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Theorem psrlmod 19222
 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 2611 . 2 (𝜑 → (Base‘𝑆) = (Base‘𝑆))
2 eqidd 2611 . 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 19210 . 2 (𝜑𝑅 = (Scalar‘𝑆))
7 eqidd 2611 . 2 (𝜑 → ( ·𝑠𝑆) = ( ·𝑠𝑆))
8 eqidd 2611 . 2 (𝜑 → (Base‘𝑅) = (Base‘𝑅))
9 eqidd 2611 . 2 (𝜑 → (+g𝑅) = (+g𝑅))
10 eqidd 2611 . 2 (𝜑 → (.r𝑅) = (.r𝑅))
11 eqidd 2611 . 2 (𝜑 → (1r𝑅) = (1r𝑅))
12 ringgrp 18375 . . . 4 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
135, 12syl 17 . . 3 (𝜑𝑅 ∈ Grp)
143, 4, 13psrgrp 19219 . 2 (𝜑𝑆 ∈ Grp)
15 eqid 2610 . . 3 ( ·𝑠𝑆) = ( ·𝑠𝑆)
16 eqid 2610 . . 3 (Base‘𝑅) = (Base‘𝑅)
17 eqid 2610 . . 3 (Base‘𝑆) = (Base‘𝑆)
1853ad2ant1 1075 . . 3 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring)
19 simp2 1055 . . 3 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑅))
20 simp3 1056 . . 3 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆))
213, 15, 16, 17, 18, 19, 20psrvscacl 19214 . 2 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥( ·𝑠𝑆)𝑦) ∈ (Base‘𝑆))
22 ovex 6577 . . . . . . 7 (ℕ0𝑚 𝐼) ∈ V
2322rabex 4740 . . . . . 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 1060 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑅))
26 fconst6g 6007 . . . . . 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 2610 . . . . . 6 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
29 simpr2 1061 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
303, 16, 28, 17, 29psrelbas 19200 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦:{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
31 simpr3 1062 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
323, 16, 28, 17, 31psrelbas 19200 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧:{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
335adantr 480 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Ring)
34 eqid 2610 . . . . . . 7 (+g𝑅) = (+g𝑅)
35 eqid 2610 . . . . . . 7 (.r𝑅) = (.r𝑅)
3616, 34, 35ringdi 18389 . . . . . 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 487 . . . . 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 6831 . . . 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 2610 . . . . . 6 (+g𝑆) = (+g𝑆)
403, 17, 34, 39, 29, 31psradd 19203 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g𝑆)𝑧) = (𝑦𝑓 (+g𝑅)𝑧))
4140oveq2d 6565 . . . 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 19212 . . . . 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 19212 . . . . 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 6567 . . . 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 2654 . . 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 480 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Grp)
473, 17, 39, 46, 29, 31psraddcl 19204 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦(+g𝑆)𝑧) ∈ (Base‘𝑆))
483, 15, 16, 17, 35, 28, 25, 47psrvsca 19212 . . 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 1268 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠𝑆)𝑦) ∈ (Base‘𝑆))
503, 15, 16, 17, 33, 25, 31psrvscacl 19214 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠𝑆)𝑧) ∈ (Base‘𝑆))
513, 17, 34, 39, 49, 50psradd 19203 . . 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 2654 . 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 1060 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑅))
54 simpr3 1062 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑧 ∈ (Base‘𝑆))
553, 15, 16, 17, 35, 28, 53, 54psrvsca 19212 . . . . 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 1061 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑅))
573, 15, 16, 17, 35, 28, 56, 54psrvsca 19212 . . . . 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 6567 . . . 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 19200 . . . . 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 6007 . . . . . 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 480 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → 𝑅 ∈ Ring)
6516, 34, 35ringdir 18390 . . . . . 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 487 . . . . 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 6832 . . . 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 6820 . . . . 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 6564 . . . 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 2651 . . 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 18401 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g𝑅)𝑦) ∈ (Base‘𝑅))
7264, 53, 56, 71syl3anc 1318 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(+g𝑅)𝑦) ∈ (Base‘𝑅))
733, 15, 16, 17, 35, 28, 72, 54psrvsca 19212 . . 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 19214 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥( ·𝑠𝑆)𝑧) ∈ (Base‘𝑆))
753, 15, 16, 17, 64, 56, 54psrvscacl 19214 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑦( ·𝑠𝑆)𝑧) ∈ (Base‘𝑆))
763, 17, 34, 39, 74, 75psradd 19203 . . 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 2654 . 2 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(+g𝑅)𝑦)( ·𝑠𝑆)𝑧) = ((𝑥( ·𝑠𝑆)𝑧)(+g𝑆)(𝑦( ·𝑠𝑆)𝑧)))
7857oveq2d 6565 . . . 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 18387 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑟 ∈ (Base‘𝑅) ∧ 𝑠 ∈ (Base‘𝑅) ∧ 𝑡 ∈ (Base‘𝑅))) → ((𝑟(.r𝑅)𝑠)(.r𝑅)𝑡) = (𝑟(.r𝑅)(𝑠(.r𝑅)𝑡)))
8064, 79sylan 487 . . . . 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 6829 . . . 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 6820 . . . . 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 6564 . . . 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 2651 . . 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 18384 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r𝑅)𝑦) ∈ (Base‘𝑅))
8664, 53, 56, 85syl3anc 1318 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → (𝑥(.r𝑅)𝑦) ∈ (Base‘𝑅))
873, 15, 16, 17, 35, 28, 86, 54psrvsca 19212 . . 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 19212 . . 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 2654 . 2 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑆))) → ((𝑥(.r𝑅)𝑦)( ·𝑠𝑆)𝑧) = (𝑥( ·𝑠𝑆)(𝑦( ·𝑠𝑆)𝑧)))
905adantr 480 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑆)) → 𝑅 ∈ Ring)
91 eqid 2610 . . . . . 6 (1r𝑅) = (1r𝑅)
9216, 91ringidcl 18391 . . . . 5 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
9390, 92syl 17 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑆)) → (1r𝑅) ∈ (Base‘𝑅))
94 simpr 476 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
953, 15, 16, 17, 35, 28, 93, 94psrvsca 19212 . . 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 19200 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑆)) → 𝑥:{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
9816, 35, 91ringlidm 18394 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → ((1r𝑅)(.r𝑅)𝑟) = 𝑟)
9990, 98sylan 487 . . . 4 (((𝜑𝑥 ∈ (Base‘𝑆)) ∧ 𝑟 ∈ (Base‘𝑅)) → ((1r𝑅)(.r𝑅)𝑟) = 𝑟)
10096, 97, 93, 99caofid0l 6823 . . 3 ((𝜑𝑥 ∈ (Base‘𝑆)) → (({𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} × {(1r𝑅)}) ∘𝑓 (.r𝑅)𝑥) = 𝑥)
10195, 100eqtrd 2644 . 2 ((𝜑𝑥 ∈ (Base‘𝑆)) → ((1r𝑅)( ·𝑠𝑆)𝑥) = 𝑥)
1021, 2, 6, 7, 8, 9, 10, 11, 5, 14, 21, 52, 77, 89, 101islmodd 18692 1 (𝜑𝑆 ∈ LMod)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  {csn 4125   × cxp 5036  ◡ccnv 5037   “ cima 5041  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793   ↑𝑚 cmap 7744  Fincfn 7841  ℕcn 10897  ℕ0cn0 11169  Basecbs 15695  +gcplusg 15768  .rcmulr 15769   ·𝑠 cvsca 15772  Grpcgrp 17245  1rcur 18324  Ringcrg 18370  LModclmod 18686   mPwSer cmps 19172 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-tset 15787  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-mgp 18313  df-ur 18325  df-ring 18372  df-lmod 18688  df-psr 19177 This theorem is referenced by:  psrassa  19235  mpllmod  19272  mplbas2  19291  opsrlmod  19437
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