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Theorem coe1mul2 20365
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 6561 . . . . 5 (𝑘 ∈ ℕ0 → (1o × {𝑘}):1o⟶ℕ0)
2 nn0ex 11891 . . . . . 6 0 ∈ V
3 1on 8098 . . . . . . 7 1o ∈ On
43elexi 3511 . . . . . 6 1o ∈ V
52, 4elmap 8424 . . . . 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 2819 . . 3 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝑘 ∈ ℕ0 ↦ (1o × {𝑘})) = (𝑘 ∈ ℕ0 ↦ (1o × {𝑘})))
9 eqid 2818 . . . 4 (1o mPwSer 𝑅) = (1o mPwSer 𝑅)
10 coe1mul2.s . . . . 5 𝑆 = (PwSer1𝑅)
11 coe1mul2.b . . . . 5 𝐵 = (Base‘𝑆)
1210, 11, 9psr1bas2 20286 . . . 4 𝐵 = (Base‘(1o mPwSer 𝑅))
13 coe1mul2.u . . . 4 · = (.r𝑅)
14 coe1mul2.t . . . . 5 = (.r𝑆)
1510, 9, 14psr1mulr 20320 . . . 4 = (.r‘(1o mPwSer 𝑅))
16 psr1baslem 20281 . . . 4 (ℕ0m 1o) = {𝑎 ∈ (ℕ0m 1o) ∣ (𝑎 “ ℕ) ∈ Fin}
17 simp2 1129 . . . 4 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → 𝐹𝐵)
18 simp3 1130 . . . 4 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → 𝐺𝐵)
199, 12, 13, 15, 16, 17, 18psrmulfval 20093 . . 3 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝐹 𝐺) = (𝑏 ∈ (ℕ0m 1o) ↦ (𝑅 Σg (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r𝑏} ↦ ((𝐹𝑐) · (𝐺‘(𝑏f𝑐)))))))
20 breq2 5061 . . . . . 6 (𝑏 = (1o × {𝑘}) → (𝑑r𝑏𝑑r ≤ (1o × {𝑘})))
2120rabbidv 3478 . . . . 5 (𝑏 = (1o × {𝑘}) → {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r𝑏} = {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})})
22 fvoveq1 7168 . . . . . 6 (𝑏 = (1o × {𝑘}) → (𝐺‘(𝑏f𝑐)) = (𝐺‘((1o × {𝑘}) ∘f𝑐)))
2322oveq2d 7161 . . . . 5 (𝑏 = (1o × {𝑘}) → ((𝐹𝑐) · (𝐺‘(𝑏f𝑐))) = ((𝐹𝑐) · (𝐺‘((1o × {𝑘}) ∘f𝑐))))
2421, 23mpteq12dv 5142 . . . 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 7161 . . 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 6883 . 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 20343 . . . 4 (𝑅 ∈ Ring → 𝑆 ∈ Ring)
2811, 14ringcl 19240 . . . 4 ((𝑆 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝐹 𝐺) ∈ 𝐵)
2927, 28syl3an1 1155 . . 3 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝐹 𝐺) ∈ 𝐵)
30 eqid 2818 . . . 4 (coe1‘(𝐹 𝐺)) = (coe1‘(𝐹 𝐺))
31 eqid 2818 . . . 4 (𝑘 ∈ ℕ0 ↦ (1o × {𝑘})) = (𝑘 ∈ ℕ0 ↦ (1o × {𝑘}))
3230, 11, 10, 31coe1fval3 20304 . . 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 2818 . . . . 5 (Base‘𝑅) = (Base‘𝑅)
35 eqid 2818 . . . . 5 (0g𝑅) = (0g𝑅)
36 simpl1 1183 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ Ring)
37 ringcmn 19260 . . . . . 6 (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
3836, 37syl 17 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ CMnd)
39 fzfi 13328 . . . . . 6 (0...𝑘) ∈ Fin
4039a1i 11 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (0...𝑘) ∈ Fin)
41 simpll1 1204 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
42 simpll2 1205 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝐹𝐵)
43 eqid 2818 . . . . . . . . . 10 (coe1𝐹) = (coe1𝐹)
4443, 11, 10, 34coe1f2 20305 . . . . . . . . 9 (𝐹𝐵 → (coe1𝐹):ℕ0⟶(Base‘𝑅))
4542, 44syl 17 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (coe1𝐹):ℕ0⟶(Base‘𝑅))
46 elfznn0 12988 . . . . . . . . 9 (𝑥 ∈ (0...𝑘) → 𝑥 ∈ ℕ0)
4746adantl 482 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝑥 ∈ ℕ0)
4845, 47ffvelrnd 6844 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → ((coe1𝐹)‘𝑥) ∈ (Base‘𝑅))
49 simpll3 1206 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝐺𝐵)
50 eqid 2818 . . . . . . . . . 10 (coe1𝐺) = (coe1𝐺)
5150, 11, 10, 34coe1f2 20305 . . . . . . . . 9 (𝐺𝐵 → (coe1𝐺):ℕ0⟶(Base‘𝑅))
5249, 51syl 17 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (coe1𝐺):ℕ0⟶(Base‘𝑅))
53 fznn0sub 12927 . . . . . . . . 9 (𝑥 ∈ (0...𝑘) → (𝑘𝑥) ∈ ℕ0)
5453adantl 482 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (𝑘𝑥) ∈ ℕ0)
5552, 54ffvelrnd 6844 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → ((coe1𝐺)‘(𝑘𝑥)) ∈ (Base‘𝑅))
5634, 13ringcl 19240 . . . . . . 7 ((𝑅 ∈ Ring ∧ ((coe1𝐹)‘𝑥) ∈ (Base‘𝑅) ∧ ((coe1𝐺)‘(𝑘𝑥)) ∈ (Base‘𝑅)) → (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))) ∈ (Base‘𝑅))
5741, 48, 55, 56syl3anc 1363 . . . . . 6 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))) ∈ (Base‘𝑅))
5857fmpttd 6871 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))):(0...𝑘)⟶(Base‘𝑅))
5939elexi 3511 . . . . . . . . 9 (0...𝑘) ∈ V
6059mptex 6977 . . . . . . . 8 (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ∈ V
61 funmpt 6386 . . . . . . . 8 Fun (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))))
62 fvex 6676 . . . . . . . 8 (0g𝑅) ∈ V
6360, 61, 623pm3.2i 1331 . . . . . . 7 ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ∈ V ∧ Fun (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ∧ (0g𝑅) ∈ V)
64 suppssdm 7832 . . . . . . . . 9 ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) supp (0g𝑅)) ⊆ dom (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))))
65 eqid 2818 . . . . . . . . . 10 (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) = (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))))
6665dmmptss 6088 . . . . . . . . 9 dom (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ⊆ (0...𝑘)
6764, 66sstri 3973 . . . . . . . 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 8836 . . . . . . 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 2818 . . . . . . 7 {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} = {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}
7372coe1mul2lem2 20364 . . . . . 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 18965 . . . 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 5060 . . . . . . . . . . 11 (𝑑 = 𝑐 → (𝑑r ≤ (1o × {𝑘}) ↔ 𝑐r ≤ (1o × {𝑘})))
7776elrab 3677 . . . . . . . . . 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 3672 . . . . . . . . . 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 20363 . . . . . . . . 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 2819 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} ↦ (𝑐‘∅)) = (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} ↦ (𝑐‘∅)))
87 eqidd 2819 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) = (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))))
88 fveq2 6663 . . . . . . . 8 (𝑥 = (𝑐‘∅) → ((coe1𝐹)‘𝑥) = ((coe1𝐹)‘(𝑐‘∅)))
89 oveq2 7153 . . . . . . . . 9 (𝑥 = (𝑐‘∅) → (𝑘𝑥) = (𝑘 − (𝑐‘∅)))
9089fveq2d 6667 . . . . . . . 8 (𝑥 = (𝑐‘∅) → ((coe1𝐺)‘(𝑘𝑥)) = ((coe1𝐺)‘(𝑘 − (𝑐‘∅))))
9188, 90oveq12d 7163 . . . . . . 7 (𝑥 = (𝑐‘∅) → (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))) = (((coe1𝐹)‘(𝑐‘∅)) · ((coe1𝐺)‘(𝑘 − (𝑐‘∅)))))
9285, 86, 87, 91fmptco 6883 . . . . . 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 1205 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → 𝐹𝐵)
9443fvcoe1 20303 . . . . . . . . 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 8105 . . . . . . . . . . . . . 14 1o = {∅}
97 0ex 5202 . . . . . . . . . . . . . 14 ∅ ∈ V
9896, 2, 97mapsnconst 8444 . . . . . . . . . . . . 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 7161 . . . . . . . . . . 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 3495 . . . . . . . . . . . . 13 𝑘 ∈ V
103102a1i 11 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → 𝑘 ∈ V)
104 fvexd 6678 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → (𝑐‘∅) ∈ V)
105101, 103, 104ofc12 7423 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → ((1o × {𝑘}) ∘f − (1o × {(𝑐‘∅)})) = (1o × {(𝑘 − (𝑐‘∅))}))
106100, 105eqtrd 2853 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → ((1o × {𝑘}) ∘f𝑐) = (1o × {(𝑘 − (𝑐‘∅))}))
107106fveq2d 6667 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → (𝐺‘((1o × {𝑘}) ∘f𝑐)) = (𝐺‘(1o × {(𝑘 − (𝑐‘∅))})))
108 simpll3 1206 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → 𝐺𝐵)
109 fznn0sub 12927 . . . . . . . . . . 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 20302 . . . . . . . . . 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 2856 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → (𝐺‘((1o × {𝑘}) ∘f𝑐)) = ((coe1𝐺)‘(𝑘 − (𝑐‘∅))))
11495, 113oveq12d 7163 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → ((𝐹𝑐) · (𝐺‘((1o × {𝑘}) ∘f𝑐))) = (((coe1𝐹)‘(𝑐‘∅)) · ((coe1𝐺)‘(𝑘 − (𝑐‘∅)))))
115114mpteq2dva 5152 . . . . . 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 2856 . . . . 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 7161 . . . 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 2853 . . 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 5152 . 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 2863 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 1079   = wceq 1528  wcel 2105  {crab 3139  Vcvv 3492  wss 3933  c0 4288  {csn 4557   class class class wbr 5057  cmpt 5137   × cxp 5546  dom cdm 5548  ccom 5552  Oncon0 6184  Fun wfun 6342  wf 6344  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  f cof 7396  r cofr 7397   supp csupp 7819  1oc1o 8084  m cmap 8395  Fincfn 8497   finSupp cfsupp 8821  0cc0 10525  cle 10664  cmin 10858  0cn0 11885  ...cfz 12880  Basecbs 16471  .rcmulr 16554  0gc0g 16701   Σg cgsu 16702  CMndccmn 18835  Ringcrg 19226   mPwSer cmps 20059  PwSer1cps1 20271  coe1cco1 20274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-ofr 7399  df-om 7570  df-1st 7678  df-2nd 7679  df-supp 7820  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fsupp 8822  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-fz 12881  df-fzo 13022  df-seq 13358  df-hash 13679  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-mulr 16567  df-sca 16569  df-vsca 16570  df-tset 16572  df-ple 16573  df-0g 16703  df-gsum 16704  df-mre 16845  df-mrc 16846  df-acs 16848  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-mhm 17944  df-submnd 17945  df-grp 18044  df-minusg 18045  df-mulg 18163  df-ghm 18294  df-cntz 18385  df-cmn 18837  df-abl 18838  df-mgp 19169  df-ur 19181  df-ring 19228  df-psr 20064  df-opsr 20068  df-psr1 20276  df-coe1 20279
This theorem is referenced by:  coe1mul  20366
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