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Theorem psrring 21937

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 2738	. 2 ⊢ (𝜑 → (Base‘𝑆) = (Base‘𝑆))
2		eqidd 2738	. 2 ⊢ (𝜑 → (+_g‘𝑆) = (+_g‘𝑆))
3		eqidd 2738	. 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		ringgrp 20185	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
8	6, 7	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ∈ Grp)
9	4, 5, 8	psrgrp 21924	. 2 ⊢ (𝜑 → 𝑆 ∈ Grp)
10		eqid 2737	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
11		eqid 2737	. . 3 ⊢ (._r‘𝑆) = (._r‘𝑆)
12	6	3ad2ant1 1134	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring)
13		simp2 1138	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
14		simp3 1139	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆))
15	4, 10, 11, 12, 13, 14	psrmulcl 21914	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(._r‘𝑆)𝑦) ∈ (Base‘𝑆))
16	5	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝐼 ∈ 𝑉)
17	6	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Ring)
18		eqid 2737	. . 3 ⊢ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
19		simpr1 1196	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑆))
20		simpr2 1197	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
21		simpr3 1198	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
22	4, 16, 17, 18, 11, 10, 19, 20, 21	psrass1 21931	. 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)))
23		eqid 2737	. . 3 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
24	4, 16, 17, 18, 11, 10, 19, 20, 21, 23	psrdi 21932	. 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)))
25	4, 16, 17, 18, 11, 10, 19, 20, 21, 23	psrdir 21933	. 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)))
26		eqid 2737	. . 3 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
27		eqid 2737	. . 3 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
28		eqid 2737	. . 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)))
29	4, 5, 6, 18, 26, 27, 28, 10	psr1cl 21928	. 2 ⊢ (𝜑 → (𝑟 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑟 = (𝐼 × {0}), (1_r‘𝑅), (0_g‘𝑅))) ∈ (Base‘𝑆))
30	5	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝐼 ∈ 𝑉)
31	6	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring)
32		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
33	4, 30, 31, 18, 26, 27, 28, 10, 11, 32	psrlidm 21929	. 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)
34	4, 30, 31, 18, 26, 27, 28, 10, 11, 32	psrridm 21930	. 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)
35	1, 2, 3, 9, 15, 22, 24, 25, 29, 33, 34	isringd 20238	1 ⊢ (𝜑 → 𝑆 ∈ Ring)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 {crab 3401 ifcif 4481 {csn 4582 ↦ cmpt 5181 × cxp 5630 ^◡ccnv 5631 “ cima 5635 ‘cfv 6500 (class class class)co 7368 ↑_m cmap 8775 Fincfn 8895 0cc0 11038 ℕcn 12157 ℕ₀cn0 12413 Basecbs 17148 +_gcplusg 17189 ._rcmulr 17190 0_gc0g 17371 Grpcgrp 18875 1_rcur 20128 Ringcrg 20180 mPwSer cmps 21872

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-ofr 7633 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-sup 9357 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-fz 13436 df-fzo 13583 df-seq 13937 df-hash 14266 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-hom 17213 df-cco 17214 df-0g 17373 df-gsum 17374 df-prds 17379 df-pws 17381 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-mhm 18720 df-submnd 18721 df-grp 18878 df-minusg 18879 df-mulg 19010 df-ghm 19154 df-cntz 19258 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-psr 21877

This theorem is referenced by: psr1 21938 psrcrng 21939 psrassa 21940 subrgpsr 21945 psrascl 21946 psrasclcl 21947 mplsubrg 21972 psdascl 22123 opsrring 22197 psrgsum 33724 mplmonprod 33730 rhmpsr 42914

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