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Theorem psrlmod 19615
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 2771 . 2 (𝜑 → (Base‘𝑆) = (Base‘𝑆))
2 eqidd 2771 . 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 19603 . 2 (𝜑𝑅 = (Scalar‘𝑆))
7 eqidd 2771 . 2 (𝜑 → ( ·𝑠𝑆) = ( ·𝑠𝑆))
8 eqidd 2771 . 2 (𝜑 → (Base‘𝑅) = (Base‘𝑅))
9 eqidd 2771 . 2 (𝜑 → (+g𝑅) = (+g𝑅))
10 eqidd 2771 . 2 (𝜑 → (.r𝑅) = (.r𝑅))
11 eqidd 2771 . 2 (𝜑 → (1r𝑅) = (1r𝑅))
12 ringgrp 18759 . . . 4 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
135, 12syl 17 . . 3 (𝜑𝑅 ∈ Grp)
143, 4, 13psrgrp 19612 . 2 (𝜑𝑆 ∈ Grp)
15 eqid 2770 . . 3 ( ·𝑠𝑆) = ( ·𝑠𝑆)
16 eqid 2770 . . 3 (Base‘𝑅) = (Base‘𝑅)
17 eqid 2770 . . 3 (Base‘𝑆) = (Base‘𝑆)
1853ad2ant1 1126 . . 3 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring)
19 simp2 1130 . . 3 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑅))
20 simp3 1131 . . 3 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆))
213, 15, 16, 17, 18, 19, 20psrvscacl 19607 . 2 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥( ·𝑠𝑆)𝑦) ∈ (Base‘𝑆))
22 ovex 6822 . . . . . . 7 (ℕ0𝑚 𝐼) ∈ V
2322rabex 4943 . . . . . 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 1232 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑅))
26 fconst6g 6234 . . . . . 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 2770 . . . . . 6 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
29 simpr2 1234 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
303, 16, 28, 17, 29psrelbas 19593 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦:{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
31 simpr3 1236 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
323, 16, 28, 17, 31psrelbas 19593 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧:{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
335adantr 466 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Ring)
34 eqid 2770 . . . . . . 7 (+g𝑅) = (+g𝑅)
35 eqid 2770 . . . . . . 7 (.r𝑅) = (.r𝑅)
3616, 34, 35ringdi 18773 . . . . . 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 561 . . . . 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 7079 . . . 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 2770 . . . . . 6 (+g𝑆) = (+g𝑆)
403, 17, 34, 39, 29, 31psradd 19596 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g𝑆)𝑧) = (𝑦𝑓 (+g𝑅)𝑧))
4140oveq2d 6808 . . . 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 19605 . . . . 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 19605 . . . . 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 6810 . . . 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 2814 . . 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 466 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Grp)
473, 17, 39, 46, 29, 31psraddcl 19597 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g𝑆)𝑧) ∈ (Base‘𝑆))
483, 15, 16, 17, 35, 28, 25, 47psrvsca 19605 . . 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 1198 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠𝑆)𝑦) ∈ (Base‘𝑆))
503, 15, 16, 17, 33, 25, 31psrvscacl 19607 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠𝑆)𝑧) ∈ (Base‘𝑆))
513, 17, 34, 39, 49, 50psradd 19596 . . 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 2814 . 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 1232 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑅))
54 simpr3 1236 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
553, 15, 16, 17, 35, 28, 53, 54psrvsca 19605 . . . . 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 1234 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑅))
573, 15, 16, 17, 35, 28, 56, 54psrvsca 19605 . . . . 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 6810 . . . 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 19593 . . . . 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 6234 . . . . . 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 466 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Ring)
6516, 34, 35ringdir 18774 . . . . . 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 561 . . . . 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 7080 . . . 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 7068 . . . . 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 6807 . . . 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 2811 . . 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 18785 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g𝑅)𝑦) ∈ (Base‘𝑅))
7264, 53, 56, 71syl3anc 1475 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(+g𝑅)𝑦) ∈ (Base‘𝑅))
733, 15, 16, 17, 35, 28, 72, 54psrvsca 19605 . . 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 19607 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠𝑆)𝑧) ∈ (Base‘𝑆))
753, 15, 16, 17, 64, 56, 54psrvscacl 19607 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦( ·𝑠𝑆)𝑧) ∈ (Base‘𝑆))
763, 17, 34, 39, 74, 75psradd 19596 . . 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 2814 . 2 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+g𝑅)𝑦)( ·𝑠𝑆)𝑧) = ((𝑥( ·𝑠𝑆)𝑧)(+g𝑆)(𝑦( ·𝑠𝑆)𝑧)))
7857oveq2d 6808 . . . 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 18771 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → ((𝑟(.r𝑅)𝑠)(.r𝑅)𝑡) = (𝑟(.r𝑅)(𝑠(.r𝑅)𝑡)))
8064, 79sylan 561 . . . . 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 7077 . . . 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 7068 . . . . 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 6807 . . . 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 2811 . . 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 18768 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r𝑅)𝑦) ∈ (Base‘𝑅))
8664, 53, 56, 85syl3anc 1475 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(.r𝑅)𝑦) ∈ (Base‘𝑅))
873, 15, 16, 17, 35, 28, 86, 54psrvsca 19605 . . 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 19605 . . 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 2814 . 2 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(.r𝑅)𝑦)( ·𝑠𝑆)𝑧) = (𝑥( ·𝑠𝑆)(𝑦( ·𝑠𝑆)𝑧)))
905adantr 466 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring)
91 eqid 2770 . . . . . 6 (1r𝑅) = (1r𝑅)
9216, 91ringidcl 18775 . . . . 5 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
9390, 92syl 17 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑆)) → (1r𝑅) ∈ (Base‘𝑅))
94 simpr 471 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
953, 15, 16, 17, 35, 28, 93, 94psrvsca 19605 . . 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 19593 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑆)) → 𝑥:{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
9816, 35, 91ringlidm 18778 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → ((1r𝑅)(.r𝑅)𝑟) = 𝑟)
9990, 98sylan 561 . . . 4 (((𝜑𝑥 ∈ (Base‘𝑆)) ∧ 𝑟 ∈ (Base‘𝑅)) → ((1r𝑅)(.r𝑅)𝑟) = 𝑟)
10096, 97, 93, 99caofid0l 7071 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆)) → (({𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {(1r𝑅)}) ∘𝑓 (.r𝑅)𝑥) = 𝑥)
10195, 100eqtrd 2804 . 2 ((𝜑𝑥 ∈ (Base‘𝑆)) → ((1r𝑅)( ·𝑠𝑆)𝑥) = 𝑥)
1021, 2, 6, 7, 8, 9, 10, 11, 5, 14, 21, 52, 77, 89, 101islmodd 19078 1 (𝜑𝑆 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1070   = wceq 1630  wcel 2144  {crab 3064  Vcvv 3349  {csn 4314   × cxp 5247  ccnv 5248  cima 5252  wf 6027  cfv 6031  (class class class)co 6792  𝑓 cof 7041  𝑚 cmap 8008  Fincfn 8108  cn 11221  0cn0 11493  Basecbs 16063  +gcplusg 16148  .rcmulr 16149   ·𝑠 cvsca 16152  Grpcgrp 17629  1rcur 18708  Ringcrg 18754  LModclmod 19072   mPwSer cmps 19565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-of 7043  df-om 7212  df-1st 7314  df-2nd 7315  df-supp 7446  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-map 8010  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-fsupp 8431  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-3 11281  df-4 11282  df-5 11283  df-6 11284  df-7 11285  df-8 11286  df-9 11287  df-n0 11494  df-z 11579  df-uz 11888  df-fz 12533  df-struct 16065  df-ndx 16066  df-slot 16067  df-base 16069  df-sets 16070  df-plusg 16161  df-mulr 16162  df-sca 16164  df-vsca 16165  df-tset 16167  df-0g 16309  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-grp 17632  df-minusg 17633  df-mgp 18697  df-ur 18709  df-ring 18756  df-lmod 19074  df-psr 19570
This theorem is referenced by:  psrassa  19628  mpllmod  19665  mplbas2  19684  opsrlmod  19830
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