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Theorem coe1mul2 19633
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 6092 . . . . 5 (𝑘 ∈ ℕ0 → (1𝑜 × {𝑘}):1𝑜⟶ℕ0)
2 nn0ex 11295 . . . . . 6 0 ∈ V
3 1on 7564 . . . . . . 7 1𝑜 ∈ On
43elexi 3211 . . . . . 6 1𝑜 ∈ V
52, 4elmap 7883 . . . . 5 ((1𝑜 × {𝑘}) ∈ (ℕ0𝑚 1𝑜) ↔ (1𝑜 × {𝑘}):1𝑜⟶ℕ0)
61, 5sylibr 224 . . . 4 (𝑘 ∈ ℕ0 → (1𝑜 × {𝑘}) ∈ (ℕ0𝑚 1𝑜))
76adantl 482 . . 3 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (1𝑜 × {𝑘}) ∈ (ℕ0𝑚 1𝑜))
8 eqidd 2622 . . 3 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝑘 ∈ ℕ0 ↦ (1𝑜 × {𝑘})) = (𝑘 ∈ ℕ0 ↦ (1𝑜 × {𝑘})))
9 eqid 2621 . . . 4 (1𝑜 mPwSer 𝑅) = (1𝑜 mPwSer 𝑅)
10 coe1mul2.s . . . . 5 𝑆 = (PwSer1𝑅)
11 coe1mul2.b . . . . 5 𝐵 = (Base‘𝑆)
1210, 11, 9psr1bas2 19554 . . . 4 𝐵 = (Base‘(1𝑜 mPwSer 𝑅))
13 coe1mul2.u . . . 4 · = (.r𝑅)
14 coe1mul2.t . . . . 5 = (.r𝑆)
1510, 9, 14psr1mulr 19588 . . . 4 = (.r‘(1𝑜 mPwSer 𝑅))
16 psr1baslem 19549 . . . 4 (ℕ0𝑚 1𝑜) = {𝑎 ∈ (ℕ0𝑚 1𝑜) ∣ (𝑎 “ ℕ) ∈ Fin}
17 simp2 1061 . . . 4 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → 𝐹𝐵)
18 simp3 1062 . . . 4 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → 𝐺𝐵)
199, 12, 13, 15, 16, 17, 18psrmulfval 19379 . . 3 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝐹 𝐺) = (𝑏 ∈ (ℕ0𝑚 1𝑜) ↦ (𝑅 Σg (𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟𝑏} ↦ ((𝐹𝑐) · (𝐺‘(𝑏𝑓𝑐)))))))
20 breq2 4655 . . . . . 6 (𝑏 = (1𝑜 × {𝑘}) → (𝑑𝑟𝑏𝑑𝑟 ≤ (1𝑜 × {𝑘})))
2120rabbidv 3187 . . . . 5 (𝑏 = (1𝑜 × {𝑘}) → {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟𝑏} = {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})})
22 oveq1 6654 . . . . . . 7 (𝑏 = (1𝑜 × {𝑘}) → (𝑏𝑓𝑐) = ((1𝑜 × {𝑘}) ∘𝑓𝑐))
2322fveq2d 6193 . . . . . 6 (𝑏 = (1𝑜 × {𝑘}) → (𝐺‘(𝑏𝑓𝑐)) = (𝐺‘((1𝑜 × {𝑘}) ∘𝑓𝑐)))
2423oveq2d 6663 . . . . 5 (𝑏 = (1𝑜 × {𝑘}) → ((𝐹𝑐) · (𝐺‘(𝑏𝑓𝑐))) = ((𝐹𝑐) · (𝐺‘((1𝑜 × {𝑘}) ∘𝑓𝑐))))
2521, 24mpteq12dv 4731 . . . 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 6663 . . 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 6394 . 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 19611 . . . 4 (𝑅 ∈ Ring → 𝑆 ∈ Ring)
2911, 14ringcl 18555 . . . 4 ((𝑆 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝐹 𝐺) ∈ 𝐵)
3028, 29syl3an1 1358 . . 3 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝐹 𝐺) ∈ 𝐵)
31 eqid 2621 . . . 4 (coe1‘(𝐹 𝐺)) = (coe1‘(𝐹 𝐺))
32 eqid 2621 . . . 4 (𝑘 ∈ ℕ0 ↦ (1𝑜 × {𝑘})) = (𝑘 ∈ ℕ0 ↦ (1𝑜 × {𝑘}))
3331, 11, 10, 32coe1fval3 19572 . . 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 2621 . . . . 5 (Base‘𝑅) = (Base‘𝑅)
36 eqid 2621 . . . . 5 (0g𝑅) = (0g𝑅)
37 simpl1 1063 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ Ring)
38 ringcmn 18575 . . . . . 6 (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
3937, 38syl 17 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ CMnd)
40 fzfi 12766 . . . . . 6 (0...𝑘) ∈ Fin
4140a1i 11 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (0...𝑘) ∈ Fin)
42 simpll1 1099 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
43 simpll2 1100 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝐹𝐵)
44 eqid 2621 . . . . . . . . . 10 (coe1𝐹) = (coe1𝐹)
4544, 11, 10, 35coe1f2 19573 . . . . . . . . 9 (𝐹𝐵 → (coe1𝐹):ℕ0⟶(Base‘𝑅))
4643, 45syl 17 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (coe1𝐹):ℕ0⟶(Base‘𝑅))
47 elfznn0 12429 . . . . . . . . 9 (𝑥 ∈ (0...𝑘) → 𝑥 ∈ ℕ0)
4847adantl 482 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝑥 ∈ ℕ0)
4946, 48ffvelrnd 6358 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → ((coe1𝐹)‘𝑥) ∈ (Base‘𝑅))
50 simpll3 1101 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝐺𝐵)
51 eqid 2621 . . . . . . . . . 10 (coe1𝐺) = (coe1𝐺)
5251, 11, 10, 35coe1f2 19573 . . . . . . . . 9 (𝐺𝐵 → (coe1𝐺):ℕ0⟶(Base‘𝑅))
5350, 52syl 17 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (coe1𝐺):ℕ0⟶(Base‘𝑅))
54 fznn0sub 12370 . . . . . . . . 9 (𝑥 ∈ (0...𝑘) → (𝑘𝑥) ∈ ℕ0)
5554adantl 482 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (𝑘𝑥) ∈ ℕ0)
5653, 55ffvelrnd 6358 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → ((coe1𝐺)‘(𝑘𝑥)) ∈ (Base‘𝑅))
5735, 13ringcl 18555 . . . . . . 7 ((𝑅 ∈ Ring ∧ ((coe1𝐹)‘𝑥) ∈ (Base‘𝑅) ∧ ((coe1𝐺)‘(𝑘𝑥)) ∈ (Base‘𝑅)) → (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))) ∈ (Base‘𝑅))
5842, 49, 56, 57syl3anc 1325 . . . . . 6 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))) ∈ (Base‘𝑅))
59 eqid 2621 . . . . . 6 (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) = (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))))
6058, 59fmptd 6383 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))):(0...𝑘)⟶(Base‘𝑅))
6140elexi 3211 . . . . . . . . 9 (0...𝑘) ∈ V
6261mptex 6483 . . . . . . . 8 (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ∈ V
63 funmpt 5924 . . . . . . . 8 Fun (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))))
64 fvex 6199 . . . . . . . 8 (0g𝑅) ∈ V
6562, 63, 643pm3.2i 1238 . . . . . . 7 ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ∈ V ∧ Fun (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ∧ (0g𝑅) ∈ V)
66 suppssdm 7305 . . . . . . . . 9 ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) supp (0g𝑅)) ⊆ dom (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))))
6759dmmptss 5629 . . . . . . . . 9 dom (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ⊆ (0...𝑘)
6866, 67sstri 3610 . . . . . . . 8 ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) supp (0g𝑅)) ⊆ (0...𝑘)
6940, 68pm3.2i 471 . . . . . . 7 ((0...𝑘) ∈ Fin ∧ ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) supp (0g𝑅)) ⊆ (0...𝑘))
70 suppssfifsupp 8287 . . . . . . 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 708 . . . . . 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 2621 . . . . . . 7 {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})} = {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}
7473coe1mul2lem2 19632 . . . . . 6 (𝑘 ∈ ℕ0 → (𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})} ↦ (𝑐‘∅)):{𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}–1-1-onto→(0...𝑘))
7574adantl 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))
7635, 36, 39, 41, 60, 72, 75gsumf1o 18311 . . . 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 4654 . . . . . . . . . . 11 (𝑑 = 𝑐 → (𝑑𝑟 ≤ (1𝑜 × {𝑘}) ↔ 𝑐𝑟 ≤ (1𝑜 × {𝑘})))
7877elrab 3361 . . . . . . . . . 10 (𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})} ↔ (𝑐 ∈ (ℕ0𝑚 1𝑜) ∧ 𝑐𝑟 ≤ (1𝑜 × {𝑘})))
7978simprbi 480 . . . . . . . . 9 (𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})} → 𝑐𝑟 ≤ (1𝑜 × {𝑘}))
8079adantl 482 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → 𝑐𝑟 ≤ (1𝑜 × {𝑘}))
81 simplr 792 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → 𝑘 ∈ ℕ0)
82 elrabi 3357 . . . . . . . . . 10 (𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})} → 𝑐 ∈ (ℕ0𝑚 1𝑜))
8382adantl 482 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → 𝑐 ∈ (ℕ0𝑚 1𝑜))
84 coe1mul2lem1 19631 . . . . . . . . 9 ((𝑘 ∈ ℕ0𝑐 ∈ (ℕ0𝑚 1𝑜)) → (𝑐𝑟 ≤ (1𝑜 × {𝑘}) ↔ (𝑐‘∅) ∈ (0...𝑘)))
8581, 83, 84syl2anc 693 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → (𝑐𝑟 ≤ (1𝑜 × {𝑘}) ↔ (𝑐‘∅) ∈ (0...𝑘)))
8680, 85mpbid 222 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → (𝑐‘∅) ∈ (0...𝑘))
87 eqidd 2622 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})} ↦ (𝑐‘∅)) = (𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})} ↦ (𝑐‘∅)))
88 eqidd 2622 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) = (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))))
89 fveq2 6189 . . . . . . . 8 (𝑥 = (𝑐‘∅) → ((coe1𝐹)‘𝑥) = ((coe1𝐹)‘(𝑐‘∅)))
90 oveq2 6655 . . . . . . . . 9 (𝑥 = (𝑐‘∅) → (𝑘𝑥) = (𝑘 − (𝑐‘∅)))
9190fveq2d 6193 . . . . . . . 8 (𝑥 = (𝑐‘∅) → ((coe1𝐺)‘(𝑘𝑥)) = ((coe1𝐺)‘(𝑘 − (𝑐‘∅))))
9289, 91oveq12d 6665 . . . . . . 7 (𝑥 = (𝑐‘∅) → (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))) = (((coe1𝐹)‘(𝑐‘∅)) · ((coe1𝐺)‘(𝑘 − (𝑐‘∅)))))
9386, 87, 88, 92fmptco 6394 . . . . . 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 1100 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → 𝐹𝐵)
9544fvcoe1 19571 . . . . . . . . 9 ((𝐹𝐵𝑐 ∈ (ℕ0𝑚 1𝑜)) → (𝐹𝑐) = ((coe1𝐹)‘(𝑐‘∅)))
9694, 83, 95syl2anc 693 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → (𝐹𝑐) = ((coe1𝐹)‘(𝑐‘∅)))
97 df1o2 7569 . . . . . . . . . . . . . 14 1𝑜 = {∅}
98 0ex 4788 . . . . . . . . . . . . . 14 ∅ ∈ V
9997, 2, 98mapsnconst 7900 . . . . . . . . . . . . 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 6663 . . . . . . . . . . 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 3201 . . . . . . . . . . . . 13 𝑘 ∈ V
104103a1i 11 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → 𝑘 ∈ V)
105 fvexd 6201 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → (𝑐‘∅) ∈ V)
106102, 104, 105ofc12 6919 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → ((1𝑜 × {𝑘}) ∘𝑓 − (1𝑜 × {(𝑐‘∅)})) = (1𝑜 × {(𝑘 − (𝑐‘∅))}))
107101, 106eqtrd 2655 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → ((1𝑜 × {𝑘}) ∘𝑓𝑐) = (1𝑜 × {(𝑘 − (𝑐‘∅))}))
108107fveq2d 6193 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → (𝐺‘((1𝑜 × {𝑘}) ∘𝑓𝑐)) = (𝐺‘(1𝑜 × {(𝑘 − (𝑐‘∅))})))
109 simpll3 1101 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → 𝐺𝐵)
110 fznn0sub 12370 . . . . . . . . . . 11 ((𝑐‘∅) ∈ (0...𝑘) → (𝑘 − (𝑐‘∅)) ∈ ℕ0)
11186, 110syl 17 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → (𝑘 − (𝑐‘∅)) ∈ ℕ0)
11251coe1fv 19570 . . . . . . . . . 10 ((𝐺𝐵 ∧ (𝑘 − (𝑐‘∅)) ∈ ℕ0) → ((coe1𝐺)‘(𝑘 − (𝑐‘∅))) = (𝐺‘(1𝑜 × {(𝑘 − (𝑐‘∅))})))
113109, 111, 112syl2anc 693 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → ((coe1𝐺)‘(𝑘 − (𝑐‘∅))) = (𝐺‘(1𝑜 × {(𝑘 − (𝑐‘∅))})))
114108, 113eqtr4d 2658 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → (𝐺‘((1𝑜 × {𝑘}) ∘𝑓𝑐)) = ((coe1𝐺)‘(𝑘 − (𝑐‘∅))))
11596, 114oveq12d 6665 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → ((𝐹𝑐) · (𝐺‘((1𝑜 × {𝑘}) ∘𝑓𝑐))) = (((coe1𝐹)‘(𝑐‘∅)) · ((coe1𝐺)‘(𝑘 − (𝑐‘∅)))))
116115mpteq2dva 4742 . . . . . 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‘∅))))))
11793, 116eqtr4d 2658 . . . . 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)))))
118117oveq2d 6663 . . . 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))))))
11976, 118eqtrd 2655 . . 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))))))
120119mpteq2dva 4742 . 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)))))))
12127, 34, 1203eqtr4d 2665 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 196  wa 384  w3a 1037   = wceq 1482  wcel 1989  {crab 2915  Vcvv 3198  wss 3572  c0 3913  {csn 4175   class class class wbr 4651  cmpt 4727   × cxp 5110  dom cdm 5112  ccom 5116  Oncon0 5721  Fun wfun 5880  wf 5882  1-1-ontowf1o 5885  cfv 5886  (class class class)co 6647  𝑓 cof 6892  𝑟 cofr 6893   supp csupp 7292  1𝑜c1o 7550  𝑚 cmap 7854  Fincfn 7952   finSupp cfsupp 8272  0cc0 9933  cle 10072  cmin 10263  0cn0 11289  ...cfz 12323  Basecbs 15851  .rcmulr 15936  0gc0g 16094   Σg cgsu 16095  CMndccmn 18187  Ringcrg 18541   mPwSer cmps 19345  PwSer1cps1 19539  coe1cco1 19542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-inf2 8535  ax-cnex 9989  ax-resscn 9990  ax-1cn 9991  ax-icn 9992  ax-addcl 9993  ax-addrcl 9994  ax-mulcl 9995  ax-mulrcl 9996  ax-mulcom 9997  ax-addass 9998  ax-mulass 9999  ax-distr 10000  ax-i2m1 10001  ax-1ne0 10002  ax-1rid 10003  ax-rnegex 10004  ax-rrecex 10005  ax-cnre 10006  ax-pre-lttri 10007  ax-pre-lttrn 10008  ax-pre-ltadd 10009  ax-pre-mulgt0 10010
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-iin 4521  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-se 5072  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-isom 5895  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-of 6894  df-ofr 6895  df-om 7063  df-1st 7165  df-2nd 7166  df-supp 7293  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-2o 7558  df-oadd 7561  df-er 7739  df-map 7856  df-pm 7857  df-ixp 7906  df-en 7953  df-dom 7954  df-sdom 7955  df-fin 7956  df-fsupp 8273  df-oi 8412  df-card 8762  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-sub 10265  df-neg 10266  df-nn 11018  df-2 11076  df-3 11077  df-4 11078  df-5 11079  df-6 11080  df-7 11081  df-8 11082  df-9 11083  df-n0 11290  df-z 11375  df-dec 11491  df-uz 11685  df-fz 12324  df-fzo 12462  df-seq 12797  df-hash 13113  df-struct 15853  df-ndx 15854  df-slot 15855  df-base 15857  df-sets 15858  df-ress 15859  df-plusg 15948  df-mulr 15949  df-sca 15951  df-vsca 15952  df-tset 15954  df-ple 15955  df-0g 16096  df-gsum 16097  df-mre 16240  df-mrc 16241  df-acs 16243  df-mgm 17236  df-sgrp 17278  df-mnd 17289  df-mhm 17329  df-submnd 17330  df-grp 17419  df-minusg 17420  df-mulg 17535  df-ghm 17652  df-cntz 17744  df-cmn 18189  df-abl 18190  df-mgp 18484  df-ur 18496  df-ring 18543  df-psr 19350  df-opsr 19354  df-psr1 19544  df-coe1 19547
This theorem is referenced by:  coe1mul  19634
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