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Theorem coe1tmmul2 19467
Description: Coefficient vector of a polynomial multiplied on the right by a term. (Contributed by Stefan O'Rear, 27-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
coe1tmmul2 (𝜑 → (coe1‘(𝐴 (𝐶 · (𝐷 𝑋)))) = (𝑥 ∈ ℕ0 ↦ if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 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 coe1tmmul2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 coe1tmmul.r . . 3 (𝜑𝑅 ∈ Ring)
2 coe1tmmul.a . . 3 (𝜑𝐴𝐵)
3 coe1tmmul.c . . . 4 (𝜑𝐶𝐾)
4 coe1tmmul.d . . . 4 (𝜑𝐷 ∈ ℕ0)
5 coe1tm.k . . . . 5 𝐾 = (Base‘𝑅)
6 coe1tm.p . . . . 5 𝑃 = (Poly1𝑅)
7 coe1tm.x . . . . 5 𝑋 = (var1𝑅)
8 coe1tm.m . . . . 5 · = ( ·𝑠𝑃)
9 coe1tm.n . . . . 5 𝑁 = (mulGrp‘𝑃)
10 coe1tm.e . . . . 5 = (.g𝑁)
11 coe1tmmul.b . . . . 5 𝐵 = (Base‘𝑃)
125, 6, 7, 8, 9, 10, 11ply1tmcl 19463 . . . 4 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐶 · (𝐷 𝑋)) ∈ 𝐵)
131, 3, 4, 12syl3anc 1318 . . 3 (𝜑 → (𝐶 · (𝐷 𝑋)) ∈ 𝐵)
14 coe1tmmul.t . . . 4 = (.r𝑃)
15 coe1tmmul.u . . . 4 × = (.r𝑅)
166, 14, 15, 11coe1mul 19461 . . 3 ((𝑅 ∈ Ring ∧ 𝐴𝐵 ∧ (𝐶 · (𝐷 𝑋)) ∈ 𝐵) → (coe1‘(𝐴 (𝐶 · (𝐷 𝑋)))) = (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))))))
171, 2, 13, 16syl3anc 1318 . 2 (𝜑 → (coe1‘(𝐴 (𝐶 · (𝐷 𝑋)))) = (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))))))
18 eqeq2 2621 . . . 4 ((((coe1𝐴)‘(𝑥𝐷)) × 𝐶) = if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 ) → ((𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = (((coe1𝐴)‘(𝑥𝐷)) × 𝐶) ↔ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 )))
19 eqeq2 2621 . . . 4 ( 0 = if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 ) → ((𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = 0 ↔ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 )))
20 coe1tm.z . . . . . . 7 0 = (0g𝑅)
211adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝑅 ∈ Ring)
22 ringmnd 18379 . . . . . . . 8 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
2321, 22syl 17 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝑅 ∈ Mnd)
24 ovex 6577 . . . . . . . 8 (0...𝑥) ∈ V
2524a1i 11 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (0...𝑥) ∈ V)
26 simprr 792 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝐷𝑥)
274adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝐷 ∈ ℕ0)
28 simprl 790 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝑥 ∈ ℕ0)
29 nn0sub 11220 . . . . . . . . . 10 ((𝐷 ∈ ℕ0𝑥 ∈ ℕ0) → (𝐷𝑥 ↔ (𝑥𝐷) ∈ ℕ0))
3027, 28, 29syl2anc 691 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝐷𝑥 ↔ (𝑥𝐷) ∈ ℕ0))
3126, 30mpbid 221 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑥𝐷) ∈ ℕ0)
3227nn0ge0d 11231 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 0 ≤ 𝐷)
33 nn0re 11178 . . . . . . . . . . 11 (𝑥 ∈ ℕ0𝑥 ∈ ℝ)
3433ad2antrl 760 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝑥 ∈ ℝ)
354nn0red 11229 . . . . . . . . . . 11 (𝜑𝐷 ∈ ℝ)
3635adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝐷 ∈ ℝ)
3734, 36subge02d 10498 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (0 ≤ 𝐷 ↔ (𝑥𝐷) ≤ 𝑥))
3832, 37mpbid 221 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑥𝐷) ≤ 𝑥)
39 fznn0 12301 . . . . . . . . 9 (𝑥 ∈ ℕ0 → ((𝑥𝐷) ∈ (0...𝑥) ↔ ((𝑥𝐷) ∈ ℕ0 ∧ (𝑥𝐷) ≤ 𝑥)))
4039ad2antrl 760 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → ((𝑥𝐷) ∈ (0...𝑥) ↔ ((𝑥𝐷) ∈ ℕ0 ∧ (𝑥𝐷) ≤ 𝑥)))
4131, 38, 40mpbir2and 959 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑥𝐷) ∈ (0...𝑥))
421ad2antrr 758 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑅 ∈ Ring)
43 eqid 2610 . . . . . . . . . . . . 13 (coe1𝐴) = (coe1𝐴)
4443, 11, 6, 5coe1f 19402 . . . . . . . . . . . 12 (𝐴𝐵 → (coe1𝐴):ℕ0𝐾)
452, 44syl 17 . . . . . . . . . . 11 (𝜑 → (coe1𝐴):ℕ0𝐾)
4645ad2antrr 758 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (coe1𝐴):ℕ0𝐾)
47 elfznn0 12302 . . . . . . . . . . 11 (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℕ0)
4847adantl 481 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 ∈ ℕ0)
4946, 48ffvelrnd 6268 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → ((coe1𝐴)‘𝑦) ∈ 𝐾)
50 eqid 2610 . . . . . . . . . . . . 13 (coe1‘(𝐶 · (𝐷 𝑋))) = (coe1‘(𝐶 · (𝐷 𝑋)))
5150, 11, 6, 5coe1f 19402 . . . . . . . . . . . 12 ((𝐶 · (𝐷 𝑋)) ∈ 𝐵 → (coe1‘(𝐶 · (𝐷 𝑋))):ℕ0𝐾)
5213, 51syl 17 . . . . . . . . . . 11 (𝜑 → (coe1‘(𝐶 · (𝐷 𝑋))):ℕ0𝐾)
5352ad2antrr 758 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (coe1‘(𝐶 · (𝐷 𝑋))):ℕ0𝐾)
54 fznn0sub 12244 . . . . . . . . . . 11 (𝑦 ∈ (0...𝑥) → (𝑥𝑦) ∈ ℕ0)
5554adantl 481 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝑥𝑦) ∈ ℕ0)
5653, 55ffvelrnd 6268 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)) ∈ 𝐾)
575, 15ringcl 18384 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ ((coe1𝐴)‘𝑦) ∈ 𝐾 ∧ ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)) ∈ 𝐾) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) ∈ 𝐾)
5842, 49, 56, 57syl3anc 1318 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) ∈ 𝐾)
59 eqid 2610 . . . . . . . 8 (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))) = (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))
6058, 59fmptd 6292 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))):(0...𝑥)⟶𝐾)
611ad2antrr 758 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → 𝑅 ∈ Ring)
623ad2antrr 758 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → 𝐶𝐾)
634ad2antrr 758 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → 𝐷 ∈ ℕ0)
64 eldifi 3694 . . . . . . . . . . . . 13 (𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)}) → 𝑦 ∈ (0...𝑥))
6564, 54syl 17 . . . . . . . . . . . 12 (𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)}) → (𝑥𝑦) ∈ ℕ0)
6665adantl 481 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → (𝑥𝑦) ∈ ℕ0)
67 eldifsn 4260 . . . . . . . . . . . 12 (𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)}) ↔ (𝑦 ∈ (0...𝑥) ∧ 𝑦 ≠ (𝑥𝐷)))
68 simplrl 796 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑥 ∈ ℕ0)
6968nn0cnd 11230 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑥 ∈ ℂ)
7047nn0cnd 11230 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℂ)
7170adantl 481 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 ∈ ℂ)
7269, 71nncand 10276 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝑥 − (𝑥𝑦)) = 𝑦)
7372eqcomd 2616 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 = (𝑥 − (𝑥𝑦)))
74 oveq2 6557 . . . . . . . . . . . . . . . 16 (𝐷 = (𝑥𝑦) → (𝑥𝐷) = (𝑥 − (𝑥𝑦)))
7574eqeq2d 2620 . . . . . . . . . . . . . . 15 (𝐷 = (𝑥𝑦) → (𝑦 = (𝑥𝐷) ↔ 𝑦 = (𝑥 − (𝑥𝑦))))
7673, 75syl5ibrcom 236 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝐷 = (𝑥𝑦) → 𝑦 = (𝑥𝐷)))
7776necon3d 2803 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝑦 ≠ (𝑥𝐷) → 𝐷 ≠ (𝑥𝑦)))
7877impr 647 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ (𝑦 ∈ (0...𝑥) ∧ 𝑦 ≠ (𝑥𝐷))) → 𝐷 ≠ (𝑥𝑦))
7967, 78sylan2b 491 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → 𝐷 ≠ (𝑥𝑦))
8020, 5, 6, 7, 8, 9, 10, 61, 62, 63, 66, 79coe1tmfv2 19466 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)) = 0 )
8180oveq2d 6565 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = (((coe1𝐴)‘𝑦) × 0 ))
825, 15, 20ringrz 18411 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ ((coe1𝐴)‘𝑦) ∈ 𝐾) → (((coe1𝐴)‘𝑦) × 0 ) = 0 )
8342, 49, 82syl2anc 691 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1𝐴)‘𝑦) × 0 ) = 0 )
8464, 83sylan2 490 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → (((coe1𝐴)‘𝑦) × 0 ) = 0 )
8581, 84eqtrd 2644 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = 0 )
8685, 25suppss2 7216 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))) supp 0 ) ⊆ {(𝑥𝐷)})
875, 20, 23, 25, 41, 60, 86gsumpt 18184 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = ((𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))‘(𝑥𝐷)))
88 fveq2 6103 . . . . . . . . 9 (𝑦 = (𝑥𝐷) → ((coe1𝐴)‘𝑦) = ((coe1𝐴)‘(𝑥𝐷)))
89 oveq2 6557 . . . . . . . . . 10 (𝑦 = (𝑥𝐷) → (𝑥𝑦) = (𝑥 − (𝑥𝐷)))
9089fveq2d 6107 . . . . . . . . 9 (𝑦 = (𝑥𝐷) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)) = ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷))))
9188, 90oveq12d 6567 . . . . . . . 8 (𝑦 = (𝑥𝐷) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = (((coe1𝐴)‘(𝑥𝐷)) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷)))))
92 ovex 6577 . . . . . . . 8 (((coe1𝐴)‘(𝑥𝐷)) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷)))) ∈ V
9391, 59, 92fvmpt 6191 . . . . . . 7 ((𝑥𝐷) ∈ (0...𝑥) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))‘(𝑥𝐷)) = (((coe1𝐴)‘(𝑥𝐷)) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷)))))
9441, 93syl 17 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))‘(𝑥𝐷)) = (((coe1𝐴)‘(𝑥𝐷)) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷)))))
9528nn0cnd 11230 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝑥 ∈ ℂ)
9627nn0cnd 11230 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝐷 ∈ ℂ)
9795, 96nncand 10276 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑥 − (𝑥𝐷)) = 𝐷)
9897fveq2d 6107 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷))) = ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷))
993adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝐶𝐾)
10020, 5, 6, 7, 8, 9, 10coe1tmfv1 19465 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷) = 𝐶)
10121, 99, 27, 100syl3anc 1318 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷) = 𝐶)
10298, 101eqtrd 2644 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷))) = 𝐶)
103102oveq2d 6565 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (((coe1𝐴)‘(𝑥𝐷)) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷)))) = (((coe1𝐴)‘(𝑥𝐷)) × 𝐶))
10487, 94, 1033eqtrd 2648 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = (((coe1𝐴)‘(𝑥𝐷)) × 𝐶))
105104anassrs 678 . . . 4 (((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = (((coe1𝐴)‘(𝑥𝐷)) × 𝐶))
1061ad2antrr 758 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝑅 ∈ Ring)
1073ad2antrr 758 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝐶𝐾)
1084ad2antrr 758 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝐷 ∈ ℕ0)
10954ad2antll 761 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (𝑥𝑦) ∈ ℕ0)
11054nn0red 11229 . . . . . . . . . . . . 13 (𝑦 ∈ (0...𝑥) → (𝑥𝑦) ∈ ℝ)
111110ad2antll 761 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (𝑥𝑦) ∈ ℝ)
11233ad2antlr 759 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝑥 ∈ ℝ)
11335ad2antrr 758 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝐷 ∈ ℝ)
11447ad2antll 761 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝑦 ∈ ℕ0)
115114nn0ge0d 11231 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 0 ≤ 𝑦)
11647nn0red 11229 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℝ)
117116ad2antll 761 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝑦 ∈ ℝ)
118112, 117subge02d 10498 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (0 ≤ 𝑦 ↔ (𝑥𝑦) ≤ 𝑥))
119115, 118mpbid 221 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (𝑥𝑦) ≤ 𝑥)
120 simprl 790 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → ¬ 𝐷𝑥)
121112, 113ltnled 10063 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (𝑥 < 𝐷 ↔ ¬ 𝐷𝑥))
122120, 121mpbird 246 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝑥 < 𝐷)
123111, 112, 113, 119, 122lelttrd 10074 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (𝑥𝑦) < 𝐷)
124111, 123gtned 10051 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝐷 ≠ (𝑥𝑦))
12520, 5, 6, 7, 8, 9, 10, 106, 107, 108, 109, 124coe1tmfv2 19466 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)) = 0 )
126125oveq2d 6565 . . . . . . . . 9 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = (((coe1𝐴)‘𝑦) × 0 ))
12745ad2antrr 758 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (coe1𝐴):ℕ0𝐾)
128127, 114ffvelrnd 6268 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → ((coe1𝐴)‘𝑦) ∈ 𝐾)
129106, 128, 82syl2anc 691 . . . . . . . . 9 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (((coe1𝐴)‘𝑦) × 0 ) = 0 )
130126, 129eqtrd 2644 . . . . . . . 8 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = 0 )
131130anassrs 678 . . . . . . 7 ((((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = 0 )
132131mpteq2dva 4672 . . . . . 6 (((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) → (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))) = (𝑦 ∈ (0...𝑥) ↦ 0 ))
133132oveq2d 6565 . . . . 5 (((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )))
1341, 22syl 17 . . . . . . 7 (𝜑𝑅 ∈ Mnd)
13520gsumz 17197 . . . . . . 7 ((𝑅 ∈ Mnd ∧ (0...𝑥) ∈ V) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
136134, 24, 135sylancl 693 . . . . . 6 (𝜑 → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
137136ad2antrr 758 . . . . 5 (((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
138133, 137eqtrd 2644 . . . 4 (((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = 0 )
13918, 19, 105, 138ifbothda 4073 . . 3 ((𝜑𝑥 ∈ ℕ0) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 ))
140139mpteq2dva 4672 . 2 (𝜑 → (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))))) = (𝑥 ∈ ℕ0 ↦ if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 )))
14117, 140eqtrd 2644 1 (𝜑 → (coe1‘(𝐴 (𝐶 · (𝐷 𝑋)))) = (𝑥 ∈ ℕ0 ↦ if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wne 2780  Vcvv 3173  cdif 3537  ifcif 4036  {csn 4125   class class class wbr 4583  cmpt 4643  wf 5800  cfv 5804  (class class class)co 6549  cc 9813  cr 9814  0cc0 9815   < clt 9953  cle 9954  cmin 10145  0cn0 11169  ...cfz 12197  Basecbs 15695  .rcmulr 15769   ·𝑠 cvsca 15772  0gc0g 15923   Σg cgsu 15924  Mndcmnd 17117  .gcmg 17363  mulGrpcmgp 18312  Ringcrg 18370  var1cv1 19367  Poly1cpl1 19368  coe1cco1 19369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-tset 15787  df-ple 15788  df-0g 15925  df-gsum 15926  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-subg 17414  df-ghm 17481  df-cntz 17573  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-subrg 18601  df-lmod 18688  df-lss 18754  df-psr 19177  df-mvr 19178  df-mpl 19179  df-opsr 19181  df-psr1 19371  df-vr1 19372  df-ply1 19373  df-coe1 19374
This theorem is referenced by:  coe1tmmul2fv  19469  coe1sclmul2  19475
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