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Theorem coe1tmmul 20364
Description: Coefficient vector of a polynomial multiplied on the left by a term. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Hypotheses
Ref Expression
coe1tm.z 0 = (0g𝑅)
coe1tm.k 𝐾 = (Base‘𝑅)
coe1tm.p 𝑃 = (Poly1𝑅)
coe1tm.x 𝑋 = (var1𝑅)
coe1tm.m · = ( ·𝑠𝑃)
coe1tm.n 𝑁 = (mulGrp‘𝑃)
coe1tm.e = (.g𝑁)
coe1tmmul.b 𝐵 = (Base‘𝑃)
coe1tmmul.t = (.r𝑃)
coe1tmmul.u × = (.r𝑅)
coe1tmmul.a (𝜑𝐴𝐵)
coe1tmmul.r (𝜑𝑅 ∈ Ring)
coe1tmmul.c (𝜑𝐶𝐾)
coe1tmmul.d (𝜑𝐷 ∈ ℕ0)
Assertion
Ref Expression
coe1tmmul (𝜑 → (coe1‘((𝐶 · (𝐷 𝑋)) 𝐴)) = (𝑥 ∈ ℕ0 ↦ if(𝐷𝑥, (𝐶 × ((coe1𝐴)‘(𝑥𝐷))), 0 )))
Distinct variable groups:   𝑥, 0   𝑥,𝐶   𝑥,𝐷   𝑥,𝐾   𝑥,   𝑥,𝐴   𝑥,𝑁   𝑥,𝑃   𝑥,𝑋   𝜑,𝑥   𝑥,𝑅   𝑥, ·   𝑥, ×   𝑥,
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem coe1tmmul
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 coe1tmmul.r . . 3 (𝜑𝑅 ∈ Ring)
2 coe1tmmul.c . . . 4 (𝜑𝐶𝐾)
3 coe1tmmul.d . . . 4 (𝜑𝐷 ∈ ℕ0)
4 coe1tm.k . . . . 5 𝐾 = (Base‘𝑅)
5 coe1tm.p . . . . 5 𝑃 = (Poly1𝑅)
6 coe1tm.x . . . . 5 𝑋 = (var1𝑅)
7 coe1tm.m . . . . 5 · = ( ·𝑠𝑃)
8 coe1tm.n . . . . 5 𝑁 = (mulGrp‘𝑃)
9 coe1tm.e . . . . 5 = (.g𝑁)
10 coe1tmmul.b . . . . 5 𝐵 = (Base‘𝑃)
114, 5, 6, 7, 8, 9, 10ply1tmcl 20359 . . . 4 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐶 · (𝐷 𝑋)) ∈ 𝐵)
121, 2, 3, 11syl3anc 1365 . . 3 (𝜑 → (𝐶 · (𝐷 𝑋)) ∈ 𝐵)
13 coe1tmmul.a . . 3 (𝜑𝐴𝐵)
14 coe1tmmul.t . . . 4 = (.r𝑃)
15 coe1tmmul.u . . . 4 × = (.r𝑅)
165, 14, 15, 10coe1mul 20357 . . 3 ((𝑅 ∈ Ring ∧ (𝐶 · (𝐷 𝑋)) ∈ 𝐵𝐴𝐵) → (coe1‘((𝐶 · (𝐷 𝑋)) 𝐴)) = (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦)))))))
171, 12, 13, 16syl3anc 1365 . 2 (𝜑 → (coe1‘((𝐶 · (𝐷 𝑋)) 𝐴)) = (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦)))))))
18 eqeq2 2837 . . . 4 ((𝐶 × ((coe1𝐴)‘(𝑥𝐷))) = if(𝐷𝑥, (𝐶 × ((coe1𝐴)‘(𝑥𝐷))), 0 ) → ((𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))))) = (𝐶 × ((coe1𝐴)‘(𝑥𝐷))) ↔ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))))) = if(𝐷𝑥, (𝐶 × ((coe1𝐴)‘(𝑥𝐷))), 0 )))
19 eqeq2 2837 . . . 4 ( 0 = if(𝐷𝑥, (𝐶 × ((coe1𝐴)‘(𝑥𝐷))), 0 ) → ((𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))))) = 0 ↔ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))))) = if(𝐷𝑥, (𝐶 × ((coe1𝐴)‘(𝑥𝐷))), 0 )))
20 coe1tm.z . . . . . 6 0 = (0g𝑅)
211ad2antrr 722 . . . . . . 7 (((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) → 𝑅 ∈ Ring)
22 ringmnd 19228 . . . . . . 7 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
2321, 22syl 17 . . . . . 6 (((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) → 𝑅 ∈ Mnd)
24 ovexd 7186 . . . . . 6 (((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) → (0...𝑥) ∈ V)
253ad2antrr 722 . . . . . . 7 (((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) → 𝐷 ∈ ℕ0)
26 simpr 485 . . . . . . 7 (((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) → 𝐷𝑥)
27 fznn0 12992 . . . . . . . 8 (𝑥 ∈ ℕ0 → (𝐷 ∈ (0...𝑥) ↔ (𝐷 ∈ ℕ0𝐷𝑥)))
2827ad2antlr 723 . . . . . . 7 (((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) → (𝐷 ∈ (0...𝑥) ↔ (𝐷 ∈ ℕ0𝐷𝑥)))
2925, 26, 28mpbir2and 709 . . . . . 6 (((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) → 𝐷 ∈ (0...𝑥))
301ad2antrr 722 . . . . . . . . 9 (((𝜑𝑥 ∈ ℕ0) ∧ 𝑦 ∈ (0...𝑥)) → 𝑅 ∈ Ring)
31 eqid 2825 . . . . . . . . . . . . 13 (coe1‘(𝐶 · (𝐷 𝑋))) = (coe1‘(𝐶 · (𝐷 𝑋)))
3231, 10, 5, 4coe1f 20298 . . . . . . . . . . . 12 ((𝐶 · (𝐷 𝑋)) ∈ 𝐵 → (coe1‘(𝐶 · (𝐷 𝑋))):ℕ0𝐾)
3312, 32syl 17 . . . . . . . . . . 11 (𝜑 → (coe1‘(𝐶 · (𝐷 𝑋))):ℕ0𝐾)
3433adantr 481 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℕ0) → (coe1‘(𝐶 · (𝐷 𝑋))):ℕ0𝐾)
35 elfznn0 12993 . . . . . . . . . 10 (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℕ0)
36 ffvelrn 6844 . . . . . . . . . 10 (((coe1‘(𝐶 · (𝐷 𝑋))):ℕ0𝐾𝑦 ∈ ℕ0) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) ∈ 𝐾)
3734, 35, 36syl2an 595 . . . . . . . . 9 (((𝜑𝑥 ∈ ℕ0) ∧ 𝑦 ∈ (0...𝑥)) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) ∈ 𝐾)
38 eqid 2825 . . . . . . . . . . . . 13 (coe1𝐴) = (coe1𝐴)
3938, 10, 5, 4coe1f 20298 . . . . . . . . . . . 12 (𝐴𝐵 → (coe1𝐴):ℕ0𝐾)
4013, 39syl 17 . . . . . . . . . . 11 (𝜑 → (coe1𝐴):ℕ0𝐾)
4140adantr 481 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℕ0) → (coe1𝐴):ℕ0𝐾)
42 fznn0sub 12932 . . . . . . . . . 10 (𝑦 ∈ (0...𝑥) → (𝑥𝑦) ∈ ℕ0)
43 ffvelrn 6844 . . . . . . . . . 10 (((coe1𝐴):ℕ0𝐾 ∧ (𝑥𝑦) ∈ ℕ0) → ((coe1𝐴)‘(𝑥𝑦)) ∈ 𝐾)
4441, 42, 43syl2an 595 . . . . . . . . 9 (((𝜑𝑥 ∈ ℕ0) ∧ 𝑦 ∈ (0...𝑥)) → ((coe1𝐴)‘(𝑥𝑦)) ∈ 𝐾)
454, 15ringcl 19233 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) ∈ 𝐾 ∧ ((coe1𝐴)‘(𝑥𝑦)) ∈ 𝐾) → (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))) ∈ 𝐾)
4630, 37, 44, 45syl3anc 1365 . . . . . . . 8 (((𝜑𝑥 ∈ ℕ0) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))) ∈ 𝐾)
4746fmpttd 6874 . . . . . . 7 ((𝜑𝑥 ∈ ℕ0) → (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦)))):(0...𝑥)⟶𝐾)
4847adantr 481 . . . . . 6 (((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) → (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦)))):(0...𝑥)⟶𝐾)
491ad3antrrr 726 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) ∧ 𝑦 ∈ ((0...𝑥) ∖ {𝐷})) → 𝑅 ∈ Ring)
502ad3antrrr 726 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) ∧ 𝑦 ∈ ((0...𝑥) ∖ {𝐷})) → 𝐶𝐾)
513ad3antrrr 726 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) ∧ 𝑦 ∈ ((0...𝑥) ∖ {𝐷})) → 𝐷 ∈ ℕ0)
52 eldifi 4106 . . . . . . . . . . . 12 (𝑦 ∈ ((0...𝑥) ∖ {𝐷}) → 𝑦 ∈ (0...𝑥))
5352, 35syl 17 . . . . . . . . . . 11 (𝑦 ∈ ((0...𝑥) ∖ {𝐷}) → 𝑦 ∈ ℕ0)
5453adantl 482 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) ∧ 𝑦 ∈ ((0...𝑥) ∖ {𝐷})) → 𝑦 ∈ ℕ0)
55 eldifsni 4720 . . . . . . . . . . . 12 (𝑦 ∈ ((0...𝑥) ∖ {𝐷}) → 𝑦𝐷)
5655necomd 3075 . . . . . . . . . . 11 (𝑦 ∈ ((0...𝑥) ∖ {𝐷}) → 𝐷𝑦)
5756adantl 482 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) ∧ 𝑦 ∈ ((0...𝑥) ∖ {𝐷})) → 𝐷𝑦)
5820, 4, 5, 6, 7, 8, 9, 49, 50, 51, 54, 57coe1tmfv2 20362 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) ∧ 𝑦 ∈ ((0...𝑥) ∖ {𝐷})) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) = 0 )
5958oveq1d 7166 . . . . . . . 8 ((((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) ∧ 𝑦 ∈ ((0...𝑥) ∖ {𝐷})) → (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))) = ( 0 × ((coe1𝐴)‘(𝑥𝑦))))
604, 15, 20ringlz 19259 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ ((coe1𝐴)‘(𝑥𝑦)) ∈ 𝐾) → ( 0 × ((coe1𝐴)‘(𝑥𝑦))) = 0 )
6130, 44, 60syl2anc 584 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℕ0) ∧ 𝑦 ∈ (0...𝑥)) → ( 0 × ((coe1𝐴)‘(𝑥𝑦))) = 0 )
6252, 61sylan2 592 . . . . . . . . 9 (((𝜑𝑥 ∈ ℕ0) ∧ 𝑦 ∈ ((0...𝑥) ∖ {𝐷})) → ( 0 × ((coe1𝐴)‘(𝑥𝑦))) = 0 )
6362adantlr 711 . . . . . . . 8 ((((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) ∧ 𝑦 ∈ ((0...𝑥) ∖ {𝐷})) → ( 0 × ((coe1𝐴)‘(𝑥𝑦))) = 0 )
6459, 63eqtrd 2860 . . . . . . 7 ((((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) ∧ 𝑦 ∈ ((0...𝑥) ∖ {𝐷})) → (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))) = 0 )
6564, 24suppss2 7858 . . . . . 6 (((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦)))) supp 0 ) ⊆ {𝐷})
664, 20, 23, 24, 29, 48, 65gsumpt 19004 . . . . 5 (((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))))) = ((𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))))‘𝐷))
67 fveq2 6666 . . . . . . . . 9 (𝑦 = 𝐷 → ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) = ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷))
68 oveq2 7159 . . . . . . . . . 10 (𝑦 = 𝐷 → (𝑥𝑦) = (𝑥𝐷))
6968fveq2d 6670 . . . . . . . . 9 (𝑦 = 𝐷 → ((coe1𝐴)‘(𝑥𝑦)) = ((coe1𝐴)‘(𝑥𝐷)))
7067, 69oveq12d 7169 . . . . . . . 8 (𝑦 = 𝐷 → (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))) = (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷) × ((coe1𝐴)‘(𝑥𝐷))))
71 eqid 2825 . . . . . . . 8 (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦)))) = (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))))
72 ovex 7184 . . . . . . . 8 (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷) × ((coe1𝐴)‘(𝑥𝐷))) ∈ V
7370, 71, 72fvmpt 6764 . . . . . . 7 (𝐷 ∈ (0...𝑥) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))))‘𝐷) = (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷) × ((coe1𝐴)‘(𝑥𝐷))))
7429, 73syl 17 . . . . . 6 (((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))))‘𝐷) = (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷) × ((coe1𝐴)‘(𝑥𝐷))))
7520, 4, 5, 6, 7, 8, 9coe1tmfv1 20361 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷) = 𝐶)
761, 2, 3, 75syl3anc 1365 . . . . . . . 8 (𝜑 → ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷) = 𝐶)
7776ad2antrr 722 . . . . . . 7 (((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷) = 𝐶)
7877oveq1d 7166 . . . . . 6 (((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) → (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷) × ((coe1𝐴)‘(𝑥𝐷))) = (𝐶 × ((coe1𝐴)‘(𝑥𝐷))))
7974, 78eqtrd 2860 . . . . 5 (((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))))‘𝐷) = (𝐶 × ((coe1𝐴)‘(𝑥𝐷))))
8066, 79eqtrd 2860 . . . 4 (((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))))) = (𝐶 × ((coe1𝐴)‘(𝑥𝐷))))
811ad3antrrr 726 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) ∧ 𝑦 ∈ (0...𝑥)) → 𝑅 ∈ Ring)
822ad3antrrr 726 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) ∧ 𝑦 ∈ (0...𝑥)) → 𝐶𝐾)
833ad3antrrr 726 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) ∧ 𝑦 ∈ (0...𝑥)) → 𝐷 ∈ ℕ0)
8435adantl 482 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 ∈ ℕ0)
85 elfzle2 12904 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0...𝑥) → 𝑦𝑥)
8685adantl 482 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℕ0) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦𝑥)
87 breq1 5065 . . . . . . . . . . . . . 14 (𝐷 = 𝑦 → (𝐷𝑥𝑦𝑥))
8886, 87syl5ibrcom 248 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℕ0) ∧ 𝑦 ∈ (0...𝑥)) → (𝐷 = 𝑦𝐷𝑥))
8988necon3bd 3034 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℕ0) ∧ 𝑦 ∈ (0...𝑥)) → (¬ 𝐷𝑥𝐷𝑦))
9089imp 407 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℕ0) ∧ 𝑦 ∈ (0...𝑥)) ∧ ¬ 𝐷𝑥) → 𝐷𝑦)
9190an32s 648 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) ∧ 𝑦 ∈ (0...𝑥)) → 𝐷𝑦)
9220, 4, 5, 6, 7, 8, 9, 81, 82, 83, 84, 91coe1tmfv2 20362 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) ∧ 𝑦 ∈ (0...𝑥)) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) = 0 )
9392oveq1d 7166 . . . . . . . 8 ((((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))) = ( 0 × ((coe1𝐴)‘(𝑥𝑦))))
9461adantlr 711 . . . . . . . 8 ((((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) ∧ 𝑦 ∈ (0...𝑥)) → ( 0 × ((coe1𝐴)‘(𝑥𝑦))) = 0 )
9593, 94eqtrd 2860 . . . . . . 7 ((((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))) = 0 )
9695mpteq2dva 5157 . . . . . 6 (((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) → (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦)))) = (𝑦 ∈ (0...𝑥) ↦ 0 ))
9796oveq2d 7167 . . . . 5 (((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))))) = (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )))
981, 22syl 17 . . . . . . 7 (𝜑𝑅 ∈ Mnd)
9998ad2antrr 722 . . . . . 6 (((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) → 𝑅 ∈ Mnd)
100 ovexd 7186 . . . . . 6 (((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) → (0...𝑥) ∈ V)
10120gsumz 17985 . . . . . 6 ((𝑅 ∈ Mnd ∧ (0...𝑥) ∈ V) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
10299, 100, 101syl2anc 584 . . . . 5 (((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
10397, 102eqtrd 2860 . . . 4 (((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))))) = 0 )
10418, 19, 80, 103ifbothda 4506 . . 3 ((𝜑𝑥 ∈ ℕ0) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))))) = if(𝐷𝑥, (𝐶 × ((coe1𝐴)‘(𝑥𝐷))), 0 ))
105104mpteq2dva 5157 . 2 (𝜑 → (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦)))))) = (𝑥 ∈ ℕ0 ↦ if(𝐷𝑥, (𝐶 × ((coe1𝐴)‘(𝑥𝐷))), 0 )))
10617, 105eqtrd 2860 1 (𝜑 → (coe1‘((𝐶 · (𝐷 𝑋)) 𝐴)) = (𝑥 ∈ ℕ0 ↦ if(𝐷𝑥, (𝐶 × ((coe1𝐴)‘(𝑥𝐷))), 0 )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wne 3020  Vcvv 3499  cdif 3936  ifcif 4469  {csn 4563   class class class wbr 5062  cmpt 5142  wf 6347  cfv 6351  (class class class)co 7151  0cc0 10529  cle 10668  cmin 10862  0cn0 11889  ...cfz 12885  Basecbs 16475  .rcmulr 16558   ·𝑠 cvsca 16561  0gc0g 16705   Σg cgsu 16706  Mndcmnd 17902  .gcmg 18156  mulGrpcmgp 19161  Ringcrg 19219  var1cv1 20263  Poly1cpl1 20264  coe1cco1 20265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-iin 4919  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-of 7402  df-ofr 7403  df-om 7572  df-1st 7683  df-2nd 7684  df-supp 7825  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8282  df-map 8401  df-pm 8402  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fsupp 8826  df-oi 8966  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12886  df-fzo 13027  df-seq 13363  df-hash 13684  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-sca 16573  df-vsca 16574  df-tset 16576  df-ple 16577  df-0g 16707  df-gsum 16708  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17892  df-mnd 17903  df-mhm 17946  df-submnd 17947  df-grp 18038  df-minusg 18039  df-sbg 18040  df-mulg 18157  df-subg 18208  df-ghm 18288  df-cntz 18379  df-cmn 18830  df-abl 18831  df-mgp 19162  df-ur 19174  df-ring 19221  df-subrg 19455  df-lmod 19558  df-lss 19626  df-psr 20057  df-mvr 20058  df-mpl 20059  df-opsr 20061  df-psr1 20267  df-vr1 20268  df-ply1 20269  df-coe1 20270
This theorem is referenced by:  coe1pwmul  20366  coe1sclmul  20369
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