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Theorem psrring 21957

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 20208	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
8	6, 7	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ∈ Grp)
9	4, 5, 8	psrgrp 21944	. 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 21934	. 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 21951	. 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 21952	. 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 21953	. 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 21948	. 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 21949	. 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 21950	. 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 20261	1 ⊢ (𝜑 → 𝑆 ∈ Ring)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 {crab 3390 ifcif 4467 {csn 4568 ↦ cmpt 5167 × cxp 5620 ^◡ccnv 5621 “ cima 5625 ‘cfv 6490 (class class class)co 7358 ↑_m cmap 8764 Fincfn 8884 0cc0 11027 ℕcn 12163 ℕ₀cn0 12426 Basecbs 17168 +_gcplusg 17209 ._rcmulr 17210 0_gc0g 17391 Grpcgrp 18898 1_rcur 20151 Ringcrg 20203 mPwSer cmps 21892

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 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104

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 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-ofr 7623 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-er 8634 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-sup 9346 df-oi 9416 df-card 9852 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-7 12238 df-8 12239 df-9 12240 df-n0 12427 df-z 12514 df-dec 12634 df-uz 12778 df-fz 13451 df-fzo 13598 df-seq 13953 df-hash 14282 df-struct 17106 df-sets 17123 df-slot 17141 df-ndx 17153 df-base 17169 df-ress 17190 df-plusg 17222 df-mulr 17223 df-sca 17225 df-vsca 17226 df-ip 17227 df-tset 17228 df-ple 17229 df-ds 17231 df-hom 17233 df-cco 17234 df-0g 17393 df-gsum 17394 df-prds 17399 df-pws 17401 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18740 df-submnd 18741 df-grp 18901 df-minusg 18902 df-mulg 19033 df-ghm 19177 df-cntz 19281 df-cmn 19746 df-abl 19747 df-mgp 20111 df-rng 20123 df-ur 20152 df-ring 20205 df-psr 21897

This theorem is referenced by: psr1 21958 psrcrng 21959 psrassa 21960 subrgpsr 21965 psrascl 21966 psrasclcl 21967 mplsubrg 21992 psdascl 22143 opsrring 22217 psrgsum 33712 mplmonprod 33718 rhmpsr 43006

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