cn0 12163 Basecbs 16840 +_gcplusg 16888 ._rcmulr 16889 Mndcmnd 18300 CMndccmn 19301 mulGrpcmgp 19635 Ringcrg 19698 CRingccrg 19699 mPwSer cmps 21017

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-ofr 7512 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-fzo 13312 df-seq 13650 df-hash 13973 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-tset 16907 df-0g 17069 df-gsum 17070 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-submnd 18346 df-grp 18495 df-minusg 18496 df-mulg 18616 df-ghm 18747 df-cntz 18838 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-psr 21022

This theorem is referenced by: mplcrng 21136 opsrcrng 21176

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