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Theorem coe1mul2 22272
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 6797 . . . . 5 (𝑘 ∈ ℕ0 → (1o × {𝑘}):1o⟶ℕ0)
2 nn0ex 12532 . . . . . 6 0 ∈ V
3 1on 8518 . . . . . . 7 1o ∈ On
43elexi 3503 . . . . . 6 1o ∈ V
52, 4elmap 8911 . . . . 5 ((1o × {𝑘}) ∈ (ℕ0m 1o) ↔ (1o × {𝑘}):1o⟶ℕ0)
61, 5sylibr 234 . . . 4 (𝑘 ∈ ℕ0 → (1o × {𝑘}) ∈ (ℕ0m 1o))
76adantl 481 . . 3 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (1o × {𝑘}) ∈ (ℕ0m 1o))
8 eqidd 2738 . . 3 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝑘 ∈ ℕ0 ↦ (1o × {𝑘})) = (𝑘 ∈ ℕ0 ↦ (1o × {𝑘})))
9 eqid 2737 . . . 4 (1o mPwSer 𝑅) = (1o mPwSer 𝑅)
10 coe1mul2.s . . . . 5 𝑆 = (PwSer1𝑅)
11 coe1mul2.b . . . . 5 𝐵 = (Base‘𝑆)
1210, 11, 9psr1bas2 22191 . . . 4 𝐵 = (Base‘(1o mPwSer 𝑅))
13 coe1mul2.u . . . 4 · = (.r𝑅)
14 coe1mul2.t . . . . 5 = (.r𝑆)
1510, 9, 14psr1mulr 22224 . . . 4 = (.r‘(1o mPwSer 𝑅))
16 psr1baslem 22186 . . . 4 (ℕ0m 1o) = {𝑎 ∈ (ℕ0m 1o) ∣ (𝑎 “ ℕ) ∈ Fin}
17 simp2 1138 . . . 4 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → 𝐹𝐵)
18 simp3 1139 . . . 4 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → 𝐺𝐵)
199, 12, 13, 15, 16, 17, 18psrmulfval 21963 . . 3 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝐹 𝐺) = (𝑏 ∈ (ℕ0m 1o) ↦ (𝑅 Σg (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r𝑏} ↦ ((𝐹𝑐) · (𝐺‘(𝑏f𝑐)))))))
20 breq2 5147 . . . . . 6 (𝑏 = (1o × {𝑘}) → (𝑑r𝑏𝑑r ≤ (1o × {𝑘})))
2120rabbidv 3444 . . . . 5 (𝑏 = (1o × {𝑘}) → {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r𝑏} = {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})})
22 fvoveq1 7454 . . . . . 6 (𝑏 = (1o × {𝑘}) → (𝐺‘(𝑏f𝑐)) = (𝐺‘((1o × {𝑘}) ∘f𝑐)))
2322oveq2d 7447 . . . . 5 (𝑏 = (1o × {𝑘}) → ((𝐹𝑐) · (𝐺‘(𝑏f𝑐))) = ((𝐹𝑐) · (𝐺‘((1o × {𝑘}) ∘f𝑐))))
2421, 23mpteq12dv 5233 . . . 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 7447 . . 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 7149 . 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 22248 . . . 4 (𝑅 ∈ Ring → 𝑆 ∈ Ring)
2811, 14ringcl 20247 . . . 4 ((𝑆 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝐹 𝐺) ∈ 𝐵)
2927, 28syl3an1 1164 . . 3 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝐹 𝐺) ∈ 𝐵)
30 eqid 2737 . . . 4 (coe1‘(𝐹 𝐺)) = (coe1‘(𝐹 𝐺))
31 eqid 2737 . . . 4 (𝑘 ∈ ℕ0 ↦ (1o × {𝑘})) = (𝑘 ∈ ℕ0 ↦ (1o × {𝑘}))
3230, 11, 10, 31coe1fval3 22210 . . 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 2737 . . . . 5 (Base‘𝑅) = (Base‘𝑅)
35 eqid 2737 . . . . 5 (0g𝑅) = (0g𝑅)
36 simpl1 1192 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ Ring)
37 ringcmn 20279 . . . . . 6 (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
3836, 37syl 17 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ CMnd)
39 fzfi 14013 . . . . . 6 (0...𝑘) ∈ Fin
4039a1i 11 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (0...𝑘) ∈ Fin)
41 simpll1 1213 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
42 simpll2 1214 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝐹𝐵)
43 eqid 2737 . . . . . . . . . 10 (coe1𝐹) = (coe1𝐹)
4443, 11, 10, 34coe1f2 22211 . . . . . . . . 9 (𝐹𝐵 → (coe1𝐹):ℕ0⟶(Base‘𝑅))
4542, 44syl 17 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (coe1𝐹):ℕ0⟶(Base‘𝑅))
46 elfznn0 13660 . . . . . . . . 9 (𝑥 ∈ (0...𝑘) → 𝑥 ∈ ℕ0)
4746adantl 481 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝑥 ∈ ℕ0)
4845, 47ffvelcdmd 7105 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → ((coe1𝐹)‘𝑥) ∈ (Base‘𝑅))
49 simpll3 1215 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝐺𝐵)
50 eqid 2737 . . . . . . . . . 10 (coe1𝐺) = (coe1𝐺)
5150, 11, 10, 34coe1f2 22211 . . . . . . . . 9 (𝐺𝐵 → (coe1𝐺):ℕ0⟶(Base‘𝑅))
5249, 51syl 17 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (coe1𝐺):ℕ0⟶(Base‘𝑅))
53 fznn0sub 13596 . . . . . . . . 9 (𝑥 ∈ (0...𝑘) → (𝑘𝑥) ∈ ℕ0)
5453adantl 481 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (𝑘𝑥) ∈ ℕ0)
5552, 54ffvelcdmd 7105 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → ((coe1𝐺)‘(𝑘𝑥)) ∈ (Base‘𝑅))
5634, 13ringcl 20247 . . . . . . 7 ((𝑅 ∈ Ring ∧ ((coe1𝐹)‘𝑥) ∈ (Base‘𝑅) ∧ ((coe1𝐺)‘(𝑘𝑥)) ∈ (Base‘𝑅)) → (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))) ∈ (Base‘𝑅))
5741, 48, 55, 56syl3anc 1373 . . . . . 6 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))) ∈ (Base‘𝑅))
5857fmpttd 7135 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))):(0...𝑘)⟶(Base‘𝑅))
5939elexi 3503 . . . . . . . . 9 (0...𝑘) ∈ V
6059mptex 7243 . . . . . . . 8 (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ∈ V
61 funmpt 6604 . . . . . . . 8 Fun (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))))
62 fvex 6919 . . . . . . . 8 (0g𝑅) ∈ V
6360, 61, 623pm3.2i 1340 . . . . . . 7 ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ∈ V ∧ Fun (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ∧ (0g𝑅) ∈ V)
64 suppssdm 8202 . . . . . . . . 9 ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) supp (0g𝑅)) ⊆ dom (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))))
65 eqid 2737 . . . . . . . . . 10 (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) = (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))))
6665dmmptss 6261 . . . . . . . . 9 dom (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ⊆ (0...𝑘)
6764, 66sstri 3993 . . . . . . . 8 ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) supp (0g𝑅)) ⊆ (0...𝑘)
6839, 67pm3.2i 470 . . . . . . 7 ((0...𝑘) ∈ Fin ∧ ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) supp (0g𝑅)) ⊆ (0...𝑘))
69 suppssfifsupp 9420 . . . . . . 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 692 . . . . . 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 2737 . . . . . . 7 {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} = {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}
7372coe1mul2lem2 22271 . . . . . 6 (𝑘 ∈ ℕ0 → (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} ↦ (𝑐‘∅)):{𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}–1-1-onto→(0...𝑘))
7473adantl 481 . . . . 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 19934 . . . 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 5146 . . . . . . . . . . 11 (𝑑 = 𝑐 → (𝑑r ≤ (1o × {𝑘}) ↔ 𝑐r ≤ (1o × {𝑘})))
7776elrab 3692 . . . . . . . . . 10 (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} ↔ (𝑐 ∈ (ℕ0m 1o) ∧ 𝑐r ≤ (1o × {𝑘})))
7877simprbi 496 . . . . . . . . 9 (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} → 𝑐r ≤ (1o × {𝑘}))
7978adantl 481 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → 𝑐r ≤ (1o × {𝑘}))
80 simplr 769 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → 𝑘 ∈ ℕ0)
81 elrabi 3687 . . . . . . . . . 10 (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} → 𝑐 ∈ (ℕ0m 1o))
8281adantl 481 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → 𝑐 ∈ (ℕ0m 1o))
83 coe1mul2lem1 22270 . . . . . . . . 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 232 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → (𝑐‘∅) ∈ (0...𝑘))
86 eqidd 2738 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} ↦ (𝑐‘∅)) = (𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})} ↦ (𝑐‘∅)))
87 eqidd 2738 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) = (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))))
88 fveq2 6906 . . . . . . . 8 (𝑥 = (𝑐‘∅) → ((coe1𝐹)‘𝑥) = ((coe1𝐹)‘(𝑐‘∅)))
89 oveq2 7439 . . . . . . . . 9 (𝑥 = (𝑐‘∅) → (𝑘𝑥) = (𝑘 − (𝑐‘∅)))
9089fveq2d 6910 . . . . . . . 8 (𝑥 = (𝑐‘∅) → ((coe1𝐺)‘(𝑘𝑥)) = ((coe1𝐺)‘(𝑘 − (𝑐‘∅))))
9188, 90oveq12d 7449 . . . . . . 7 (𝑥 = (𝑐‘∅) → (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))) = (((coe1𝐹)‘(𝑐‘∅)) · ((coe1𝐺)‘(𝑘 − (𝑐‘∅)))))
9285, 86, 87, 91fmptco 7149 . . . . . 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 1214 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → 𝐹𝐵)
9443fvcoe1 22209 . . . . . . . . 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 8513 . . . . . . . . . . . . . 14 1o = {∅}
97 0ex 5307 . . . . . . . . . . . . . 14 ∅ ∈ V
9896, 2, 97mapsnconst 8932 . . . . . . . . . . . . 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 7447 . . . . . . . . . . 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 3484 . . . . . . . . . . . . 13 𝑘 ∈ V
103102a1i 11 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → 𝑘 ∈ V)
104 fvexd 6921 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → (𝑐‘∅) ∈ V)
105101, 103, 104ofc12 7727 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → ((1o × {𝑘}) ∘f − (1o × {(𝑐‘∅)})) = (1o × {(𝑘 − (𝑐‘∅))}))
106100, 105eqtrd 2777 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → ((1o × {𝑘}) ∘f𝑐) = (1o × {(𝑘 − (𝑐‘∅))}))
107106fveq2d 6910 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → (𝐺‘((1o × {𝑘}) ∘f𝑐)) = (𝐺‘(1o × {(𝑘 − (𝑐‘∅))})))
108 simpll3 1215 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → 𝐺𝐵)
109 fznn0sub 13596 . . . . . . . . . . 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 22208 . . . . . . . . . 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 2780 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → (𝐺‘((1o × {𝑘}) ∘f𝑐)) = ((coe1𝐺)‘(𝑘 − (𝑐‘∅))))
11495, 113oveq12d 7449 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0m 1o) ∣ 𝑑r ≤ (1o × {𝑘})}) → ((𝐹𝑐) · (𝐺‘((1o × {𝑘}) ∘f𝑐))) = (((coe1𝐹)‘(𝑐‘∅)) · ((coe1𝐺)‘(𝑘 − (𝑐‘∅)))))
115114mpteq2dva 5242 . . . . . 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 2780 . . . . 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 7447 . . . 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 2777 . . 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 5242 . 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 2787 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  wss 3951  c0 4333  {csn 4626   class class class wbr 5143  cmpt 5225   × cxp 5683  dom cdm 5685  ccom 5689  Oncon0 6384  Fun wfun 6555  wf 6557  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  f cof 7695  r cofr 7696   supp csupp 8185  1oc1o 8499  m cmap 8866  Fincfn 8985   finSupp cfsupp 9401  0cc0 11155  cle 11296  cmin 11492  0cn0 12526  ...cfz 13547  Basecbs 17247  .rcmulr 17298  0gc0g 17484   Σg cgsu 17485  CMndccmn 19798  Ringcrg 20230   mPwSer cmps 21924  PwSer1cps1 22176  coe1cco1 22179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-mulg 19086  df-ghm 19231  df-cntz 19335  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-psr 21929  df-opsr 21933  df-psr1 22181  df-coe1 22184
This theorem is referenced by:  coe1mul  22273
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