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Theorem psrlmod 19724
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 2800 . 2 (𝜑 → (Base‘𝑆) = (Base‘𝑆))
2 eqidd 2800 . 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 19712 . 2 (𝜑𝑅 = (Scalar‘𝑆))
7 eqidd 2800 . 2 (𝜑 → ( ·𝑠𝑆) = ( ·𝑠𝑆))
8 eqidd 2800 . 2 (𝜑 → (Base‘𝑅) = (Base‘𝑅))
9 eqidd 2800 . 2 (𝜑 → (+g𝑅) = (+g𝑅))
10 eqidd 2800 . 2 (𝜑 → (.r𝑅) = (.r𝑅))
11 eqidd 2800 . 2 (𝜑 → (1r𝑅) = (1r𝑅))
12 ringgrp 18868 . . . 4 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
135, 12syl 17 . . 3 (𝜑𝑅 ∈ Grp)
143, 4, 13psrgrp 19721 . 2 (𝜑𝑆 ∈ Grp)
15 eqid 2799 . . 3 ( ·𝑠𝑆) = ( ·𝑠𝑆)
16 eqid 2799 . . 3 (Base‘𝑅) = (Base‘𝑅)
17 eqid 2799 . . 3 (Base‘𝑆) = (Base‘𝑆)
1853ad2ant1 1164 . . 3 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring)
19 simp2 1168 . . 3 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑅))
20 simp3 1169 . . 3 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆))
213, 15, 16, 17, 18, 19, 20psrvscacl 19716 . 2 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥( ·𝑠𝑆)𝑦) ∈ (Base‘𝑆))
22 ovex 6910 . . . . . . 7 (ℕ0𝑚 𝐼) ∈ V
2322rabex 5007 . . . . . 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 1249 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑅))
26 fconst6g 6309 . . . . . 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 2799 . . . . . 6 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
29 simpr2 1251 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
303, 16, 28, 17, 29psrelbas 19702 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦:{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
31 simpr3 1253 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
323, 16, 28, 17, 31psrelbas 19702 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧:{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
335adantr 473 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Ring)
34 eqid 2799 . . . . . . 7 (+g𝑅) = (+g𝑅)
35 eqid 2799 . . . . . . 7 (.r𝑅) = (.r𝑅)
3616, 34, 35ringdi 18882 . . . . . 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 576 . . . . 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 7167 . . . 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 2799 . . . . . 6 (+g𝑆) = (+g𝑆)
403, 17, 34, 39, 29, 31psradd 19705 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g𝑆)𝑧) = (𝑦𝑓 (+g𝑅)𝑧))
4140oveq2d 6894 . . . 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 19714 . . . . 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 19714 . . . . 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 6896 . . . 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 2843 . . 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 473 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Grp)
473, 17, 39, 46, 29, 31psraddcl 19706 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g𝑆)𝑧) ∈ (Base‘𝑆))
483, 15, 16, 17, 35, 28, 25, 47psrvsca 19714 . . 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 1236 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠𝑆)𝑦) ∈ (Base‘𝑆))
503, 15, 16, 17, 33, 25, 31psrvscacl 19716 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠𝑆)𝑧) ∈ (Base‘𝑆))
513, 17, 34, 39, 49, 50psradd 19705 . . 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 2843 . 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 1249 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑅))
54 simpr3 1253 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
553, 15, 16, 17, 35, 28, 53, 54psrvsca 19714 . . . . 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 1251 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑅))
573, 15, 16, 17, 35, 28, 56, 54psrvsca 19714 . . . . 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 6896 . . . 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 19702 . . . . 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 6309 . . . . . 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 473 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Ring)
6516, 34, 35ringdir 18883 . . . . . 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 576 . . . . 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 7168 . . . 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 7156 . . . . 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 6893 . . . 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 2840 . . 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 18894 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g𝑅)𝑦) ∈ (Base‘𝑅))
7264, 53, 56, 71syl3anc 1491 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(+g𝑅)𝑦) ∈ (Base‘𝑅))
733, 15, 16, 17, 35, 28, 72, 54psrvsca 19714 . . 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 19716 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠𝑆)𝑧) ∈ (Base‘𝑆))
753, 15, 16, 17, 64, 56, 54psrvscacl 19716 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦( ·𝑠𝑆)𝑧) ∈ (Base‘𝑆))
763, 17, 34, 39, 74, 75psradd 19705 . . 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 2843 . 2 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+g𝑅)𝑦)( ·𝑠𝑆)𝑧) = ((𝑥( ·𝑠𝑆)𝑧)(+g𝑆)(𝑦( ·𝑠𝑆)𝑧)))
7857oveq2d 6894 . . . 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 18880 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → ((𝑟(.r𝑅)𝑠)(.r𝑅)𝑡) = (𝑟(.r𝑅)(𝑠(.r𝑅)𝑡)))
8064, 79sylan 576 . . . . 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 7165 . . . 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 7156 . . . . 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 6893 . . . 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 2840 . . 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 18877 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r𝑅)𝑦) ∈ (Base‘𝑅))
8664, 53, 56, 85syl3anc 1491 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(.r𝑅)𝑦) ∈ (Base‘𝑅))
873, 15, 16, 17, 35, 28, 86, 54psrvsca 19714 . . 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 19714 . . 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 2843 . 2 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(.r𝑅)𝑦)( ·𝑠𝑆)𝑧) = (𝑥( ·𝑠𝑆)(𝑦( ·𝑠𝑆)𝑧)))
905adantr 473 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring)
91 eqid 2799 . . . . . 6 (1r𝑅) = (1r𝑅)
9216, 91ringidcl 18884 . . . . 5 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
9390, 92syl 17 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑆)) → (1r𝑅) ∈ (Base‘𝑅))
94 simpr 478 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
953, 15, 16, 17, 35, 28, 93, 94psrvsca 19714 . . 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 19702 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑆)) → 𝑥:{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
9816, 35, 91ringlidm 18887 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → ((1r𝑅)(.r𝑅)𝑟) = 𝑟)
9990, 98sylan 576 . . . 4 (((𝜑𝑥 ∈ (Base‘𝑆)) ∧ 𝑟 ∈ (Base‘𝑅)) → ((1r𝑅)(.r𝑅)𝑟) = 𝑟)
10096, 97, 93, 99caofid0l 7159 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆)) → (({𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {(1r𝑅)}) ∘𝑓 (.r𝑅)𝑥) = 𝑥)
10195, 100eqtrd 2833 . 2 ((𝜑𝑥 ∈ (Base‘𝑆)) → ((1r𝑅)( ·𝑠𝑆)𝑥) = 𝑥)
1021, 2, 6, 7, 8, 9, 10, 11, 5, 14, 21, 52, 77, 89, 101islmodd 19187 1 (𝜑𝑆 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wcel 2157  {crab 3093  Vcvv 3385  {csn 4368   × cxp 5310  ccnv 5311  cima 5315  wf 6097  cfv 6101  (class class class)co 6878  𝑓 cof 7129  𝑚 cmap 8095  Fincfn 8195  cn 11312  0cn0 11580  Basecbs 16184  +gcplusg 16267  .rcmulr 16268   ·𝑠 cvsca 16271  Grpcgrp 17738  1rcur 18817  Ringcrg 18863  LModclmod 19181   mPwSer cmps 19674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-of 7131  df-om 7300  df-1st 7401  df-2nd 7402  df-supp 7533  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-er 7982  df-map 8097  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-fsupp 8518  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-nn 11313  df-2 11376  df-3 11377  df-4 11378  df-5 11379  df-6 11380  df-7 11381  df-8 11382  df-9 11383  df-n0 11581  df-z 11667  df-uz 11931  df-fz 12581  df-struct 16186  df-ndx 16187  df-slot 16188  df-base 16190  df-sets 16191  df-plusg 16280  df-mulr 16281  df-sca 16283  df-vsca 16284  df-tset 16286  df-0g 16417  df-mgm 17557  df-sgrp 17599  df-mnd 17610  df-grp 17741  df-minusg 17742  df-mgp 18806  df-ur 18818  df-ring 18865  df-lmod 19183  df-psr 19679
This theorem is referenced by:  psrassa  19737  mpllmod  19774  mplbas2  19793  opsrlmod  19938
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