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Theorem coe1mul2 19402
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 5988 . . . . 5 (𝑘 ∈ ℕ0 → (1𝑜 × {𝑘}):1𝑜⟶ℕ0)
2 nn0ex 11141 . . . . . 6 0 ∈ V
3 1on 7427 . . . . . . 7 1𝑜 ∈ On
43elexi 3181 . . . . . 6 1𝑜 ∈ V
52, 4elmap 7745 . . . . 5 ((1𝑜 × {𝑘}) ∈ (ℕ0𝑚 1𝑜) ↔ (1𝑜 × {𝑘}):1𝑜⟶ℕ0)
61, 5sylibr 222 . . . 4 (𝑘 ∈ ℕ0 → (1𝑜 × {𝑘}) ∈ (ℕ0𝑚 1𝑜))
76adantl 480 . . 3 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (1𝑜 × {𝑘}) ∈ (ℕ0𝑚 1𝑜))
8 eqidd 2606 . . 3 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝑘 ∈ ℕ0 ↦ (1𝑜 × {𝑘})) = (𝑘 ∈ ℕ0 ↦ (1𝑜 × {𝑘})))
9 eqid 2605 . . . 4 (1𝑜 mPwSer 𝑅) = (1𝑜 mPwSer 𝑅)
10 coe1mul2.s . . . . 5 𝑆 = (PwSer1𝑅)
11 coe1mul2.b . . . . 5 𝐵 = (Base‘𝑆)
1210, 11, 9psr1bas2 19323 . . . 4 𝐵 = (Base‘(1𝑜 mPwSer 𝑅))
13 coe1mul2.u . . . 4 · = (.r𝑅)
14 coe1mul2.t . . . . 5 = (.r𝑆)
1510, 9, 14psr1mulr 19357 . . . 4 = (.r‘(1𝑜 mPwSer 𝑅))
16 psr1baslem 19318 . . . 4 (ℕ0𝑚 1𝑜) = {𝑎 ∈ (ℕ0𝑚 1𝑜) ∣ (𝑎 “ ℕ) ∈ Fin}
17 simp2 1054 . . . 4 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → 𝐹𝐵)
18 simp3 1055 . . . 4 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → 𝐺𝐵)
199, 12, 13, 15, 16, 17, 18psrmulfval 19148 . . 3 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝐹 𝐺) = (𝑏 ∈ (ℕ0𝑚 1𝑜) ↦ (𝑅 Σg (𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟𝑏} ↦ ((𝐹𝑐) · (𝐺‘(𝑏𝑓𝑐)))))))
20 breq2 4577 . . . . . 6 (𝑏 = (1𝑜 × {𝑘}) → (𝑑𝑟𝑏𝑑𝑟 ≤ (1𝑜 × {𝑘})))
2120rabbidv 3159 . . . . 5 (𝑏 = (1𝑜 × {𝑘}) → {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟𝑏} = {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})})
22 oveq1 6530 . . . . . . 7 (𝑏 = (1𝑜 × {𝑘}) → (𝑏𝑓𝑐) = ((1𝑜 × {𝑘}) ∘𝑓𝑐))
2322fveq2d 6088 . . . . . 6 (𝑏 = (1𝑜 × {𝑘}) → (𝐺‘(𝑏𝑓𝑐)) = (𝐺‘((1𝑜 × {𝑘}) ∘𝑓𝑐)))
2423oveq2d 6539 . . . . 5 (𝑏 = (1𝑜 × {𝑘}) → ((𝐹𝑐) · (𝐺‘(𝑏𝑓𝑐))) = ((𝐹𝑐) · (𝐺‘((1𝑜 × {𝑘}) ∘𝑓𝑐))))
2521, 24mpteq12dv 4653 . . . 4 (𝑏 = (1𝑜 × {𝑘}) → (𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟𝑏} ↦ ((𝐹𝑐) · (𝐺‘(𝑏𝑓𝑐)))) = (𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})} ↦ ((𝐹𝑐) · (𝐺‘((1𝑜 × {𝑘}) ∘𝑓𝑐)))))
2625oveq2d 6539 . . 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))))))
277, 8, 19, 26fmptco 6284 . 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)))))))
2810psr1ring 19380 . . . 4 (𝑅 ∈ Ring → 𝑆 ∈ Ring)
2911, 14ringcl 18326 . . . 4 ((𝑆 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝐹 𝐺) ∈ 𝐵)
3028, 29syl3an1 1350 . . 3 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝐹 𝐺) ∈ 𝐵)
31 eqid 2605 . . . 4 (coe1‘(𝐹 𝐺)) = (coe1‘(𝐹 𝐺))
32 eqid 2605 . . . 4 (𝑘 ∈ ℕ0 ↦ (1𝑜 × {𝑘})) = (𝑘 ∈ ℕ0 ↦ (1𝑜 × {𝑘}))
3331, 11, 10, 32coe1fval3 19341 . . 3 ((𝐹 𝐺) ∈ 𝐵 → (coe1‘(𝐹 𝐺)) = ((𝐹 𝐺) ∘ (𝑘 ∈ ℕ0 ↦ (1𝑜 × {𝑘}))))
3430, 33syl 17 . 2 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (coe1‘(𝐹 𝐺)) = ((𝐹 𝐺) ∘ (𝑘 ∈ ℕ0 ↦ (1𝑜 × {𝑘}))))
35 eqid 2605 . . . . 5 (Base‘𝑅) = (Base‘𝑅)
36 eqid 2605 . . . . 5 (0g𝑅) = (0g𝑅)
37 simpl1 1056 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ Ring)
38 ringcmn 18346 . . . . . 6 (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
3937, 38syl 17 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ CMnd)
40 fzfi 12584 . . . . . 6 (0...𝑘) ∈ Fin
4140a1i 11 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (0...𝑘) ∈ Fin)
42 simpll1 1092 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
43 simpll2 1093 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝐹𝐵)
44 eqid 2605 . . . . . . . . . 10 (coe1𝐹) = (coe1𝐹)
4544, 11, 10, 35coe1f2 19342 . . . . . . . . 9 (𝐹𝐵 → (coe1𝐹):ℕ0⟶(Base‘𝑅))
4643, 45syl 17 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (coe1𝐹):ℕ0⟶(Base‘𝑅))
47 elfznn0 12253 . . . . . . . . 9 (𝑥 ∈ (0...𝑘) → 𝑥 ∈ ℕ0)
4847adantl 480 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝑥 ∈ ℕ0)
4946, 48ffvelrnd 6249 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → ((coe1𝐹)‘𝑥) ∈ (Base‘𝑅))
50 simpll3 1094 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝐺𝐵)
51 eqid 2605 . . . . . . . . . 10 (coe1𝐺) = (coe1𝐺)
5251, 11, 10, 35coe1f2 19342 . . . . . . . . 9 (𝐺𝐵 → (coe1𝐺):ℕ0⟶(Base‘𝑅))
5350, 52syl 17 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (coe1𝐺):ℕ0⟶(Base‘𝑅))
54 fznn0sub 12195 . . . . . . . . 9 (𝑥 ∈ (0...𝑘) → (𝑘𝑥) ∈ ℕ0)
5554adantl 480 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (𝑘𝑥) ∈ ℕ0)
5653, 55ffvelrnd 6249 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → ((coe1𝐺)‘(𝑘𝑥)) ∈ (Base‘𝑅))
5735, 13ringcl 18326 . . . . . . 7 ((𝑅 ∈ Ring ∧ ((coe1𝐹)‘𝑥) ∈ (Base‘𝑅) ∧ ((coe1𝐺)‘(𝑘𝑥)) ∈ (Base‘𝑅)) → (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))) ∈ (Base‘𝑅))
5842, 49, 56, 57syl3anc 1317 . . . . . 6 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))) ∈ (Base‘𝑅))
59 eqid 2605 . . . . . 6 (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) = (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))))
6058, 59fmptd 6273 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))):(0...𝑘)⟶(Base‘𝑅))
6140elexi 3181 . . . . . . . . 9 (0...𝑘) ∈ V
6261mptex 6364 . . . . . . . 8 (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ∈ V
63 funmpt 5822 . . . . . . . 8 Fun (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))))
64 fvex 6094 . . . . . . . 8 (0g𝑅) ∈ V
6562, 63, 643pm3.2i 1231 . . . . . . 7 ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ∈ V ∧ Fun (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ∧ (0g𝑅) ∈ V)
66 suppssdm 7168 . . . . . . . . 9 ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) supp (0g𝑅)) ⊆ dom (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))))
6759dmmptss 5530 . . . . . . . . 9 dom (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ⊆ (0...𝑘)
6866, 67sstri 3572 . . . . . . . 8 ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) supp (0g𝑅)) ⊆ (0...𝑘)
6940, 68pm3.2i 469 . . . . . . 7 ((0...𝑘) ∈ Fin ∧ ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) supp (0g𝑅)) ⊆ (0...𝑘))
70 suppssfifsupp 8146 . . . . . . 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))
7165, 69, 70mp2an 703 . . . . . 6 (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) finSupp (0g𝑅)
7271a1i 11 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) finSupp (0g𝑅))
73 eqid 2605 . . . . . . 7 {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})} = {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}
7473coe1mul2lem2 19401 . . . . . 6 (𝑘 ∈ ℕ0 → (𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})} ↦ (𝑐‘∅)):{𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}–1-1-onto→(0...𝑘))
7574adantl 480 . . . . 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))
7635, 36, 39, 41, 60, 72, 75gsumf1o 18082 . . . 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‘∅)))))
77 breq1 4576 . . . . . . . . . . 11 (𝑑 = 𝑐 → (𝑑𝑟 ≤ (1𝑜 × {𝑘}) ↔ 𝑐𝑟 ≤ (1𝑜 × {𝑘})))
7877elrab 3326 . . . . . . . . . 10 (𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})} ↔ (𝑐 ∈ (ℕ0𝑚 1𝑜) ∧ 𝑐𝑟 ≤ (1𝑜 × {𝑘})))
7978simprbi 478 . . . . . . . . 9 (𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})} → 𝑐𝑟 ≤ (1𝑜 × {𝑘}))
8079adantl 480 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → 𝑐𝑟 ≤ (1𝑜 × {𝑘}))
81 simplr 787 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → 𝑘 ∈ ℕ0)
82 elrabi 3323 . . . . . . . . . 10 (𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})} → 𝑐 ∈ (ℕ0𝑚 1𝑜))
8382adantl 480 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → 𝑐 ∈ (ℕ0𝑚 1𝑜))
84 coe1mul2lem1 19400 . . . . . . . . 9 ((𝑘 ∈ ℕ0𝑐 ∈ (ℕ0𝑚 1𝑜)) → (𝑐𝑟 ≤ (1𝑜 × {𝑘}) ↔ (𝑐‘∅) ∈ (0...𝑘)))
8581, 83, 84syl2anc 690 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → (𝑐𝑟 ≤ (1𝑜 × {𝑘}) ↔ (𝑐‘∅) ∈ (0...𝑘)))
8680, 85mpbid 220 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → (𝑐‘∅) ∈ (0...𝑘))
87 eqidd 2606 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})} ↦ (𝑐‘∅)) = (𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})} ↦ (𝑐‘∅)))
88 eqidd 2606 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) = (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))))
89 fveq2 6084 . . . . . . . 8 (𝑥 = (𝑐‘∅) → ((coe1𝐹)‘𝑥) = ((coe1𝐹)‘(𝑐‘∅)))
90 oveq2 6531 . . . . . . . . 9 (𝑥 = (𝑐‘∅) → (𝑘𝑥) = (𝑘 − (𝑐‘∅)))
9190fveq2d 6088 . . . . . . . 8 (𝑥 = (𝑐‘∅) → ((coe1𝐺)‘(𝑘𝑥)) = ((coe1𝐺)‘(𝑘 − (𝑐‘∅))))
9289, 91oveq12d 6541 . . . . . . 7 (𝑥 = (𝑐‘∅) → (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))) = (((coe1𝐹)‘(𝑐‘∅)) · ((coe1𝐺)‘(𝑘 − (𝑐‘∅)))))
9386, 87, 88, 92fmptco 6284 . . . . . 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‘∅))))))
94 simpll2 1093 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → 𝐹𝐵)
9544fvcoe1 19340 . . . . . . . . 9 ((𝐹𝐵𝑐 ∈ (ℕ0𝑚 1𝑜)) → (𝐹𝑐) = ((coe1𝐹)‘(𝑐‘∅)))
9694, 83, 95syl2anc 690 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → (𝐹𝑐) = ((coe1𝐹)‘(𝑐‘∅)))
97 df1o2 7432 . . . . . . . . . . . . . 14 1𝑜 = {∅}
98 0ex 4709 . . . . . . . . . . . . . 14 ∅ ∈ V
9997, 2, 98mapsnconst 7762 . . . . . . . . . . . . 13 (𝑐 ∈ (ℕ0𝑚 1𝑜) → 𝑐 = (1𝑜 × {(𝑐‘∅)}))
10083, 99syl 17 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → 𝑐 = (1𝑜 × {(𝑐‘∅)}))
101100oveq2d 6539 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → ((1𝑜 × {𝑘}) ∘𝑓𝑐) = ((1𝑜 × {𝑘}) ∘𝑓 − (1𝑜 × {(𝑐‘∅)})))
1023a1i 11 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → 1𝑜 ∈ On)
103 vex 3171 . . . . . . . . . . . . 13 𝑘 ∈ V
104103a1i 11 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → 𝑘 ∈ V)
105 fvex 6094 . . . . . . . . . . . . 13 (𝑐‘∅) ∈ V
106105a1i 11 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → (𝑐‘∅) ∈ V)
107102, 104, 106ofc12 6793 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → ((1𝑜 × {𝑘}) ∘𝑓 − (1𝑜 × {(𝑐‘∅)})) = (1𝑜 × {(𝑘 − (𝑐‘∅))}))
108101, 107eqtrd 2639 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → ((1𝑜 × {𝑘}) ∘𝑓𝑐) = (1𝑜 × {(𝑘 − (𝑐‘∅))}))
109108fveq2d 6088 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → (𝐺‘((1𝑜 × {𝑘}) ∘𝑓𝑐)) = (𝐺‘(1𝑜 × {(𝑘 − (𝑐‘∅))})))
110 simpll3 1094 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → 𝐺𝐵)
111 fznn0sub 12195 . . . . . . . . . . 11 ((𝑐‘∅) ∈ (0...𝑘) → (𝑘 − (𝑐‘∅)) ∈ ℕ0)
11286, 111syl 17 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → (𝑘 − (𝑐‘∅)) ∈ ℕ0)
11351coe1fv 19339 . . . . . . . . . 10 ((𝐺𝐵 ∧ (𝑘 − (𝑐‘∅)) ∈ ℕ0) → ((coe1𝐺)‘(𝑘 − (𝑐‘∅))) = (𝐺‘(1𝑜 × {(𝑘 − (𝑐‘∅))})))
114110, 112, 113syl2anc 690 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → ((coe1𝐺)‘(𝑘 − (𝑐‘∅))) = (𝐺‘(1𝑜 × {(𝑘 − (𝑐‘∅))})))
115109, 114eqtr4d 2642 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → (𝐺‘((1𝑜 × {𝑘}) ∘𝑓𝑐)) = ((coe1𝐺)‘(𝑘 − (𝑐‘∅))))
11696, 115oveq12d 6541 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → ((𝐹𝑐) · (𝐺‘((1𝑜 × {𝑘}) ∘𝑓𝑐))) = (((coe1𝐹)‘(𝑐‘∅)) · ((coe1𝐺)‘(𝑘 − (𝑐‘∅)))))
117116mpteq2dva 4662 . . . . . 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‘∅))))))
11893, 117eqtr4d 2642 . . . . 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)))))
119118oveq2d 6539 . . . 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))))))
12076, 119eqtrd 2639 . . 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))))))
121120mpteq2dva 4662 . 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)))))))
12227, 34, 1213eqtr4d 2649 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 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  {crab 2895  Vcvv 3168  wss 3535  c0 3869  {csn 4120   class class class wbr 4573  cmpt 4633   × cxp 5022  dom cdm 5024  ccom 5028  Oncon0 5622  Fun wfun 5780  wf 5782  1-1-ontowf1o 5785  cfv 5786  (class class class)co 6523  𝑓 cof 6766  𝑟 cofr 6767   supp csupp 7155  1𝑜c1o 7413  𝑚 cmap 7717  Fincfn 7814   finSupp cfsupp 8131  0cc0 9788  cle 9927  cmin 10113  0cn0 11135  ...cfz 12148  Basecbs 15637  .rcmulr 15711  0gc0g 15865   Σg cgsu 15866  CMndccmn 17958  Ringcrg 18312   mPwSer cmps 19114  PwSer1cps1 19308  coe1cco1 19311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-inf2 8394  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-iin 4448  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-se 4984  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-isom 5795  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-of 6768  df-ofr 6769  df-om 6931  df-1st 7032  df-2nd 7033  df-supp 7156  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-2o 7421  df-oadd 7424  df-er 7602  df-map 7719  df-pm 7720  df-ixp 7768  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-fsupp 8132  df-oi 8271  df-card 8621  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-nn 10864  df-2 10922  df-3 10923  df-4 10924  df-5 10925  df-6 10926  df-7 10927  df-8 10928  df-9 10929  df-n0 11136  df-z 11207  df-dec 11322  df-uz 11516  df-fz 12149  df-fzo 12286  df-seq 12615  df-hash 12931  df-struct 15639  df-ndx 15640  df-slot 15641  df-base 15642  df-sets 15643  df-ress 15644  df-plusg 15723  df-mulr 15724  df-sca 15726  df-vsca 15727  df-tset 15729  df-ple 15730  df-0g 15867  df-gsum 15868  df-mre 16011  df-mrc 16012  df-acs 16014  df-mgm 17007  df-sgrp 17049  df-mnd 17060  df-mhm 17100  df-submnd 17101  df-grp 17190  df-minusg 17191  df-mulg 17306  df-ghm 17423  df-cntz 17515  df-cmn 17960  df-abl 17961  df-mgp 18255  df-ur 18267  df-ring 18314  df-psr 19119  df-opsr 19123  df-psr1 19313  df-coe1 19316
This theorem is referenced by:  coe1mul  19403
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