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Theorem psrassa 21935

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

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

**Assertion**
Ref	Expression
psrassa	⊢ (𝜑 → 𝑆 ∈ AssAlg)

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

Step	Hyp	Ref	Expression
1		eqidd 2726	. 2 ⊢ (𝜑 → (Base‘𝑆) = (Base‘𝑆))
2		psrcnrg.s	. . 3 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
3		psrcnrg.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
4		psrcnrg.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ CRing)
5	2, 3, 4	psrsca 21909	. 2 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑆))
6		eqidd 2726	. 2 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑅))
7		eqidd 2726	. 2 ⊢ (𝜑 → ( ·_𝑠 ‘𝑆) = ( ·_𝑠 ‘𝑆))
8		eqidd 2726	. 2 ⊢ (𝜑 → (._r‘𝑆) = (._r‘𝑆))
9	4	crngringd 20198	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
10	2, 3, 9	psrlmod 21922	. 2 ⊢ (𝜑 → 𝑆 ∈ LMod)
11	2, 3, 9	psrring 21932	. 2 ⊢ (𝜑 → 𝑆 ∈ Ring)
12	3	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝐼 ∈ 𝑉)
13	9	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Ring)
14		eqid 2725	. . . 4 ⊢ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
15		eqid 2725	. . . 4 ⊢ (._r‘𝑆) = (._r‘𝑆)
16		eqid 2725	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
17		simpr2 1192	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
18		simpr3 1193	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
19	4	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ CRing)
20		eqid 2725	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
21		eqid 2725	. . . 4 ⊢ ( ·_𝑠 ‘𝑆) = ( ·_𝑠 ‘𝑆)
22		simpr1 1191	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑅))
23	2, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22	psrass23 21931	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (((𝑥( ·_𝑠 ‘𝑆)𝑦)(._r‘𝑆)𝑧) = (𝑥( ·_𝑠 ‘𝑆)(𝑦(._r‘𝑆)𝑧)) ∧ (𝑦(._r‘𝑆)(𝑥( ·_𝑠 ‘𝑆)𝑧)) = (𝑥( ·_𝑠 ‘𝑆)(𝑦(._r‘𝑆)𝑧))))
24	23	simpld 493	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥( ·_𝑠 ‘𝑆)𝑦)(._r‘𝑆)𝑧) = (𝑥( ·_𝑠 ‘𝑆)(𝑦(._r‘𝑆)𝑧)))
25	23	simprd 494	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(._r‘𝑆)(𝑥( ·_𝑠 ‘𝑆)𝑧)) = (𝑥( ·_𝑠 ‘𝑆)(𝑦(._r‘𝑆)𝑧)))
26	1, 5, 6, 7, 8, 10, 11, 24, 25	isassad 21815	1 ⊢ (𝜑 → 𝑆 ∈ AssAlg)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 {crab 3418 ^◡ccnv 5677 “ cima 5681 ‘cfv 6549 (class class class)co 7419 ↑_m cmap 8845 Fincfn 8964 ℕcn 12245 ℕ₀cn0 12505 Basecbs 17183 ._rcmulr 17237 ·_𝑠 cvsca 17240 Ringcrg 20185 CRingccrg 20186 AssAlgcasa 21801 mPwSer cmps 21854

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-ofr 7686 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-sup 9467 df-oi 9535 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-fz 13520 df-fzo 13663 df-seq 14003 df-hash 14326 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ds 17258 df-hom 17260 df-cco 17261 df-0g 17426 df-gsum 17427 df-prds 17432 df-pws 17434 df-mre 17569 df-mrc 17570 df-acs 17572 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18743 df-submnd 18744 df-grp 18901 df-minusg 18902 df-mulg 19032 df-ghm 19176 df-cntz 19280 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-cring 20188 df-lmod 20757 df-assa 21804 df-psr 21859

This theorem is referenced by: mplassa 21984 mplbas2 22002 opsrassa 22026 mplind 22036 evlseu 22051

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