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Theorem coe1tmmul 20373
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 20368 . . . 4 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐶 · (𝐷 𝑋)) ∈ 𝐵)
121, 2, 3, 11syl3anc 1363 . . 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 20366 . . 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 1363 . 2 (𝜑 → (coe1‘((𝐶 · (𝐷 𝑋)) 𝐴)) = (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦)))))))
18 eqeq2 2830 . . . 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 2830 . . . 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 19235 . . . . . . 7 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
2321, 22syl 17 . . . . . 6 (((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) → 𝑅 ∈ Mnd)
24 ovexd 7180 . . . . . 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 12987 . . . . . . . 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 2818 . . . . . . . . . . . . 13 (coe1‘(𝐶 · (𝐷 𝑋))) = (coe1‘(𝐶 · (𝐷 𝑋)))
3231, 10, 5, 4coe1f 20307 . . . . . . . . . . . 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 12988 . . . . . . . . . 10 (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℕ0)
36 ffvelrn 6841 . . . . . . . . . 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 2818 . . . . . . . . . . . . 13 (coe1𝐴) = (coe1𝐴)
3938, 10, 5, 4coe1f 20307 . . . . . . . . . . . 12 (𝐴𝐵 → (coe1𝐴):ℕ0𝐾)
4013, 39syl 17 . . . . . . . . . . 11 (𝜑 → (coe1𝐴):ℕ0𝐾)
4140adantr 481 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℕ0) → (coe1𝐴):ℕ0𝐾)
42 fznn0sub 12927 . . . . . . . . . 10 (𝑦 ∈ (0...𝑥) → (𝑥𝑦) ∈ ℕ0)
43 ffvelrn 6841 . . . . . . . . . 10 (((coe1𝐴):ℕ0𝐾 ∧ (𝑥𝑦) ∈ ℕ0) → ((coe1𝐴)‘(𝑥𝑦)) ∈ 𝐾)
4441, 42, 43syl2an 595 . . . . . . . . 9 (((𝜑𝑥 ∈ ℕ0) ∧ 𝑦 ∈ (0...𝑥)) → ((coe1𝐴)‘(𝑥𝑦)) ∈ 𝐾)
454, 15ringcl 19240 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) ∈ 𝐾 ∧ ((coe1𝐴)‘(𝑥𝑦)) ∈ 𝐾) → (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))) ∈ 𝐾)
4630, 37, 44, 45syl3anc 1363 . . . . . . . 8 (((𝜑𝑥 ∈ ℕ0) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))) ∈ 𝐾)
4746fmpttd 6871 . . . . . . 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 4100 . . . . . . . . . . . 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 4714 . . . . . . . . . . . 12 (𝑦 ∈ ((0...𝑥) ∖ {𝐷}) → 𝑦𝐷)
5655necomd 3068 . . . . . . . . . . 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 20371 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) ∧ 𝑦 ∈ ((0...𝑥) ∖ {𝐷})) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) = 0 )
5958oveq1d 7160 . . . . . . . 8 ((((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) ∧ 𝑦 ∈ ((0...𝑥) ∖ {𝐷})) → (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))) = ( 0 × ((coe1𝐴)‘(𝑥𝑦))))
604, 15, 20ringlz 19266 . . . . . . . . . . 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 2853 . . . . . . 7 ((((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) ∧ 𝑦 ∈ ((0...𝑥) ∖ {𝐷})) → (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))) = 0 )
6564, 24suppss2 7853 . . . . . 6 (((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦)))) supp 0 ) ⊆ {𝐷})
664, 20, 23, 24, 29, 48, 65gsumpt 19011 . . . . 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 6663 . . . . . . . . 9 (𝑦 = 𝐷 → ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) = ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷))
68 oveq2 7153 . . . . . . . . . 10 (𝑦 = 𝐷 → (𝑥𝑦) = (𝑥𝐷))
6968fveq2d 6667 . . . . . . . . 9 (𝑦 = 𝐷 → ((coe1𝐴)‘(𝑥𝑦)) = ((coe1𝐴)‘(𝑥𝐷)))
7067, 69oveq12d 7163 . . . . . . . 8 (𝑦 = 𝐷 → (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))) = (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷) × ((coe1𝐴)‘(𝑥𝐷))))
71 eqid 2818 . . . . . . . 8 (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦)))) = (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))))
72 ovex 7178 . . . . . . . 8 (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷) × ((coe1𝐴)‘(𝑥𝐷))) ∈ V
7370, 71, 72fvmpt 6761 . . . . . . 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 20370 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷) = 𝐶)
761, 2, 3, 75syl3anc 1363 . . . . . . . 8 (𝜑 → ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷) = 𝐶)
7776ad2antrr 722 . . . . . . 7 (((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷) = 𝐶)
7877oveq1d 7160 . . . . . 6 (((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) → (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷) × ((coe1𝐴)‘(𝑥𝐷))) = (𝐶 × ((coe1𝐴)‘(𝑥𝐷))))
7974, 78eqtrd 2853 . . . . 5 (((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))))‘𝐷) = (𝐶 × ((coe1𝐴)‘(𝑥𝐷))))
8066, 79eqtrd 2853 . . . 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 12899 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0...𝑥) → 𝑦𝑥)
8685adantl 482 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℕ0) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦𝑥)
87 breq1 5060 . . . . . . . . . . . . . 14 (𝐷 = 𝑦 → (𝐷𝑥𝑦𝑥))
8886, 87syl5ibrcom 248 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℕ0) ∧ 𝑦 ∈ (0...𝑥)) → (𝐷 = 𝑦𝐷𝑥))
8988necon3bd 3027 . . . . . . . . . . . 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 20371 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) ∧ 𝑦 ∈ (0...𝑥)) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) = 0 )
9392oveq1d 7160 . . . . . . . 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 2853 . . . . . . 7 ((((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))) = 0 )
9695mpteq2dva 5152 . . . . . 6 (((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) → (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦)))) = (𝑦 ∈ (0...𝑥) ↦ 0 ))
9796oveq2d 7161 . . . . 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 7180 . . . . . 6 (((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) → (0...𝑥) ∈ V)
10120gsumz 17988 . . . . . 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 2853 . . . 4 (((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))))) = 0 )
10418, 19, 80, 103ifbothda 4500 . . 3 ((𝜑𝑥 ∈ ℕ0) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 𝑋)))‘𝑦) × ((coe1𝐴)‘(𝑥𝑦))))) = if(𝐷𝑥, (𝐶 × ((coe1𝐴)‘(𝑥𝐷))), 0 ))
105104mpteq2dva 5152 . 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 2853 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 1528  wcel 2105  wne 3013  Vcvv 3492  cdif 3930  ifcif 4463  {csn 4557   class class class wbr 5057  cmpt 5137  wf 6344  cfv 6348  (class class class)co 7145  0cc0 10525  cle 10664  cmin 10858  0cn0 11885  ...cfz 12880  Basecbs 16471  .rcmulr 16554   ·𝑠 cvsca 16557  0gc0g 16701   Σg cgsu 16702  Mndcmnd 17899  .gcmg 18162  mulGrpcmgp 19168  Ringcrg 19226  var1cv1 20272  Poly1cpl1 20273  coe1cco1 20274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-ofr 7399  df-om 7570  df-1st 7678  df-2nd 7679  df-supp 7820  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fsupp 8822  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-fz 12881  df-fzo 13022  df-seq 13358  df-hash 13679  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-mulr 16567  df-sca 16569  df-vsca 16570  df-tset 16572  df-ple 16573  df-0g 16703  df-gsum 16704  df-mre 16845  df-mrc 16846  df-acs 16848  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-mhm 17944  df-submnd 17945  df-grp 18044  df-minusg 18045  df-sbg 18046  df-mulg 18163  df-subg 18214  df-ghm 18294  df-cntz 18385  df-cmn 18837  df-abl 18838  df-mgp 19169  df-ur 19181  df-ring 19228  df-subrg 19462  df-lmod 19565  df-lss 19633  df-psr 20064  df-mvr 20065  df-mpl 20066  df-opsr 20068  df-psr1 20276  df-vr1 20277  df-ply1 20278  df-coe1 20279
This theorem is referenced by:  coe1pwmul  20375  coe1sclmul  20378
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