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Theorem psrassa 21889

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 2731	. 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 21863	. 2 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑆))
6		eqidd 2731	. 2 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑅))
7		eqidd 2731	. 2 ⊢ (𝜑 → ( ·_𝑠 ‘𝑆) = ( ·_𝑠 ‘𝑆))
8		eqidd 2731	. 2 ⊢ (𝜑 → (._r‘𝑆) = (._r‘𝑆))
9	4	crngringd 20162	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
10	2, 3, 9	psrlmod 21876	. 2 ⊢ (𝜑 → 𝑆 ∈ LMod)
11	2, 3, 9	psrring 21886	. 2 ⊢ (𝜑 → 𝑆 ∈ Ring)
12	3	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝐼 ∈ 𝑉)
13	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Ring)
14		eqid 2730	. . . 4 ⊢ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
15		eqid 2730	. . . 4 ⊢ (._r‘𝑆) = (._r‘𝑆)
16		eqid 2730	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
17		simpr2 1196	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
18		simpr3 1197	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
19	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ CRing)
20		eqid 2730	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
21		eqid 2730	. . . 4 ⊢ ( ·_𝑠 ‘𝑆) = ( ·_𝑠 ‘𝑆)
22		simpr1 1195	. . . 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 21885	. . 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 494	. 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 495	. 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 21781	1 ⊢ (𝜑 → 𝑆 ∈ AssAlg)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 {crab 3408 ^◡ccnv 5640 “ cima 5644 ‘cfv 6514 (class class class)co 7390 ↑_m cmap 8802 Fincfn 8921 ℕcn 12193 ℕ₀cn0 12449 Basecbs 17186 ._rcmulr 17228 ·_𝑠 cvsca 17231 Ringcrg 20149 CRingccrg 20150 AssAlgcasa 21766 mPwSer cmps 21820

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-ofr 7657 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-sup 9400 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-fz 13476 df-fzo 13623 df-seq 13974 df-hash 14303 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-hom 17251 df-cco 17252 df-0g 17411 df-gsum 17412 df-prds 17417 df-pws 17419 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-mhm 18717 df-submnd 18718 df-grp 18875 df-minusg 18876 df-mulg 19007 df-ghm 19152 df-cntz 19256 df-cmn 19719 df-abl 19720 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-cring 20152 df-lmod 20775 df-assa 21769 df-psr 21825

This theorem is referenced by: mplassa 21938 mplbas2 21956 opsrassa 21974 mplind 21984 evlseu 21997

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