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Theorem psrlmod 21170
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 2739 . 2 (𝜑 → (Base‘𝑆) = (Base‘𝑆))
2 eqidd 2739 . 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 21158 . 2 (𝜑𝑅 = (Scalar‘𝑆))
7 eqidd 2739 . 2 (𝜑 → ( ·𝑠𝑆) = ( ·𝑠𝑆))
8 eqidd 2739 . 2 (𝜑 → (Base‘𝑅) = (Base‘𝑅))
9 eqidd 2739 . 2 (𝜑 → (+g𝑅) = (+g𝑅))
10 eqidd 2739 . 2 (𝜑 → (.r𝑅) = (.r𝑅))
11 eqidd 2739 . 2 (𝜑 → (1r𝑅) = (1r𝑅))
12 ringgrp 19788 . . . 4 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
135, 12syl 17 . . 3 (𝜑𝑅 ∈ Grp)
143, 4, 13psrgrp 21167 . 2 (𝜑𝑆 ∈ Grp)
15 eqid 2738 . . 3 ( ·𝑠𝑆) = ( ·𝑠𝑆)
16 eqid 2738 . . 3 (Base‘𝑅) = (Base‘𝑅)
17 eqid 2738 . . 3 (Base‘𝑆) = (Base‘𝑆)
1853ad2ant1 1132 . . 3 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring)
19 simp2 1136 . . 3 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑅))
20 simp3 1137 . . 3 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆))
213, 15, 16, 17, 18, 19, 20psrvscacl 21162 . 2 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥( ·𝑠𝑆)𝑦) ∈ (Base‘𝑆))
22 ovex 7308 . . . . . . 7 (ℕ0m 𝐼) ∈ V
2322rabex 5256 . . . . . 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 1193 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑅))
26 fconst6g 6663 . . . . . 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 2738 . . . . . 6 {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
29 simpr2 1194 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
303, 16, 28, 17, 29psrelbas 21148 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦:{𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
31 simpr3 1195 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
323, 16, 28, 17, 31psrelbas 21148 . . . . 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 2738 . . . . . . 7 (+g𝑅) = (+g𝑅)
35 eqid 2738 . . . . . . 7 (.r𝑅) = (.r𝑅)
3616, 34, 35ringdi 19805 . . . . . 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 7572 . . . 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 2738 . . . . . 6 (+g𝑆) = (+g𝑆)
403, 17, 34, 39, 29, 31psradd 21151 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g𝑆)𝑧) = (𝑦f (+g𝑅)𝑧))
4140oveq2d 7291 . . . 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 21160 . . . . 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 21160 . . . . 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 7293 . . . 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 2788 . . 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 21152 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g𝑆)𝑧) ∈ (Base‘𝑆))
483, 15, 16, 17, 35, 28, 25, 47psrvsca 21160 . . 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 1183 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠𝑆)𝑦) ∈ (Base‘𝑆))
503, 15, 16, 17, 33, 25, 31psrvscacl 21162 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠𝑆)𝑧) ∈ (Base‘𝑆))
513, 17, 34, 39, 49, 50psradd 21151 . . 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 2788 . 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 1193 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑅))
54 simpr3 1195 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
553, 15, 16, 17, 35, 28, 53, 54psrvsca 21160 . . . . 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 1194 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑅))
573, 15, 16, 17, 35, 28, 56, 54psrvsca 21160 . . . . 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 7293 . . . 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 21148 . . . . 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 6663 . . . . . 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 19806 . . . . . 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 7573 . . . 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 7561 . . . . 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 7290 . . . 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 2785 . . 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 19817 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g𝑅)𝑦) ∈ (Base‘𝑅))
7264, 53, 56, 71syl3anc 1370 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(+g𝑅)𝑦) ∈ (Base‘𝑅))
733, 15, 16, 17, 35, 28, 72, 54psrvsca 21160 . . 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 21162 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠𝑆)𝑧) ∈ (Base‘𝑆))
753, 15, 16, 17, 64, 56, 54psrvscacl 21162 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦( ·𝑠𝑆)𝑧) ∈ (Base‘𝑆))
763, 17, 34, 39, 74, 75psradd 21151 . . 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 2788 . 2 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+g𝑅)𝑦)( ·𝑠𝑆)𝑧) = ((𝑥( ·𝑠𝑆)𝑧)(+g𝑆)(𝑦( ·𝑠𝑆)𝑧)))
7857oveq2d 7291 . . . 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 19803 . . . . . 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 7570 . . . 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 7561 . . . . 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 7290 . . . 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 2785 . . 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 19800 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r𝑅)𝑦) ∈ (Base‘𝑅))
8664, 53, 56, 85syl3anc 1370 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(.r𝑅)𝑦) ∈ (Base‘𝑅))
873, 15, 16, 17, 35, 28, 86, 54psrvsca 21160 . . 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 21160 . . 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 2788 . 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 2738 . . . . . 6 (1r𝑅) = (1r𝑅)
9216, 91ringidcl 19807 . . . . 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 21160 . . 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 21148 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑆)) → 𝑥:{𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
9816, 35, 91ringlidm 19810 . . . . 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 7564 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆)) → (({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {(1r𝑅)}) ∘f (.r𝑅)𝑥) = 𝑥)
10195, 100eqtrd 2778 . 2 ((𝜑𝑥 ∈ (Base‘𝑆)) → ((1r𝑅)( ·𝑠𝑆)𝑥) = 𝑥)
1021, 2, 6, 7, 8, 9, 10, 11, 5, 14, 21, 52, 77, 89, 101islmodd 20129 1 (𝜑𝑆 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  {crab 3068  Vcvv 3432  {csn 4561   × cxp 5587  ccnv 5588  cima 5592  wf 6429  cfv 6433  (class class class)co 7275  f cof 7531  m cmap 8615  Fincfn 8733  cn 11973  0cn0 12233  Basecbs 16912  +gcplusg 16962  .rcmulr 16963   ·𝑠 cvsca 16966  Grpcgrp 18577  1rcur 19737  Ringcrg 19783  LModclmod 20123   mPwSer cmps 21107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-plusg 16975  df-mulr 16976  df-sca 16978  df-vsca 16979  df-tset 16981  df-0g 17152  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-grp 18580  df-minusg 18581  df-mgp 19721  df-ur 19738  df-ring 19785  df-lmod 20125  df-psr 21112
This theorem is referenced by:  psrassa  21183  mpllmod  21223  mplbas2  21243  opsrlmod  21417
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