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Theorem psrring 22282

Description: The ring of power series is a ring. (Contributed by Mario Carneiro, 29-Dec-2014.)

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

**Assertion**
Ref	Expression
psrring	⊢ (𝜑 → 𝑆 ∈ Ring)

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

Step	Hyp	Ref	Expression
1		eqidd 2761	. 2 ⊢ (𝜑 → (Base‘𝑆) = (Base‘𝑆))
2		eqidd 2761	. 2 ⊢ (𝜑 → (+_g‘𝑆) = (+_g‘𝑆))
3		eqidd 2761	. 2 ⊢ (𝜑 → (._r‘𝑆) = (._r‘𝑆))
4		psrring.s	. . 3 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
5		psrring.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
6		psrring.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring)
7	6	ringgrpd 20474	. . 3 ⊢ (𝜑 → 𝑅 ∈ Grp)
8	4, 5, 7	psrgrp 22269	. 2 ⊢ (𝜑 → 𝑆 ∈ Grp)
9		eqid 2760	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
10		eqid 2760	. . 3 ⊢ (._r‘𝑆) = (._r‘𝑆)
11	6	3ad2ant1 1151	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring)
12		simp2 1155	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
13		simp3 1156	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆))
14	4, 9, 10, 11, 12, 13	psrmulcl 22259	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(._r‘𝑆)𝑦) ∈ (Base‘𝑆))
15	5	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝐼 ∈ 𝑉)
16	6	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Ring)
17		eqid 2760	. . 3 ⊢ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
18		simpr1 1213	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑆))
19		simpr2 1214	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
20		simpr3 1215	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
21	4, 15, 16, 17, 10, 9, 18, 19, 20	psrass1 22276	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(._r‘𝑆)𝑦)(._r‘𝑆)𝑧) = (𝑥(._r‘𝑆)(𝑦(._r‘𝑆)𝑧)))
22		eqid 2760	. . 3 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
23	4, 15, 16, 17, 10, 9, 18, 19, 20, 22	psrdi 22277	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(._r‘𝑆)(𝑦(+_g‘𝑆)𝑧)) = ((𝑥(._r‘𝑆)𝑦)(+_g‘𝑆)(𝑥(._r‘𝑆)𝑧)))
24	4, 15, 16, 17, 10, 9, 18, 19, 20, 22	psrdir 22278	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+_g‘𝑆)𝑦)(._r‘𝑆)𝑧) = ((𝑥(._r‘𝑆)𝑧)(+_g‘𝑆)(𝑦(._r‘𝑆)𝑧)))
25		eqid 2760	. . 3 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
26		eqid 2760	. . 3 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
27		eqid 2760	. . 3 ⊢ (𝑟 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑟 = (𝐼 × {0}), (1_r‘𝑅), (0_g‘𝑅))) = (𝑟 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑟 = (𝐼 × {0}), (1_r‘𝑅), (0_g‘𝑅)))
28	4, 5, 6, 17, 25, 26, 27, 9	psr1cl 22273	. 2 ⊢ (𝜑 → (𝑟 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑟 = (𝐼 × {0}), (1_r‘𝑅), (0_g‘𝑅))) ∈ (Base‘𝑆))
29	5	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝐼 ∈ 𝑉)
30	6	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring)
31		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
32	4, 29, 30, 17, 25, 26, 27, 9, 10, 31	psrlidm 22274	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝑟 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑟 = (𝐼 × {0}), (1_r‘𝑅), (0_g‘𝑅)))(._r‘𝑆)𝑥) = 𝑥)
33	4, 29, 30, 17, 25, 26, 27, 9, 10, 31	psrridm 22275	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → (𝑥(._r‘𝑆)(𝑟 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑟 = (𝐼 × {0}), (1_r‘𝑅), (0_g‘𝑅)))) = 𝑥)
34	1, 2, 3, 8, 14, 21, 23, 24, 28, 32, 33	isringd 20527	1 ⊢ (𝜑 → 𝑆 ∈ Ring)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 {crab 3412 ifcif 4481 {csn 4584 ↦ cmpt 5185 × cxp 5648 ^◡ccnv 5649 “ cima 5653 ‘cfv 6538 (class class class)co 7420 ↑_m cmap 8850 Fincfn 8976 0cc0 11203 ℕcn 12338 ℕ₀cn0 12609 Basecbs 17392 +_gcplusg 17433 ._rcmulr 17434 0_gc0g 17615 1_rcur 20412 Ringcrg 20464 mPwSer cmps 22217

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-ofr 7694 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8180 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-1o 8479 df-2o 8480 df-er 8720 df-map 8852 df-pm 8853 df-ixp 8929 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-fsupp 9357 df-sup 9437 df-oi 9507 df-card 10023 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-nn 12339 df-2 12408 df-3 12409 df-4 12410 df-5 12411 df-6 12412 df-7 12413 df-8 12414 df-9 12415 df-n0 12610 df-z 12697 df-dec 12818 df-uz 12969 df-fz 13643 df-fzo 13792 df-seq 14148 df-hash 14478 df-struct 17330 df-sets 17347 df-slot 17365 df-ndx 17377 df-base 17393 df-ress 17414 df-plusg 17446 df-mulr 17447 df-sca 17449 df-vsca 17450 df-ip 17451 df-tset 17452 df-ple 17453 df-ds 17455 df-hom 17457 df-cco 17458 df-0g 17617 df-gsum 17618 df-prds 17623 df-pws 17625 df-mre 17761 df-mrc 17762 df-acs 17764 df-mgm 18821 df-sgrp 18913 df-mnd 18929 df-mhm 18983 df-submnd 18984 df-grp 19152 df-minusg 19153 df-mulg 19283 df-ghm 19433 df-cntz 19536 df-cmn 20001 df-abl 20002 df-mgp 20366 df-rng 20380 df-ur 20413 df-ring 20466 df-psr 22222

This theorem is used by: psr1 22283 psrcrng 22284 psrassa 22285 subrgpsr 22290 psrascl 22291 psrasclcl 22292 mplsubrg 22317 psdascl 22494 opsrring 22567 psrnzr 34190 psrgsum 34226 mplmonprod 34232 rhmpsr 43704

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