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Theorem coe1mul2 20354
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 6564 . . . . 5 (𝑘 ∈ ℕ0 → (1o × {𝑘}):1o⟶ℕ0)
2 nn0ex 11895 . . . . . 6 0 ∈ V
3 1on 8103 . . . . . . 7 1o ∈ On
43elexi 3518 . . . . . 6 1o ∈ V
52, 4elmap 8428 . . . . 5 ((1o × {𝑘}) ∈ (ℕ0m 1o) ↔ (1o × {𝑘}):1o⟶ℕ0)
61, 5sylibr 235 . . . 4 (𝑘 ∈ ℕ0 → (1o × {𝑘}) ∈ (ℕ0m 1o))
76adantl 482 . . 3 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (1o × {𝑘}) ∈ (ℕ0m 1o))
8 eqidd 2826 . . 3 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝑘 ∈ ℕ0 ↦ (1o × {𝑘})) = (𝑘 ∈ ℕ0 ↦ (1o × {𝑘})))
9 eqid 2825 . . . 4 (1o mPwSer 𝑅) = (1o mPwSer 𝑅)
10 coe1mul2.s . . . . 5 𝑆 = (PwSer1𝑅)
11 coe1mul2.b . . . . 5 𝐵 = (Base‘𝑆)
1210, 11, 9psr1bas2 20275 . . . 4 𝐵 = (Base‘(1o mPwSer 𝑅))
13 coe1mul2.u . . . 4 · = (.r𝑅)
14 coe1mul2.t . . . . 5 = (.r𝑆)
1510, 9, 14psr1mulr 20309 . . . 4 = (.r‘(1o mPwSer 𝑅))
16 psr1baslem 20270 . . . 4 (ℕ0m 1o) = {𝑎 ∈ (ℕ0m 1o) ∣ (𝑎 “ ℕ) ∈ Fin}
17 simp2 1131 . . . 4 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → 𝐹𝐵)
18 simp3 1132 . . . 4 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → 𝐺𝐵)
199, 12, 13, 15, 16, 17, 18psrmulfval 20086 . . 3 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝐹 𝐺) = (𝑏 ∈ (ℕ0m 1o) ↦ (𝑅 Σg (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r𝑏} ↦ ((𝐹𝑐) · (𝐺‘(𝑏f𝑐)))))))
20 breq2 5066 . . . . . 6 (𝑏 = (1o × {𝑘}) → (𝑑r𝑏𝑑r ≤ (1o × {𝑘})))
2120rabbidv 3485 . . . . 5 (𝑏 = (1o × {𝑘}) → {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r𝑏} = {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})})
22 fvoveq1 7174 . . . . . 6 (𝑏 = (1o × {𝑘}) → (𝐺‘(𝑏f𝑐)) = (𝐺‘((1o × {𝑘}) ∘f𝑐)))
2322oveq2d 7167 . . . . 5 (𝑏 = (1o × {𝑘}) → ((𝐹𝑐) · (𝐺‘(𝑏f𝑐))) = ((𝐹𝑐) · (𝐺‘((1o × {𝑘}) ∘f𝑐))))
2421, 23mpteq12dv 5147 . . . 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 7167 . . 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 6886 . 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 20332 . . . 4 (𝑅 ∈ Ring → 𝑆 ∈ Ring)
2811, 14ringcl 19233 . . . 4 ((𝑆 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝐹 𝐺) ∈ 𝐵)
2927, 28syl3an1 1157 . . 3 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝐹 𝐺) ∈ 𝐵)
30 eqid 2825 . . . 4 (coe1‘(𝐹 𝐺)) = (coe1‘(𝐹 𝐺))
31 eqid 2825 . . . 4 (𝑘 ∈ ℕ0 ↦ (1o × {𝑘})) = (𝑘 ∈ ℕ0 ↦ (1o × {𝑘}))
3230, 11, 10, 31coe1fval3 20293 . . 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 2825 . . . . 5 (Base‘𝑅) = (Base‘𝑅)
35 eqid 2825 . . . . 5 (0g𝑅) = (0g𝑅)
36 simpl1 1185 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ Ring)
37 ringcmn 19253 . . . . . 6 (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
3836, 37syl 17 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ CMnd)
39 fzfi 13333 . . . . . 6 (0...𝑘) ∈ Fin
4039a1i 11 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (0...𝑘) ∈ Fin)
41 simpll1 1206 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
42 simpll2 1207 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝐹𝐵)
43 eqid 2825 . . . . . . . . . 10 (coe1𝐹) = (coe1𝐹)
4443, 11, 10, 34coe1f2 20294 . . . . . . . . 9 (𝐹𝐵 → (coe1𝐹):ℕ0⟶(Base‘𝑅))
4542, 44syl 17 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (coe1𝐹):ℕ0⟶(Base‘𝑅))
46 elfznn0 12993 . . . . . . . . 9 (𝑥 ∈ (0...𝑘) → 𝑥 ∈ ℕ0)
4746adantl 482 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝑥 ∈ ℕ0)
4845, 47ffvelrnd 6847 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → ((coe1𝐹)‘𝑥) ∈ (Base‘𝑅))
49 simpll3 1208 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝐺𝐵)
50 eqid 2825 . . . . . . . . . 10 (coe1𝐺) = (coe1𝐺)
5150, 11, 10, 34coe1f2 20294 . . . . . . . . 9 (𝐺𝐵 → (coe1𝐺):ℕ0⟶(Base‘𝑅))
5249, 51syl 17 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (coe1𝐺):ℕ0⟶(Base‘𝑅))
53 fznn0sub 12932 . . . . . . . . 9 (𝑥 ∈ (0...𝑘) → (𝑘𝑥) ∈ ℕ0)
5453adantl 482 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (𝑘𝑥) ∈ ℕ0)
5552, 54ffvelrnd 6847 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → ((coe1𝐺)‘(𝑘𝑥)) ∈ (Base‘𝑅))
5634, 13ringcl 19233 . . . . . . 7 ((𝑅 ∈ Ring ∧ ((coe1𝐹)‘𝑥) ∈ (Base‘𝑅) ∧ ((coe1𝐺)‘(𝑘𝑥)) ∈ (Base‘𝑅)) → (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))) ∈ (Base‘𝑅))
5741, 48, 55, 56syl3anc 1365 . . . . . 6 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))) ∈ (Base‘𝑅))
5857fmpttd 6874 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))):(0...𝑘)⟶(Base‘𝑅))
5939elexi 3518 . . . . . . . . 9 (0...𝑘) ∈ V
6059mptex 6984 . . . . . . . 8 (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ∈ V
61 funmpt 6389 . . . . . . . 8 Fun (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))))
62 fvex 6679 . . . . . . . 8 (0g𝑅) ∈ V
6360, 61, 623pm3.2i 1333 . . . . . . 7 ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ∈ V ∧ Fun (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ∧ (0g𝑅) ∈ V)
64 suppssdm 7837 . . . . . . . . 9 ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) supp (0g𝑅)) ⊆ dom (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))))
65 eqid 2825 . . . . . . . . . 10 (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) = (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))))
6665dmmptss 6092 . . . . . . . . 9 dom (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ⊆ (0...𝑘)
6764, 66sstri 3979 . . . . . . . 8 ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) supp (0g𝑅)) ⊆ (0...𝑘)
6839, 67pm3.2i 471 . . . . . . 7 ((0...𝑘) ∈ Fin ∧ ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) supp (0g𝑅)) ⊆ (0...𝑘))
69 suppssfifsupp 8840 . . . . . . 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 688 . . . . . 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 2825 . . . . . . 7 {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} = {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}
7372coe1mul2lem2 20353 . . . . . 6 (𝑘 ∈ ℕ0 → (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} ↦ (𝑐‘∅)):{𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}–1-1-onto→(0...𝑘))
7473adantl 482 . . . . 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 18958 . . . 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 5065 . . . . . . . . . . 11 (𝑑 = 𝑐 → (𝑑r ≤ (1o × {𝑘}) ↔ 𝑐r ≤ (1o × {𝑘})))
7776elrab 3683 . . . . . . . . . 10 (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} ↔ (𝑐 ∈ (ℕ0m 1o) ∧ 𝑐r ≤ (1o × {𝑘})))
7877simprbi 497 . . . . . . . . 9 (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} → 𝑐r ≤ (1o × {𝑘}))
7978adantl 482 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → 𝑐r ≤ (1o × {𝑘}))
80 simplr 765 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → 𝑘 ∈ ℕ0)
81 elrabi 3678 . . . . . . . . . 10 (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} → 𝑐 ∈ (ℕ0m 1o))
8281adantl 482 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → 𝑐 ∈ (ℕ0m 1o))
83 coe1mul2lem1 20352 . . . . . . . . 9 ((𝑘 ∈ ℕ0𝑐 ∈ (ℕ0m 1o)) → (𝑐r ≤ (1o × {𝑘}) ↔ (𝑐‘∅) ∈ (0...𝑘)))
8480, 82, 83syl2anc 584 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → (𝑐r ≤ (1o × {𝑘}) ↔ (𝑐‘∅) ∈ (0...𝑘)))
8579, 84mpbid 233 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → (𝑐‘∅) ∈ (0...𝑘))
86 eqidd 2826 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} ↦ (𝑐‘∅)) = (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} ↦ (𝑐‘∅)))
87 eqidd 2826 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) = (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))))
88 fveq2 6666 . . . . . . . 8 (𝑥 = (𝑐‘∅) → ((coe1𝐹)‘𝑥) = ((coe1𝐹)‘(𝑐‘∅)))
89 oveq2 7159 . . . . . . . . 9 (𝑥 = (𝑐‘∅) → (𝑘𝑥) = (𝑘 − (𝑐‘∅)))
9089fveq2d 6670 . . . . . . . 8 (𝑥 = (𝑐‘∅) → ((coe1𝐺)‘(𝑘𝑥)) = ((coe1𝐺)‘(𝑘 − (𝑐‘∅))))
9188, 90oveq12d 7169 . . . . . . 7 (𝑥 = (𝑐‘∅) → (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))) = (((coe1𝐹)‘(𝑐‘∅)) · ((coe1𝐺)‘(𝑘 − (𝑐‘∅)))))
9285, 86, 87, 91fmptco 6886 . . . . . 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 1207 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → 𝐹𝐵)
9443fvcoe1 20292 . . . . . . . . 9 ((𝐹𝐵𝑐 ∈ (ℕ0m 1o)) → (𝐹𝑐) = ((coe1𝐹)‘(𝑐‘∅)))
9593, 82, 94syl2anc 584 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → (𝐹𝑐) = ((coe1𝐹)‘(𝑐‘∅)))
96 df1o2 8110 . . . . . . . . . . . . . 14 1o = {∅}
97 0ex 5207 . . . . . . . . . . . . . 14 ∅ ∈ V
9896, 2, 97mapsnconst 8448 . . . . . . . . . . . . 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 7167 . . . . . . . . . . 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 3502 . . . . . . . . . . . . 13 𝑘 ∈ V
103102a1i 11 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → 𝑘 ∈ V)
104 fvexd 6681 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → (𝑐‘∅) ∈ V)
105101, 103, 104ofc12 7427 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → ((1o × {𝑘}) ∘f − (1o × {(𝑐‘∅)})) = (1o × {(𝑘 − (𝑐‘∅))}))
106100, 105eqtrd 2860 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → ((1o × {𝑘}) ∘f𝑐) = (1o × {(𝑘 − (𝑐‘∅))}))
107106fveq2d 6670 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → (𝐺‘((1o × {𝑘}) ∘f𝑐)) = (𝐺‘(1o × {(𝑘 − (𝑐‘∅))})))
108 simpll3 1208 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → 𝐺𝐵)
109 fznn0sub 12932 . . . . . . . . . . 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 20291 . . . . . . . . . 10 ((𝐺𝐵 ∧ (𝑘 − (𝑐‘∅)) ∈ ℕ0) → ((coe1𝐺)‘(𝑘 − (𝑐‘∅))) = (𝐺‘(1o × {(𝑘 − (𝑐‘∅))})))
112108, 110, 111syl2anc 584 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → ((coe1𝐺)‘(𝑘 − (𝑐‘∅))) = (𝐺‘(1o × {(𝑘 − (𝑐‘∅))})))
113107, 112eqtr4d 2863 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → (𝐺‘((1o × {𝑘}) ∘f𝑐)) = ((coe1𝐺)‘(𝑘 − (𝑐‘∅))))
11495, 113oveq12d 7169 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → ((𝐹𝑐) · (𝐺‘((1o × {𝑘}) ∘f𝑐))) = (((coe1𝐹)‘(𝑐‘∅)) · ((coe1𝐺)‘(𝑘 − (𝑐‘∅)))))
115114mpteq2dva 5157 . . . . . 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 2863 . . . . 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 7167 . . . 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 2860 . . 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 5157 . 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 2870 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 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  {crab 3146  Vcvv 3499  wss 3939  c0 4294  {csn 4563   class class class wbr 5062  cmpt 5142   × cxp 5551  dom cdm 5553  ccom 5557  Oncon0 6188  Fun wfun 6345  wf 6347  1-1-ontowf1o 6350  cfv 6351  (class class class)co 7151  f cof 7400  r cofr 7401   supp csupp 7824  1oc1o 8089  m cmap 8399  Fincfn 8501   finSupp cfsupp 8825  0cc0 10529  cle 10668  cmin 10862  0cn0 11889  ...cfz 12885  Basecbs 16475  .rcmulr 16558  0gc0g 16705   Σg cgsu 16706  CMndccmn 18828  Ringcrg 19219   mPwSer cmps 20052  PwSer1cps1 20260  coe1cco1 20263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-iin 4919  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-of 7402  df-ofr 7403  df-om 7572  df-1st 7683  df-2nd 7684  df-supp 7825  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8282  df-map 8401  df-pm 8402  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fsupp 8826  df-oi 8966  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12886  df-fzo 13027  df-seq 13363  df-hash 13684  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-sca 16573  df-vsca 16574  df-tset 16576  df-ple 16577  df-0g 16707  df-gsum 16708  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17892  df-mnd 17903  df-mhm 17946  df-submnd 17947  df-grp 18038  df-minusg 18039  df-mulg 18157  df-ghm 18288  df-cntz 18379  df-cmn 18830  df-abl 18831  df-mgp 19162  df-ur 19174  df-ring 19221  df-psr 20057  df-opsr 20061  df-psr1 20265  df-coe1 20268
This theorem is referenced by:  coe1mul  20355
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