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Theorem coe1mul2 20998
 Description: The coefficient vector of multiplication in the univariate power series ring. (Contributed by Stefan O'Rear, 25-Mar-2015.)
Hypotheses
Ref Expression
coe1mul2.s 𝑆 = (PwSer1𝑅)
coe1mul2.t = (.r𝑆)
coe1mul2.u · = (.r𝑅)
coe1mul2.b 𝐵 = (Base‘𝑆)
Assertion
Ref Expression
coe1mul2 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (coe1‘(𝐹 𝐺)) = (𝑘 ∈ ℕ0 ↦ (𝑅 Σg (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))))))
Distinct variable groups:   𝑥,𝑘,𝐵   𝑘,𝐹,𝑥   · ,𝑘,𝑥   𝑘,𝐺,𝑥   𝑅,𝑘,𝑥   ,𝑘
Allowed substitution hints:   𝑆(𝑥,𝑘)   (𝑥)

Proof of Theorem coe1mul2
Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fconst6g 6557 . . . . 5 (𝑘 ∈ ℕ0 → (1o × {𝑘}):1o⟶ℕ0)
2 nn0ex 11945 . . . . . 6 0 ∈ V
3 1on 8124 . . . . . . 7 1o ∈ On
43elexi 3429 . . . . . 6 1o ∈ V
52, 4elmap 8458 . . . . 5 ((1o × {𝑘}) ∈ (ℕ0m 1o) ↔ (1o × {𝑘}):1o⟶ℕ0)
61, 5sylibr 237 . . . 4 (𝑘 ∈ ℕ0 → (1o × {𝑘}) ∈ (ℕ0m 1o))
76adantl 485 . . 3 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (1o × {𝑘}) ∈ (ℕ0m 1o))
8 eqidd 2759 . . 3 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝑘 ∈ ℕ0 ↦ (1o × {𝑘})) = (𝑘 ∈ ℕ0 ↦ (1o × {𝑘})))
9 eqid 2758 . . . 4 (1o mPwSer 𝑅) = (1o mPwSer 𝑅)
10 coe1mul2.s . . . . 5 𝑆 = (PwSer1𝑅)
11 coe1mul2.b . . . . 5 𝐵 = (Base‘𝑆)
1210, 11, 9psr1bas2 20919 . . . 4 𝐵 = (Base‘(1o mPwSer 𝑅))
13 coe1mul2.u . . . 4 · = (.r𝑅)
14 coe1mul2.t . . . . 5 = (.r𝑆)
1510, 9, 14psr1mulr 20953 . . . 4 = (.r‘(1o mPwSer 𝑅))
16 psr1baslem 20914 . . . 4 (ℕ0m 1o) = {𝑎 ∈ (ℕ0m 1o) ∣ (𝑎 “ ℕ) ∈ Fin}
17 simp2 1134 . . . 4 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → 𝐹𝐵)
18 simp3 1135 . . . 4 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → 𝐺𝐵)
199, 12, 13, 15, 16, 17, 18psrmulfval 20718 . . 3 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝐹 𝐺) = (𝑏 ∈ (ℕ0m 1o) ↦ (𝑅 Σg (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r𝑏} ↦ ((𝐹𝑐) · (𝐺‘(𝑏f𝑐)))))))
20 breq2 5039 . . . . . 6 (𝑏 = (1o × {𝑘}) → (𝑑r𝑏𝑑r ≤ (1o × {𝑘})))
2120rabbidv 3392 . . . . 5 (𝑏 = (1o × {𝑘}) → {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r𝑏} = {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})})
22 fvoveq1 7178 . . . . . 6 (𝑏 = (1o × {𝑘}) → (𝐺‘(𝑏f𝑐)) = (𝐺‘((1o × {𝑘}) ∘f𝑐)))
2322oveq2d 7171 . . . . 5 (𝑏 = (1o × {𝑘}) → ((𝐹𝑐) · (𝐺‘(𝑏f𝑐))) = ((𝐹𝑐) · (𝐺‘((1o × {𝑘}) ∘f𝑐))))
2421, 23mpteq12dv 5120 . . . 4 (𝑏 = (1o × {𝑘}) → (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r𝑏} ↦ ((𝐹𝑐) · (𝐺‘(𝑏f𝑐)))) = (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} ↦ ((𝐹𝑐) · (𝐺‘((1o × {𝑘}) ∘f𝑐)))))
2524oveq2d 7171 . . 3 (𝑏 = (1o × {𝑘}) → (𝑅 Σg (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r𝑏} ↦ ((𝐹𝑐) · (𝐺‘(𝑏f𝑐))))) = (𝑅 Σg (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} ↦ ((𝐹𝑐) · (𝐺‘((1o × {𝑘}) ∘f𝑐))))))
267, 8, 19, 25fmptco 6887 . 2 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → ((𝐹 𝐺) ∘ (𝑘 ∈ ℕ0 ↦ (1o × {𝑘}))) = (𝑘 ∈ ℕ0 ↦ (𝑅 Σg (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} ↦ ((𝐹𝑐) · (𝐺‘((1o × {𝑘}) ∘f𝑐)))))))
2710psr1ring 20976 . . . 4 (𝑅 ∈ Ring → 𝑆 ∈ Ring)
2811, 14ringcl 19387 . . . 4 ((𝑆 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝐹 𝐺) ∈ 𝐵)
2927, 28syl3an1 1160 . . 3 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝐹 𝐺) ∈ 𝐵)
30 eqid 2758 . . . 4 (coe1‘(𝐹 𝐺)) = (coe1‘(𝐹 𝐺))
31 eqid 2758 . . . 4 (𝑘 ∈ ℕ0 ↦ (1o × {𝑘})) = (𝑘 ∈ ℕ0 ↦ (1o × {𝑘}))
3230, 11, 10, 31coe1fval3 20937 . . 3 ((𝐹 𝐺) ∈ 𝐵 → (coe1‘(𝐹 𝐺)) = ((𝐹 𝐺) ∘ (𝑘 ∈ ℕ0 ↦ (1o × {𝑘}))))
3329, 32syl 17 . 2 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (coe1‘(𝐹 𝐺)) = ((𝐹 𝐺) ∘ (𝑘 ∈ ℕ0 ↦ (1o × {𝑘}))))
34 eqid 2758 . . . . 5 (Base‘𝑅) = (Base‘𝑅)
35 eqid 2758 . . . . 5 (0g𝑅) = (0g𝑅)
36 simpl1 1188 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ Ring)
37 ringcmn 19407 . . . . . 6 (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
3836, 37syl 17 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ CMnd)
39 fzfi 13394 . . . . . 6 (0...𝑘) ∈ Fin
4039a1i 11 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (0...𝑘) ∈ Fin)
41 simpll1 1209 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
42 simpll2 1210 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝐹𝐵)
43 eqid 2758 . . . . . . . . . 10 (coe1𝐹) = (coe1𝐹)
4443, 11, 10, 34coe1f2 20938 . . . . . . . . 9 (𝐹𝐵 → (coe1𝐹):ℕ0⟶(Base‘𝑅))
4542, 44syl 17 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (coe1𝐹):ℕ0⟶(Base‘𝑅))
46 elfznn0 13054 . . . . . . . . 9 (𝑥 ∈ (0...𝑘) → 𝑥 ∈ ℕ0)
4746adantl 485 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝑥 ∈ ℕ0)
4845, 47ffvelrnd 6848 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → ((coe1𝐹)‘𝑥) ∈ (Base‘𝑅))
49 simpll3 1211 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝐺𝐵)
50 eqid 2758 . . . . . . . . . 10 (coe1𝐺) = (coe1𝐺)
5150, 11, 10, 34coe1f2 20938 . . . . . . . . 9 (𝐺𝐵 → (coe1𝐺):ℕ0⟶(Base‘𝑅))
5249, 51syl 17 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (coe1𝐺):ℕ0⟶(Base‘𝑅))
53 fznn0sub 12993 . . . . . . . . 9 (𝑥 ∈ (0...𝑘) → (𝑘𝑥) ∈ ℕ0)
5453adantl 485 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (𝑘𝑥) ∈ ℕ0)
5552, 54ffvelrnd 6848 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → ((coe1𝐺)‘(𝑘𝑥)) ∈ (Base‘𝑅))
5634, 13ringcl 19387 . . . . . . 7 ((𝑅 ∈ Ring ∧ ((coe1𝐹)‘𝑥) ∈ (Base‘𝑅) ∧ ((coe1𝐺)‘(𝑘𝑥)) ∈ (Base‘𝑅)) → (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))) ∈ (Base‘𝑅))
5741, 48, 55, 56syl3anc 1368 . . . . . 6 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))) ∈ (Base‘𝑅))
5857fmpttd 6875 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))):(0...𝑘)⟶(Base‘𝑅))
5939elexi 3429 . . . . . . . . 9 (0...𝑘) ∈ V
6059mptex 6982 . . . . . . . 8 (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ∈ V
61 funmpt 6377 . . . . . . . 8 Fun (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))))
62 fvex 6675 . . . . . . . 8 (0g𝑅) ∈ V
6360, 61, 623pm3.2i 1336 . . . . . . 7 ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ∈ V ∧ Fun (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ∧ (0g𝑅) ∈ V)
64 suppssdm 7856 . . . . . . . . 9 ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) supp (0g𝑅)) ⊆ dom (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))))
65 eqid 2758 . . . . . . . . . 10 (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) = (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))))
6665dmmptss 6074 . . . . . . . . 9 dom (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ⊆ (0...𝑘)
6764, 66sstri 3903 . . . . . . . 8 ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) supp (0g𝑅)) ⊆ (0...𝑘)
6839, 67pm3.2i 474 . . . . . . 7 ((0...𝑘) ∈ Fin ∧ ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) supp (0g𝑅)) ⊆ (0...𝑘))
69 suppssfifsupp 8886 . . . . . . 7 ((((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ∈ V ∧ Fun (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ∧ (0g𝑅) ∈ V) ∧ ((0...𝑘) ∈ Fin ∧ ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) supp (0g𝑅)) ⊆ (0...𝑘))) → (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) finSupp (0g𝑅))
7063, 68, 69mp2an 691 . . . . . 6 (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) finSupp (0g𝑅)
7170a1i 11 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) finSupp (0g𝑅))
72 eqid 2758 . . . . . . 7 {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} = {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}
7372coe1mul2lem2 20997 . . . . . 6 (𝑘 ∈ ℕ0 → (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} ↦ (𝑐‘∅)):{𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}–1-1-onto→(0...𝑘))
7473adantl 485 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} ↦ (𝑐‘∅)):{𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}–1-1-onto→(0...𝑘))
7534, 35, 38, 40, 58, 71, 74gsumf1o 19109 . . . 4 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑅 Σg (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))))) = (𝑅 Σg ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ∘ (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} ↦ (𝑐‘∅)))))
76 breq1 5038 . . . . . . . . . . 11 (𝑑 = 𝑐 → (𝑑r ≤ (1o × {𝑘}) ↔ 𝑐r ≤ (1o × {𝑘})))
7776elrab 3604 . . . . . . . . . 10 (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} ↔ (𝑐 ∈ (ℕ0m 1o) ∧ 𝑐r ≤ (1o × {𝑘})))
7877simprbi 500 . . . . . . . . 9 (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} → 𝑐r ≤ (1o × {𝑘}))
7978adantl 485 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → 𝑐r ≤ (1o × {𝑘}))
80 simplr 768 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → 𝑘 ∈ ℕ0)
81 elrabi 3598 . . . . . . . . . 10 (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} → 𝑐 ∈ (ℕ0m 1o))
8281adantl 485 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → 𝑐 ∈ (ℕ0m 1o))
83 coe1mul2lem1 20996 . . . . . . . . 9 ((𝑘 ∈ ℕ0𝑐 ∈ (ℕ0m 1o)) → (𝑐r ≤ (1o × {𝑘}) ↔ (𝑐‘∅) ∈ (0...𝑘)))
8480, 82, 83syl2anc 587 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → (𝑐r ≤ (1o × {𝑘}) ↔ (𝑐‘∅) ∈ (0...𝑘)))
8579, 84mpbid 235 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → (𝑐‘∅) ∈ (0...𝑘))
86 eqidd 2759 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} ↦ (𝑐‘∅)) = (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} ↦ (𝑐‘∅)))
87 eqidd 2759 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) = (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))))
88 fveq2 6662 . . . . . . . 8 (𝑥 = (𝑐‘∅) → ((coe1𝐹)‘𝑥) = ((coe1𝐹)‘(𝑐‘∅)))
89 oveq2 7163 . . . . . . . . 9 (𝑥 = (𝑐‘∅) → (𝑘𝑥) = (𝑘 − (𝑐‘∅)))
9089fveq2d 6666 . . . . . . . 8 (𝑥 = (𝑐‘∅) → ((coe1𝐺)‘(𝑘𝑥)) = ((coe1𝐺)‘(𝑘 − (𝑐‘∅))))
9188, 90oveq12d 7173 . . . . . . 7 (𝑥 = (𝑐‘∅) → (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))) = (((coe1𝐹)‘(𝑐‘∅)) · ((coe1𝐺)‘(𝑘 − (𝑐‘∅)))))
9285, 86, 87, 91fmptco 6887 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ∘ (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} ↦ (𝑐‘∅))) = (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} ↦ (((coe1𝐹)‘(𝑐‘∅)) · ((coe1𝐺)‘(𝑘 − (𝑐‘∅))))))
93 simpll2 1210 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → 𝐹𝐵)
9443fvcoe1 20936 . . . . . . . . 9 ((𝐹𝐵𝑐 ∈ (ℕ0m 1o)) → (𝐹𝑐) = ((coe1𝐹)‘(𝑐‘∅)))
9593, 82, 94syl2anc 587 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → (𝐹𝑐) = ((coe1𝐹)‘(𝑐‘∅)))
96 df1o2 8131 . . . . . . . . . . . . . 14 1o = {∅}
97 0ex 5180 . . . . . . . . . . . . . 14 ∅ ∈ V
9896, 2, 97mapsnconst 8479 . . . . . . . . . . . . 13 (𝑐 ∈ (ℕ0m 1o) → 𝑐 = (1o × {(𝑐‘∅)}))
9982, 98syl 17 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → 𝑐 = (1o × {(𝑐‘∅)}))
10099oveq2d 7171 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → ((1o × {𝑘}) ∘f𝑐) = ((1o × {𝑘}) ∘f − (1o × {(𝑐‘∅)})))
1013a1i 11 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → 1o ∈ On)
102 vex 3413 . . . . . . . . . . . . 13 𝑘 ∈ V
103102a1i 11 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → 𝑘 ∈ V)
104 fvexd 6677 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → (𝑐‘∅) ∈ V)
105101, 103, 104ofc12 7437 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → ((1o × {𝑘}) ∘f − (1o × {(𝑐‘∅)})) = (1o × {(𝑘 − (𝑐‘∅))}))
106100, 105eqtrd 2793 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → ((1o × {𝑘}) ∘f𝑐) = (1o × {(𝑘 − (𝑐‘∅))}))
107106fveq2d 6666 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → (𝐺‘((1o × {𝑘}) ∘f𝑐)) = (𝐺‘(1o × {(𝑘 − (𝑐‘∅))})))
108 simpll3 1211 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → 𝐺𝐵)
109 fznn0sub 12993 . . . . . . . . . . 11 ((𝑐‘∅) ∈ (0...𝑘) → (𝑘 − (𝑐‘∅)) ∈ ℕ0)
11085, 109syl 17 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → (𝑘 − (𝑐‘∅)) ∈ ℕ0)
11150coe1fv 20935 . . . . . . . . . 10 ((𝐺𝐵 ∧ (𝑘 − (𝑐‘∅)) ∈ ℕ0) → ((coe1𝐺)‘(𝑘 − (𝑐‘∅))) = (𝐺‘(1o × {(𝑘 − (𝑐‘∅))})))
112108, 110, 111syl2anc 587 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → ((coe1𝐺)‘(𝑘 − (𝑐‘∅))) = (𝐺‘(1o × {(𝑘 − (𝑐‘∅))})))
113107, 112eqtr4d 2796 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → (𝐺‘((1o × {𝑘}) ∘f𝑐)) = ((coe1𝐺)‘(𝑘 − (𝑐‘∅))))
11495, 113oveq12d 7173 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → ((𝐹𝑐) · (𝐺‘((1o × {𝑘}) ∘f𝑐))) = (((coe1𝐹)‘(𝑐‘∅)) · ((coe1𝐺)‘(𝑘 − (𝑐‘∅)))))
115114mpteq2dva 5130 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} ↦ ((𝐹𝑐) · (𝐺‘((1o × {𝑘}) ∘f𝑐)))) = (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} ↦ (((coe1𝐹)‘(𝑐‘∅)) · ((coe1𝐺)‘(𝑘 − (𝑐‘∅))))))
11692, 115eqtr4d 2796 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ∘ (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} ↦ (𝑐‘∅))) = (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} ↦ ((𝐹𝑐) · (𝐺‘((1o × {𝑘}) ∘f𝑐)))))
117116oveq2d 7171 . . . 4 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑅 Σg ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ∘ (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} ↦ (𝑐‘∅)))) = (𝑅 Σg (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} ↦ ((𝐹𝑐) · (𝐺‘((1o × {𝑘}) ∘f𝑐))))))
11875, 117eqtrd 2793 . . 3 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑅 Σg (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))))) = (𝑅 Σg (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} ↦ ((𝐹𝑐) · (𝐺‘((1o × {𝑘}) ∘f𝑐))))))
119118mpteq2dva 5130 . 2 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝑘 ∈ ℕ0 ↦ (𝑅 Σg (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))))) = (𝑘 ∈ ℕ0 ↦ (𝑅 Σg (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} ↦ ((𝐹𝑐) · (𝐺‘((1o × {𝑘}) ∘f𝑐)))))))
12026, 33, 1193eqtr4d 2803 1 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (coe1‘(𝐹 𝐺)) = (𝑘 ∈ ℕ0 ↦ (𝑅 Σg (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  {crab 3074  Vcvv 3409   ⊆ wss 3860  ∅c0 4227  {csn 4525   class class class wbr 5035   ↦ cmpt 5115   × cxp 5525  dom cdm 5527   ∘ ccom 5531  Oncon0 6173  Fun wfun 6333  ⟶wf 6335  –1-1-onto→wf1o 6338  ‘cfv 6339  (class class class)co 7155   ∘f cof 7408   ∘r cofr 7409   supp csupp 7840  1oc1o 8110   ↑m cmap 8421  Fincfn 8532   finSupp cfsupp 8871  0cc0 10580   ≤ cle 10719   − cmin 10913  ℕ0cn0 11939  ...cfz 12944  Basecbs 16546  .rcmulr 16629  0gc0g 16776   Σg cgsu 16777  CMndccmn 18978  Ringcrg 19370   mPwSer cmps 20671  PwSer1cps1 20904  coe1cco1 20907 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464  ax-cnex 10636  ax-resscn 10637  ax-1cn 10638  ax-icn 10639  ax-addcl 10640  ax-addrcl 10641  ax-mulcl 10642  ax-mulrcl 10643  ax-mulcom 10644  ax-addass 10645  ax-mulass 10646  ax-distr 10647  ax-i2m1 10648  ax-1ne0 10649  ax-1rid 10650  ax-rnegex 10651  ax-rrecex 10652  ax-cnre 10653  ax-pre-lttri 10654  ax-pre-lttrn 10655  ax-pre-ltadd 10656  ax-pre-mulgt0 10657 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-iin 4889  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-se 5487  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-of 7410  df-ofr 7411  df-om 7585  df-1st 7698  df-2nd 7699  df-supp 7841  df-wrecs 7962  df-recs 8023  df-rdg 8061  df-1o 8117  df-er 8304  df-map 8423  df-pm 8424  df-ixp 8485  df-en 8533  df-dom 8534  df-sdom 8535  df-fin 8536  df-fsupp 8872  df-oi 9012  df-card 9406  df-pnf 10720  df-mnf 10721  df-xr 10722  df-ltxr 10723  df-le 10724  df-sub 10915  df-neg 10916  df-nn 11680  df-2 11742  df-3 11743  df-4 11744  df-5 11745  df-6 11746  df-7 11747  df-8 11748  df-9 11749  df-n0 11940  df-z 12026  df-dec 12143  df-uz 12288  df-fz 12945  df-fzo 13088  df-seq 13424  df-hash 13746  df-struct 16548  df-ndx 16549  df-slot 16550  df-base 16552  df-sets 16553  df-ress 16554  df-plusg 16641  df-mulr 16642  df-sca 16644  df-vsca 16645  df-tset 16647  df-ple 16648  df-0g 16778  df-gsum 16779  df-mre 16920  df-mrc 16921  df-acs 16923  df-mgm 17923  df-sgrp 17972  df-mnd 17983  df-mhm 18027  df-submnd 18028  df-grp 18177  df-minusg 18178  df-mulg 18297  df-ghm 18428  df-cntz 18519  df-cmn 18980  df-abl 18981  df-mgp 19313  df-ur 19325  df-ring 19372  df-psr 20676  df-opsr 20680  df-psr1 20909  df-coe1 20912 This theorem is referenced by:  coe1mul  20999
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