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Theorem coe1mul2 19912
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 6276 . . . . 5 (𝑘 ∈ ℕ0 → (1𝑜 × {𝑘}):1𝑜⟶ℕ0)
2 nn0ex 11545 . . . . . 6 0 ∈ V
3 1on 7771 . . . . . . 7 1𝑜 ∈ On
43elexi 3366 . . . . . 6 1𝑜 ∈ V
52, 4elmap 8089 . . . . 5 ((1𝑜 × {𝑘}) ∈ (ℕ0𝑚 1𝑜) ↔ (1𝑜 × {𝑘}):1𝑜⟶ℕ0)
61, 5sylibr 225 . . . 4 (𝑘 ∈ ℕ0 → (1𝑜 × {𝑘}) ∈ (ℕ0𝑚 1𝑜))
76adantl 473 . . 3 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (1𝑜 × {𝑘}) ∈ (ℕ0𝑚 1𝑜))
8 eqidd 2766 . . 3 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝑘 ∈ ℕ0 ↦ (1𝑜 × {𝑘})) = (𝑘 ∈ ℕ0 ↦ (1𝑜 × {𝑘})))
9 eqid 2765 . . . 4 (1𝑜 mPwSer 𝑅) = (1𝑜 mPwSer 𝑅)
10 coe1mul2.s . . . . 5 𝑆 = (PwSer1𝑅)
11 coe1mul2.b . . . . 5 𝐵 = (Base‘𝑆)
1210, 11, 9psr1bas2 19833 . . . 4 𝐵 = (Base‘(1𝑜 mPwSer 𝑅))
13 coe1mul2.u . . . 4 · = (.r𝑅)
14 coe1mul2.t . . . . 5 = (.r𝑆)
1510, 9, 14psr1mulr 19867 . . . 4 = (.r‘(1𝑜 mPwSer 𝑅))
16 psr1baslem 19828 . . . 4 (ℕ0𝑚 1𝑜) = {𝑎 ∈ (ℕ0𝑚 1𝑜) ∣ (𝑎 “ ℕ) ∈ Fin}
17 simp2 1167 . . . 4 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → 𝐹𝐵)
18 simp3 1168 . . . 4 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → 𝐺𝐵)
199, 12, 13, 15, 16, 17, 18psrmulfval 19659 . . 3 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝐹 𝐺) = (𝑏 ∈ (ℕ0𝑚 1𝑜) ↦ (𝑅 Σg (𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟𝑏} ↦ ((𝐹𝑐) · (𝐺‘(𝑏𝑓𝑐)))))))
20 breq2 4813 . . . . . 6 (𝑏 = (1𝑜 × {𝑘}) → (𝑑𝑟𝑏𝑑𝑟 ≤ (1𝑜 × {𝑘})))
2120rabbidv 3338 . . . . 5 (𝑏 = (1𝑜 × {𝑘}) → {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟𝑏} = {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})})
22 fvoveq1 6865 . . . . . 6 (𝑏 = (1𝑜 × {𝑘}) → (𝐺‘(𝑏𝑓𝑐)) = (𝐺‘((1𝑜 × {𝑘}) ∘𝑓𝑐)))
2322oveq2d 6858 . . . . 5 (𝑏 = (1𝑜 × {𝑘}) → ((𝐹𝑐) · (𝐺‘(𝑏𝑓𝑐))) = ((𝐹𝑐) · (𝐺‘((1𝑜 × {𝑘}) ∘𝑓𝑐))))
2421, 23mpteq12dv 4892 . . . 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 6858 . . 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 6587 . 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 19890 . . . 4 (𝑅 ∈ Ring → 𝑆 ∈ Ring)
2811, 14ringcl 18828 . . . 4 ((𝑆 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝐹 𝐺) ∈ 𝐵)
2927, 28syl3an1 1202 . . 3 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (𝐹 𝐺) ∈ 𝐵)
30 eqid 2765 . . . 4 (coe1‘(𝐹 𝐺)) = (coe1‘(𝐹 𝐺))
31 eqid 2765 . . . 4 (𝑘 ∈ ℕ0 ↦ (1𝑜 × {𝑘})) = (𝑘 ∈ ℕ0 ↦ (1𝑜 × {𝑘}))
3230, 11, 10, 31coe1fval3 19851 . . 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 2765 . . . . 5 (Base‘𝑅) = (Base‘𝑅)
35 eqid 2765 . . . . 5 (0g𝑅) = (0g𝑅)
36 simpl1 1242 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ Ring)
37 ringcmn 18848 . . . . . 6 (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
3836, 37syl 17 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ CMnd)
39 fzfi 12979 . . . . . 6 (0...𝑘) ∈ Fin
4039a1i 11 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (0...𝑘) ∈ Fin)
41 simpll1 1269 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
42 simpll2 1271 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝐹𝐵)
43 eqid 2765 . . . . . . . . . 10 (coe1𝐹) = (coe1𝐹)
4443, 11, 10, 34coe1f2 19852 . . . . . . . . 9 (𝐹𝐵 → (coe1𝐹):ℕ0⟶(Base‘𝑅))
4542, 44syl 17 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (coe1𝐹):ℕ0⟶(Base‘𝑅))
46 elfznn0 12640 . . . . . . . . 9 (𝑥 ∈ (0...𝑘) → 𝑥 ∈ ℕ0)
4746adantl 473 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝑥 ∈ ℕ0)
4845, 47ffvelrnd 6550 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → ((coe1𝐹)‘𝑥) ∈ (Base‘𝑅))
49 simpll3 1273 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝐺𝐵)
50 eqid 2765 . . . . . . . . . 10 (coe1𝐺) = (coe1𝐺)
5150, 11, 10, 34coe1f2 19852 . . . . . . . . 9 (𝐺𝐵 → (coe1𝐺):ℕ0⟶(Base‘𝑅))
5249, 51syl 17 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (coe1𝐺):ℕ0⟶(Base‘𝑅))
53 fznn0sub 12580 . . . . . . . . 9 (𝑥 ∈ (0...𝑘) → (𝑘𝑥) ∈ ℕ0)
5453adantl 473 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (𝑘𝑥) ∈ ℕ0)
5552, 54ffvelrnd 6550 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → ((coe1𝐺)‘(𝑘𝑥)) ∈ (Base‘𝑅))
5634, 13ringcl 18828 . . . . . . 7 ((𝑅 ∈ Ring ∧ ((coe1𝐹)‘𝑥) ∈ (Base‘𝑅) ∧ ((coe1𝐺)‘(𝑘𝑥)) ∈ (Base‘𝑅)) → (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))) ∈ (Base‘𝑅))
5741, 48, 55, 56syl3anc 1490 . . . . . 6 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))) ∈ (Base‘𝑅))
5857fmpttd 6575 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))):(0...𝑘)⟶(Base‘𝑅))
5939elexi 3366 . . . . . . . . 9 (0...𝑘) ∈ V
6059mptex 6679 . . . . . . . 8 (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ∈ V
61 funmpt 6106 . . . . . . . 8 Fun (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))))
62 fvex 6388 . . . . . . . 8 (0g𝑅) ∈ V
6360, 61, 623pm3.2i 1438 . . . . . . 7 ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ∈ V ∧ Fun (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ∧ (0g𝑅) ∈ V)
64 suppssdm 7510 . . . . . . . . 9 ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) supp (0g𝑅)) ⊆ dom (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))))
65 eqid 2765 . . . . . . . . . 10 (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) = (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))))
6665dmmptss 5817 . . . . . . . . 9 dom (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) ⊆ (0...𝑘)
6764, 66sstri 3770 . . . . . . . 8 ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) supp (0g𝑅)) ⊆ (0...𝑘)
6839, 67pm3.2i 462 . . . . . . 7 ((0...𝑘) ∈ Fin ∧ ((𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) supp (0g𝑅)) ⊆ (0...𝑘))
69 suppssfifsupp 8497 . . . . . . 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 683 . . . . . 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 2765 . . . . . . 7 {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})} = {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}
7372coe1mul2lem2 19911 . . . . . 6 (𝑘 ∈ ℕ0 → (𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})} ↦ (𝑐‘∅)):{𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}–1-1-onto→(0...𝑘))
7473adantl 473 . . . . 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 18583 . . . 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 4812 . . . . . . . . . . 11 (𝑑 = 𝑐 → (𝑑𝑟 ≤ (1𝑜 × {𝑘}) ↔ 𝑐𝑟 ≤ (1𝑜 × {𝑘})))
7776elrab 3519 . . . . . . . . . 10 (𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})} ↔ (𝑐 ∈ (ℕ0𝑚 1𝑜) ∧ 𝑐𝑟 ≤ (1𝑜 × {𝑘})))
7877simprbi 490 . . . . . . . . 9 (𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})} → 𝑐𝑟 ≤ (1𝑜 × {𝑘}))
7978adantl 473 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → 𝑐𝑟 ≤ (1𝑜 × {𝑘}))
80 simplr 785 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → 𝑘 ∈ ℕ0)
81 elrabi 3514 . . . . . . . . . 10 (𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})} → 𝑐 ∈ (ℕ0𝑚 1𝑜))
8281adantl 473 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → 𝑐 ∈ (ℕ0𝑚 1𝑜))
83 coe1mul2lem1 19910 . . . . . . . . 9 ((𝑘 ∈ ℕ0𝑐 ∈ (ℕ0𝑚 1𝑜)) → (𝑐𝑟 ≤ (1𝑜 × {𝑘}) ↔ (𝑐‘∅) ∈ (0...𝑘)))
8480, 82, 83syl2anc 579 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → (𝑐𝑟 ≤ (1𝑜 × {𝑘}) ↔ (𝑐‘∅) ∈ (0...𝑘)))
8579, 84mpbid 223 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → (𝑐‘∅) ∈ (0...𝑘))
86 eqidd 2766 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})} ↦ (𝑐‘∅)) = (𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})} ↦ (𝑐‘∅)))
87 eqidd 2766 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))) = (𝑥 ∈ (0...𝑘) ↦ (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥)))))
88 fveq2 6375 . . . . . . . 8 (𝑥 = (𝑐‘∅) → ((coe1𝐹)‘𝑥) = ((coe1𝐹)‘(𝑐‘∅)))
89 oveq2 6850 . . . . . . . . 9 (𝑥 = (𝑐‘∅) → (𝑘𝑥) = (𝑘 − (𝑐‘∅)))
9089fveq2d 6379 . . . . . . . 8 (𝑥 = (𝑐‘∅) → ((coe1𝐺)‘(𝑘𝑥)) = ((coe1𝐺)‘(𝑘 − (𝑐‘∅))))
9188, 90oveq12d 6860 . . . . . . 7 (𝑥 = (𝑐‘∅) → (((coe1𝐹)‘𝑥) · ((coe1𝐺)‘(𝑘𝑥))) = (((coe1𝐹)‘(𝑐‘∅)) · ((coe1𝐺)‘(𝑘 − (𝑐‘∅)))))
9285, 86, 87, 91fmptco 6587 . . . . . 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 1271 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → 𝐹𝐵)
9443fvcoe1 19850 . . . . . . . . 9 ((𝐹𝐵𝑐 ∈ (ℕ0𝑚 1𝑜)) → (𝐹𝑐) = ((coe1𝐹)‘(𝑐‘∅)))
9593, 82, 94syl2anc 579 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → (𝐹𝑐) = ((coe1𝐹)‘(𝑐‘∅)))
96 df1o2 7777 . . . . . . . . . . . . . 14 1𝑜 = {∅}
97 0ex 4950 . . . . . . . . . . . . . 14 ∅ ∈ V
9896, 2, 97mapsnconst 8108 . . . . . . . . . . . . 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 6858 . . . . . . . . . . 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 3353 . . . . . . . . . . . . 13 𝑘 ∈ V
103102a1i 11 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → 𝑘 ∈ V)
104 fvexd 6390 . . . . . . . . . . . 12 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → (𝑐‘∅) ∈ V)
105101, 103, 104ofc12 7120 . . . . . . . . . . 11 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → ((1𝑜 × {𝑘}) ∘𝑓 − (1𝑜 × {(𝑐‘∅)})) = (1𝑜 × {(𝑘 − (𝑐‘∅))}))
106100, 105eqtrd 2799 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → ((1𝑜 × {𝑘}) ∘𝑓𝑐) = (1𝑜 × {(𝑘 − (𝑐‘∅))}))
107106fveq2d 6379 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → (𝐺‘((1𝑜 × {𝑘}) ∘𝑓𝑐)) = (𝐺‘(1𝑜 × {(𝑘 − (𝑐‘∅))})))
108 simpll3 1273 . . . . . . . . . 10 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → 𝐺𝐵)
109 fznn0sub 12580 . . . . . . . . . . 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 19849 . . . . . . . . . 10 ((𝐺𝐵 ∧ (𝑘 − (𝑐‘∅)) ∈ ℕ0) → ((coe1𝐺)‘(𝑘 − (𝑐‘∅))) = (𝐺‘(1𝑜 × {(𝑘 − (𝑐‘∅))})))
112108, 110, 111syl2anc 579 . . . . . . . . 9 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → ((coe1𝐺)‘(𝑘 − (𝑐‘∅))) = (𝐺‘(1𝑜 × {(𝑘 − (𝑐‘∅))})))
113107, 112eqtr4d 2802 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → (𝐺‘((1𝑜 × {𝑘}) ∘𝑓𝑐)) = ((coe1𝐺)‘(𝑘 − (𝑐‘∅))))
11495, 113oveq12d 6860 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ 𝑑𝑟 ≤ (1𝑜 × {𝑘})}) → ((𝐹𝑐) · (𝐺‘((1𝑜 × {𝑘}) ∘𝑓𝑐))) = (((coe1𝐹)‘(𝑐‘∅)) · ((coe1𝐺)‘(𝑘 − (𝑐‘∅)))))
115114mpteq2dva 4903 . . . . . 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 2802 . . . . 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 6858 . . . 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 2799 . . 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 4903 . 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 2809 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  {crab 3059  Vcvv 3350  wss 3732  c0 4079  {csn 4334   class class class wbr 4809  cmpt 4888   × cxp 5275  dom cdm 5277  ccom 5281  Oncon0 5908  Fun wfun 6062  wf 6064  1-1-ontowf1o 6067  cfv 6068  (class class class)co 6842  𝑓 cof 7093  𝑟 cofr 7094   supp csupp 7497  1𝑜c1o 7757  𝑚 cmap 8060  Fincfn 8160   finSupp cfsupp 8482  0cc0 10189  cle 10329  cmin 10520  0cn0 11538  ...cfz 12533  Basecbs 16130  .rcmulr 16215  0gc0g 16366   Σg cgsu 16367  CMndccmn 18459  Ringcrg 18814   mPwSer cmps 19625  PwSer1cps1 19818  coe1cco1 19821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-ofr 7096  df-om 7264  df-1st 7366  df-2nd 7367  df-supp 7498  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-ixp 8114  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fsupp 8483  df-oi 8622  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-z 11625  df-dec 11741  df-uz 11887  df-fz 12534  df-fzo 12674  df-seq 13009  df-hash 13322  df-struct 16132  df-ndx 16133  df-slot 16134  df-base 16136  df-sets 16137  df-ress 16138  df-plusg 16227  df-mulr 16228  df-sca 16230  df-vsca 16231  df-tset 16233  df-ple 16234  df-0g 16368  df-gsum 16369  df-mre 16512  df-mrc 16513  df-acs 16515  df-mgm 17508  df-sgrp 17550  df-mnd 17561  df-mhm 17601  df-submnd 17602  df-grp 17692  df-minusg 17693  df-mulg 17808  df-ghm 17922  df-cntz 18013  df-cmn 18461  df-abl 18462  df-mgp 18757  df-ur 18769  df-ring 18816  df-psr 19630  df-opsr 19634  df-psr1 19823  df-coe1 19826
This theorem is referenced by:  coe1mul  19913
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