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Theorem psrlmod 20109
Description: The ring of power series is a left module. (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
psrring.s 𝑆 = (𝐼 mPwSer 𝑅)
psrring.i (𝜑𝐼𝑉)
psrring.r (𝜑𝑅 ∈ Ring)
Assertion
Ref Expression
psrlmod (𝜑𝑆 ∈ LMod)

Proof of Theorem psrlmod
Dummy variables 𝑥 𝑓 𝑦 𝑧 𝑟 𝑠 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2819 . 2 (𝜑 → (Base‘𝑆) = (Base‘𝑆))
2 eqidd 2819 . 2 (𝜑 → (+g𝑆) = (+g𝑆))
3 psrring.s . . 3 𝑆 = (𝐼 mPwSer 𝑅)
4 psrring.i . . 3 (𝜑𝐼𝑉)
5 psrring.r . . 3 (𝜑𝑅 ∈ Ring)
63, 4, 5psrsca 20097 . 2 (𝜑𝑅 = (Scalar‘𝑆))
7 eqidd 2819 . 2 (𝜑 → ( ·𝑠𝑆) = ( ·𝑠𝑆))
8 eqidd 2819 . 2 (𝜑 → (Base‘𝑅) = (Base‘𝑅))
9 eqidd 2819 . 2 (𝜑 → (+g𝑅) = (+g𝑅))
10 eqidd 2819 . 2 (𝜑 → (.r𝑅) = (.r𝑅))
11 eqidd 2819 . 2 (𝜑 → (1r𝑅) = (1r𝑅))
12 ringgrp 19231 . . . 4 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
135, 12syl 17 . . 3 (𝜑𝑅 ∈ Grp)
143, 4, 13psrgrp 20106 . 2 (𝜑𝑆 ∈ Grp)
15 eqid 2818 . . 3 ( ·𝑠𝑆) = ( ·𝑠𝑆)
16 eqid 2818 . . 3 (Base‘𝑅) = (Base‘𝑅)
17 eqid 2818 . . 3 (Base‘𝑆) = (Base‘𝑆)
1853ad2ant1 1125 . . 3 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring)
19 simp2 1129 . . 3 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑅))
20 simp3 1130 . . 3 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆))
213, 15, 16, 17, 18, 19, 20psrvscacl 20101 . 2 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥( ·𝑠𝑆)𝑦) ∈ (Base‘𝑆))
22 ovex 7178 . . . . . . 7 (ℕ0m 𝐼) ∈ V
2322rabex 5226 . . . . . 6 {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∈ V
2423a1i 11 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∈ V)
25 simpr1 1186 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑅))
26 fconst6g 6561 . . . . . 6 (𝑥 ∈ (Base‘𝑅) → ({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑥}):{𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
2725, 26syl 17 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑥}):{𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
28 eqid 2818 . . . . . 6 {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
29 simpr2 1187 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
303, 16, 28, 17, 29psrelbas 20087 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦:{𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
31 simpr3 1188 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
323, 16, 28, 17, 31psrelbas 20087 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧:{𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
335adantr 481 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Ring)
34 eqid 2818 . . . . . . 7 (+g𝑅) = (+g𝑅)
35 eqid 2818 . . . . . . 7 (.r𝑅) = (.r𝑅)
3616, 34, 35ringdi 19245 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → (𝑟(.r𝑅)(𝑠(+g𝑅)𝑡)) = ((𝑟(.r𝑅)𝑠)(+g𝑅)(𝑟(.r𝑅)𝑡)))
3733, 36sylan 580 . . . . 5 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → (𝑟(.r𝑅)(𝑠(+g𝑅)𝑡)) = ((𝑟(.r𝑅)𝑠)(+g𝑅)(𝑟(.r𝑅)𝑡)))
3824, 27, 30, 32, 37caofdi 7434 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)(𝑦f (+g𝑅)𝑧)) = ((({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)𝑦) ∘f (+g𝑅)(({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)𝑧)))
39 eqid 2818 . . . . . 6 (+g𝑆) = (+g𝑆)
403, 17, 34, 39, 29, 31psradd 20090 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g𝑆)𝑧) = (𝑦f (+g𝑅)𝑧))
4140oveq2d 7161 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)(𝑦(+g𝑆)𝑧)) = (({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)(𝑦f (+g𝑅)𝑧)))
423, 15, 16, 17, 35, 28, 25, 29psrvsca 20099 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠𝑆)𝑦) = (({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)𝑦))
433, 15, 16, 17, 35, 28, 25, 31psrvsca 20099 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠𝑆)𝑧) = (({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)𝑧))
4442, 43oveq12d 7163 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥( ·𝑠𝑆)𝑦) ∘f (+g𝑅)(𝑥( ·𝑠𝑆)𝑧)) = ((({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)𝑦) ∘f (+g𝑅)(({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)𝑧)))
4538, 41, 443eqtr4d 2863 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)(𝑦(+g𝑆)𝑧)) = ((𝑥( ·𝑠𝑆)𝑦) ∘f (+g𝑅)(𝑥( ·𝑠𝑆)𝑧)))
4613adantr 481 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Grp)
473, 17, 39, 46, 29, 31psraddcl 20091 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g𝑆)𝑧) ∈ (Base‘𝑆))
483, 15, 16, 17, 35, 28, 25, 47psrvsca 20099 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠𝑆)(𝑦(+g𝑆)𝑧)) = (({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)(𝑦(+g𝑆)𝑧)))
49213adant3r3 1176 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠𝑆)𝑦) ∈ (Base‘𝑆))
503, 15, 16, 17, 33, 25, 31psrvscacl 20101 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠𝑆)𝑧) ∈ (Base‘𝑆))
513, 17, 34, 39, 49, 50psradd 20090 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥( ·𝑠𝑆)𝑦)(+g𝑆)(𝑥( ·𝑠𝑆)𝑧)) = ((𝑥( ·𝑠𝑆)𝑦) ∘f (+g𝑅)(𝑥( ·𝑠𝑆)𝑧)))
5245, 48, 513eqtr4d 2863 . 2 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠𝑆)(𝑦(+g𝑆)𝑧)) = ((𝑥( ·𝑠𝑆)𝑦)(+g𝑆)(𝑥( ·𝑠𝑆)𝑧)))
53 simpr1 1186 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑅))
54 simpr3 1188 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
553, 15, 16, 17, 35, 28, 53, 54psrvsca 20099 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠𝑆)𝑧) = (({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)𝑧))
56 simpr2 1187 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑅))
573, 15, 16, 17, 35, 28, 56, 54psrvsca 20099 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦( ·𝑠𝑆)𝑧) = (({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑦}) ∘f (.r𝑅)𝑧))
5855, 57oveq12d 7163 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥( ·𝑠𝑆)𝑧) ∘f (+g𝑅)(𝑦( ·𝑠𝑆)𝑧)) = ((({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)𝑧) ∘f (+g𝑅)(({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑦}) ∘f (.r𝑅)𝑧)))
5923a1i 11 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∈ V)
603, 16, 28, 17, 54psrelbas 20087 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧:{𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
6153, 26syl 17 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑥}):{𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
62 fconst6g 6561 . . . . . 6 (𝑦 ∈ (Base‘𝑅) → ({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑦}):{𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
6356, 62syl 17 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑦}):{𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
645adantr 481 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Ring)
6516, 34, 35ringdir 19246 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → ((𝑟(+g𝑅)𝑠)(.r𝑅)𝑡) = ((𝑟(.r𝑅)𝑡)(+g𝑅)(𝑠(.r𝑅)𝑡)))
6664, 65sylan 580 . . . . 5 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → ((𝑟(+g𝑅)𝑠)(.r𝑅)𝑡) = ((𝑟(.r𝑅)𝑡)(+g𝑅)(𝑠(.r𝑅)𝑡)))
6759, 60, 61, 63, 66caofdir 7435 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (+g𝑅)({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑦})) ∘f (.r𝑅)𝑧) = ((({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)𝑧) ∘f (+g𝑅)(({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑦}) ∘f (.r𝑅)𝑧)))
6859, 53, 56ofc12 7423 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (+g𝑅)({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑦})) = ({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {(𝑥(+g𝑅)𝑦)}))
6968oveq1d 7160 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (+g𝑅)({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑦})) ∘f (.r𝑅)𝑧) = (({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {(𝑥(+g𝑅)𝑦)}) ∘f (.r𝑅)𝑧))
7058, 67, 693eqtr2rd 2860 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {(𝑥(+g𝑅)𝑦)}) ∘f (.r𝑅)𝑧) = ((𝑥( ·𝑠𝑆)𝑧) ∘f (+g𝑅)(𝑦( ·𝑠𝑆)𝑧)))
7116, 34ringacl 19257 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g𝑅)𝑦) ∈ (Base‘𝑅))
7264, 53, 56, 71syl3anc 1363 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(+g𝑅)𝑦) ∈ (Base‘𝑅))
733, 15, 16, 17, 35, 28, 72, 54psrvsca 20099 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+g𝑅)𝑦)( ·𝑠𝑆)𝑧) = (({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {(𝑥(+g𝑅)𝑦)}) ∘f (.r𝑅)𝑧))
743, 15, 16, 17, 64, 53, 54psrvscacl 20101 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠𝑆)𝑧) ∈ (Base‘𝑆))
753, 15, 16, 17, 64, 56, 54psrvscacl 20101 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦( ·𝑠𝑆)𝑧) ∈ (Base‘𝑆))
763, 17, 34, 39, 74, 75psradd 20090 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥( ·𝑠𝑆)𝑧)(+g𝑆)(𝑦( ·𝑠𝑆)𝑧)) = ((𝑥( ·𝑠𝑆)𝑧) ∘f (+g𝑅)(𝑦( ·𝑠𝑆)𝑧)))
7770, 73, 763eqtr4d 2863 . 2 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+g𝑅)𝑦)( ·𝑠𝑆)𝑧) = ((𝑥( ·𝑠𝑆)𝑧)(+g𝑆)(𝑦( ·𝑠𝑆)𝑧)))
7857oveq2d 7161 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)(𝑦( ·𝑠𝑆)𝑧)) = (({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)(({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑦}) ∘f (.r𝑅)𝑧)))
7916, 35ringass 19243 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → ((𝑟(.r𝑅)𝑠)(.r𝑅)𝑡) = (𝑟(.r𝑅)(𝑠(.r𝑅)𝑡)))
8064, 79sylan 580 . . . . 5 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → ((𝑟(.r𝑅)𝑠)(.r𝑅)𝑡) = (𝑟(.r𝑅)(𝑠(.r𝑅)𝑡)))
8159, 61, 63, 60, 80caofass 7432 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑦})) ∘f (.r𝑅)𝑧) = (({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)(({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑦}) ∘f (.r𝑅)𝑧)))
8259, 53, 56ofc12 7423 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑦})) = ({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {(𝑥(.r𝑅)𝑦)}))
8382oveq1d 7160 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑦})) ∘f (.r𝑅)𝑧) = (({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {(𝑥(.r𝑅)𝑦)}) ∘f (.r𝑅)𝑧))
8478, 81, 833eqtr2rd 2860 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {(𝑥(.r𝑅)𝑦)}) ∘f (.r𝑅)𝑧) = (({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)(𝑦( ·𝑠𝑆)𝑧)))
8516, 35ringcl 19240 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r𝑅)𝑦) ∈ (Base‘𝑅))
8664, 53, 56, 85syl3anc 1363 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(.r𝑅)𝑦) ∈ (Base‘𝑅))
873, 15, 16, 17, 35, 28, 86, 54psrvsca 20099 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(.r𝑅)𝑦)( ·𝑠𝑆)𝑧) = (({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {(𝑥(.r𝑅)𝑦)}) ∘f (.r𝑅)𝑧))
883, 15, 16, 17, 35, 28, 53, 75psrvsca 20099 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠𝑆)(𝑦( ·𝑠𝑆)𝑧)) = (({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {𝑥}) ∘f (.r𝑅)(𝑦( ·𝑠𝑆)𝑧)))
8984, 87, 883eqtr4d 2863 . 2 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(.r𝑅)𝑦)( ·𝑠𝑆)𝑧) = (𝑥( ·𝑠𝑆)(𝑦( ·𝑠𝑆)𝑧)))
905adantr 481 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring)
91 eqid 2818 . . . . . 6 (1r𝑅) = (1r𝑅)
9216, 91ringidcl 19247 . . . . 5 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
9390, 92syl 17 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑆)) → (1r𝑅) ∈ (Base‘𝑅))
94 simpr 485 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
953, 15, 16, 17, 35, 28, 93, 94psrvsca 20099 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆)) → ((1r𝑅)( ·𝑠𝑆)𝑥) = (({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {(1r𝑅)}) ∘f (.r𝑅)𝑥))
9623a1i 11 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑆)) → {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∈ V)
973, 16, 28, 17, 94psrelbas 20087 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑆)) → 𝑥:{𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
9816, 35, 91ringlidm 19250 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → ((1r𝑅)(.r𝑅)𝑟) = 𝑟)
9990, 98sylan 580 . . . 4 (((𝜑𝑥 ∈ (Base‘𝑆)) ∧ 𝑟 ∈ (Base‘𝑅)) → ((1r𝑅)(.r𝑅)𝑟) = 𝑟)
10096, 97, 93, 99caofid0l 7426 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆)) → (({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {(1r𝑅)}) ∘f (.r𝑅)𝑥) = 𝑥)
10195, 100eqtrd 2853 . 2 ((𝜑𝑥 ∈ (Base‘𝑆)) → ((1r𝑅)( ·𝑠𝑆)𝑥) = 𝑥)
1021, 2, 6, 7, 8, 9, 10, 11, 5, 14, 21, 52, 77, 89, 101islmodd 19569 1 (𝜑𝑆 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  {crab 3139  Vcvv 3492  {csn 4557   × cxp 5546  ccnv 5547  cima 5551  wf 6344  cfv 6348  (class class class)co 7145  f cof 7396  m cmap 8395  Fincfn 8497  cn 11626  0cn0 11885  Basecbs 16471  +gcplusg 16553  .rcmulr 16554   ·𝑠 cvsca 16557  Grpcgrp 18041  1rcur 19180  Ringcrg 19226  LModclmod 19563   mPwSer cmps 20059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7570  df-1st 7678  df-2nd 7679  df-supp 7820  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fsupp 8822  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-plusg 16566  df-mulr 16567  df-sca 16569  df-vsca 16570  df-tset 16572  df-0g 16703  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-grp 18044  df-minusg 18045  df-mgp 19169  df-ur 19181  df-ring 19228  df-lmod 19565  df-psr 20064
This theorem is referenced by:  psrassa  20122  mpllmod  20159  mplbas2  20179  opsrlmod  20342
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