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Theorem psrcrng 20187

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 19302	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5	3, 4	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
6	1, 2, 5	psrring 20185	. 2 ⊢ (𝜑 → 𝑆 ∈ Ring)
7		eqid 2821	. . . . 5 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
8		eqid 2821	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
9	7, 8	mgpbas 19239	. . . 4 ⊢ (Base‘𝑆) = (Base‘(mulGrp‘𝑆))
10	9	a1i 11	. . 3 ⊢ (𝜑 → (Base‘𝑆) = (Base‘(mulGrp‘𝑆)))
11		eqid 2821	. . . . 5 ⊢ (._r‘𝑆) = (._r‘𝑆)
12	7, 11	mgpplusg 19237	. . . 4 ⊢ (._r‘𝑆) = (+_g‘(mulGrp‘𝑆))
13	12	a1i 11	. . 3 ⊢ (𝜑 → (._r‘𝑆) = (+_g‘(mulGrp‘𝑆)))
14	7	ringmgp 19297	. . . 4 ⊢ (𝑆 ∈ Ring → (mulGrp‘𝑆) ∈ Mnd)
15	6, 14	syl 17	. . 3 ⊢ (𝜑 → (mulGrp‘𝑆) ∈ Mnd)
16	2	3ad2ant1 1129	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝐼 ∈ 𝑉)
17	5	3ad2ant1 1129	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring)
18		eqid 2821	. . . 4 ⊢ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
19		simp2 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
20		simp3 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆))
21	3	3ad2ant1 1129	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ CRing)
22	1, 16, 17, 18, 11, 8, 19, 20, 21	psrcom 20183	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(._r‘𝑆)𝑦) = (𝑦(._r‘𝑆)𝑥))
23	10, 13, 15, 22	iscmnd 18913	. 2 ⊢ (𝜑 → (mulGrp‘𝑆) ∈ CMnd)
24	7	iscrng 19298	. 2 ⊢ (𝑆 ∈ CRing ↔ (𝑆 ∈ Ring ∧ (mulGrp‘𝑆) ∈ CMnd))
25	6, 23, 24	sylanbrc 585	1 ⊢ (𝜑 → 𝑆 ∈ CRing)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 {crab 3142 ^◡ccnv 5549 “ cima 5553 ‘cfv 6350 (class class class)co 7150 ↑_m cmap 8400 Fincfn 8503 ℕcn 11632 ℕ₀cn0 11891 Basecbs 16477 +_gcplusg 16559 ._rcmulr 16560 Mndcmnd 17905 CMndccmn 18900 mulGrpcmgp 19233 Ringcrg 19291 CRingccrg 19292 mPwSer cmps 20125

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-ofr 7404 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-tset 16578 df-0g 16709 df-gsum 16710 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-grp 18100 df-minusg 18101 df-mulg 18219 df-ghm 18350 df-cntz 18441 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-psr 20130

This theorem is referenced by: mplcrng 20228 opsrcrng 20262

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