MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  coe1tmmul2 Structured version   Visualization version   GIF version

Theorem coe1tmmul2 19413
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 19409 . . . 4 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐶 · (𝐷 𝑋)) ∈ 𝐵)
131, 3, 4, 12syl3anc 1317 . . 3 (𝜑 → (𝐶 · (𝐷 𝑋)) ∈ 𝐵)
14 coe1tmmul.t . . . 4 = (.r𝑃)
15 coe1tmmul.u . . . 4 × = (.r𝑅)
166, 14, 15, 11coe1mul 19407 . . 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 1317 . 2 (𝜑 → (coe1‘(𝐴 (𝐶 · (𝐷 𝑋)))) = (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))))))
18 eqeq2 2620 . . . 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 2620 . . . 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 479 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝑅 ∈ Ring)
22 ringmnd 18325 . . . . . . . 8 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
2321, 22syl 17 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝑅 ∈ Mnd)
24 ovex 6555 . . . . . . . 8 (0...𝑥) ∈ V
2524a1i 11 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (0...𝑥) ∈ V)
26 simprr 791 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝐷𝑥)
274adantr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝐷 ∈ ℕ0)
28 simprl 789 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝑥 ∈ ℕ0)
29 nn0sub 11190 . . . . . . . . . 10 ((𝐷 ∈ ℕ0𝑥 ∈ ℕ0) → (𝐷𝑥 ↔ (𝑥𝐷) ∈ ℕ0))
3027, 28, 29syl2anc 690 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝐷𝑥 ↔ (𝑥𝐷) ∈ ℕ0))
3126, 30mpbid 220 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑥𝐷) ∈ ℕ0)
3227nn0ge0d 11201 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 0 ≤ 𝐷)
33 nn0re 11148 . . . . . . . . . . 11 (𝑥 ∈ ℕ0𝑥 ∈ ℝ)
3433ad2antrl 759 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝑥 ∈ ℝ)
354nn0red 11199 . . . . . . . . . . 11 (𝜑𝐷 ∈ ℝ)
3635adantr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝐷 ∈ ℝ)
3734, 36subge02d 10468 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (0 ≤ 𝐷 ↔ (𝑥𝐷) ≤ 𝑥))
3832, 37mpbid 220 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑥𝐷) ≤ 𝑥)
39 fznn0 12256 . . . . . . . . 9 (𝑥 ∈ ℕ0 → ((𝑥𝐷) ∈ (0...𝑥) ↔ ((𝑥𝐷) ∈ ℕ0 ∧ (𝑥𝐷) ≤ 𝑥)))
4039ad2antrl 759 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → ((𝑥𝐷) ∈ (0...𝑥) ↔ ((𝑥𝐷) ∈ ℕ0 ∧ (𝑥𝐷) ≤ 𝑥)))
4131, 38, 40mpbir2and 958 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑥𝐷) ∈ (0...𝑥))
421ad2antrr 757 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑅 ∈ Ring)
43 eqid 2609 . . . . . . . . . . . . 13 (coe1𝐴) = (coe1𝐴)
4443, 11, 6, 5coe1f 19348 . . . . . . . . . . . 12 (𝐴𝐵 → (coe1𝐴):ℕ0𝐾)
452, 44syl 17 . . . . . . . . . . 11 (𝜑 → (coe1𝐴):ℕ0𝐾)
4645ad2antrr 757 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (coe1𝐴):ℕ0𝐾)
47 elfznn0 12257 . . . . . . . . . . 11 (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℕ0)
4847adantl 480 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 ∈ ℕ0)
4946, 48ffvelrnd 6253 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → ((coe1𝐴)‘𝑦) ∈ 𝐾)
50 eqid 2609 . . . . . . . . . . . . 13 (coe1‘(𝐶 · (𝐷 𝑋))) = (coe1‘(𝐶 · (𝐷 𝑋)))
5150, 11, 6, 5coe1f 19348 . . . . . . . . . . . 12 ((𝐶 · (𝐷 𝑋)) ∈ 𝐵 → (coe1‘(𝐶 · (𝐷 𝑋))):ℕ0𝐾)
5213, 51syl 17 . . . . . . . . . . 11 (𝜑 → (coe1‘(𝐶 · (𝐷 𝑋))):ℕ0𝐾)
5352ad2antrr 757 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (coe1‘(𝐶 · (𝐷 𝑋))):ℕ0𝐾)
54 fznn0sub 12199 . . . . . . . . . . 11 (𝑦 ∈ (0...𝑥) → (𝑥𝑦) ∈ ℕ0)
5554adantl 480 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝑥𝑦) ∈ ℕ0)
5653, 55ffvelrnd 6253 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)) ∈ 𝐾)
575, 15ringcl 18330 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ ((coe1𝐴)‘𝑦) ∈ 𝐾 ∧ ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)) ∈ 𝐾) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) ∈ 𝐾)
5842, 49, 56, 57syl3anc 1317 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) ∈ 𝐾)
59 eqid 2609 . . . . . . . 8 (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))) = (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))
6058, 59fmptd 6277 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))):(0...𝑥)⟶𝐾)
611ad2antrr 757 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → 𝑅 ∈ Ring)
623ad2antrr 757 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → 𝐶𝐾)
634ad2antrr 757 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → 𝐷 ∈ ℕ0)
64 eldifi 3693 . . . . . . . . . . . . 13 (𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)}) → 𝑦 ∈ (0...𝑥))
6564, 54syl 17 . . . . . . . . . . . 12 (𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)}) → (𝑥𝑦) ∈ ℕ0)
6665adantl 480 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → (𝑥𝑦) ∈ ℕ0)
67 eldifsn 4259 . . . . . . . . . . . 12 (𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)}) ↔ (𝑦 ∈ (0...𝑥) ∧ 𝑦 ≠ (𝑥𝐷)))
68 simplrl 795 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑥 ∈ ℕ0)
6968nn0cnd 11200 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑥 ∈ ℂ)
7047nn0cnd 11200 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℂ)
7170adantl 480 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 ∈ ℂ)
7269, 71nncand 10248 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝑥 − (𝑥𝑦)) = 𝑦)
7372eqcomd 2615 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 = (𝑥 − (𝑥𝑦)))
74 oveq2 6535 . . . . . . . . . . . . . . . 16 (𝐷 = (𝑥𝑦) → (𝑥𝐷) = (𝑥 − (𝑥𝑦)))
7574eqeq2d 2619 . . . . . . . . . . . . . . 15 (𝐷 = (𝑥𝑦) → (𝑦 = (𝑥𝐷) ↔ 𝑦 = (𝑥 − (𝑥𝑦))))
7673, 75syl5ibrcom 235 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝐷 = (𝑥𝑦) → 𝑦 = (𝑥𝐷)))
7776necon3d 2802 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝑦 ≠ (𝑥𝐷) → 𝐷 ≠ (𝑥𝑦)))
7877impr 646 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ (𝑦 ∈ (0...𝑥) ∧ 𝑦 ≠ (𝑥𝐷))) → 𝐷 ≠ (𝑥𝑦))
7967, 78sylan2b 490 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → 𝐷 ≠ (𝑥𝑦))
8020, 5, 6, 7, 8, 9, 10, 61, 62, 63, 66, 79coe1tmfv2 19412 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)) = 0 )
8180oveq2d 6543 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = (((coe1𝐴)‘𝑦) × 0 ))
825, 15, 20ringrz 18357 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ ((coe1𝐴)‘𝑦) ∈ 𝐾) → (((coe1𝐴)‘𝑦) × 0 ) = 0 )
8342, 49, 82syl2anc 690 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1𝐴)‘𝑦) × 0 ) = 0 )
8464, 83sylan2 489 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → (((coe1𝐴)‘𝑦) × 0 ) = 0 )
8581, 84eqtrd 2643 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = 0 )
8685, 25suppss2 7193 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))) supp 0 ) ⊆ {(𝑥𝐷)})
875, 20, 23, 25, 41, 60, 86gsumpt 18130 . . . . . 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 6088 . . . . . . . . 9 (𝑦 = (𝑥𝐷) → ((coe1𝐴)‘𝑦) = ((coe1𝐴)‘(𝑥𝐷)))
89 oveq2 6535 . . . . . . . . . 10 (𝑦 = (𝑥𝐷) → (𝑥𝑦) = (𝑥 − (𝑥𝐷)))
9089fveq2d 6092 . . . . . . . . 9 (𝑦 = (𝑥𝐷) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)) = ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷))))
9188, 90oveq12d 6545 . . . . . . . 8 (𝑦 = (𝑥𝐷) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = (((coe1𝐴)‘(𝑥𝐷)) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷)))))
92 ovex 6555 . . . . . . . 8 (((coe1𝐴)‘(𝑥𝐷)) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷)))) ∈ V
9391, 59, 92fvmpt 6176 . . . . . . 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 11200 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝑥 ∈ ℂ)
9627nn0cnd 11200 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝐷 ∈ ℂ)
9795, 96nncand 10248 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑥 − (𝑥𝐷)) = 𝐷)
9897fveq2d 6092 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷))) = ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷))
993adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝐶𝐾)
10020, 5, 6, 7, 8, 9, 10coe1tmfv1 19411 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷) = 𝐶)
10121, 99, 27, 100syl3anc 1317 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷) = 𝐶)
10298, 101eqtrd 2643 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷))) = 𝐶)
103102oveq2d 6543 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (((coe1𝐴)‘(𝑥𝐷)) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷)))) = (((coe1𝐴)‘(𝑥𝐷)) × 𝐶))
10487, 94, 1033eqtrd 2647 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = (((coe1𝐴)‘(𝑥𝐷)) × 𝐶))
105104anassrs 677 . . . 4 (((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = (((coe1𝐴)‘(𝑥𝐷)) × 𝐶))
1061ad2antrr 757 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝑅 ∈ Ring)
1073ad2antrr 757 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝐶𝐾)
1084ad2antrr 757 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝐷 ∈ ℕ0)
10954ad2antll 760 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (𝑥𝑦) ∈ ℕ0)
11054nn0red 11199 . . . . . . . . . . . . 13 (𝑦 ∈ (0...𝑥) → (𝑥𝑦) ∈ ℝ)
111110ad2antll 760 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (𝑥𝑦) ∈ ℝ)
11233ad2antlr 758 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝑥 ∈ ℝ)
11335ad2antrr 757 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝐷 ∈ ℝ)
11447ad2antll 760 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝑦 ∈ ℕ0)
115114nn0ge0d 11201 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 0 ≤ 𝑦)
11647nn0red 11199 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℝ)
117116ad2antll 760 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝑦 ∈ ℝ)
118112, 117subge02d 10468 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (0 ≤ 𝑦 ↔ (𝑥𝑦) ≤ 𝑥))
119115, 118mpbid 220 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (𝑥𝑦) ≤ 𝑥)
120 simprl 789 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → ¬ 𝐷𝑥)
121112, 113ltnled 10035 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (𝑥 < 𝐷 ↔ ¬ 𝐷𝑥))
122120, 121mpbird 245 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝑥 < 𝐷)
123111, 112, 113, 119, 122lelttrd 10046 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (𝑥𝑦) < 𝐷)
124111, 123gtned 10023 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝐷 ≠ (𝑥𝑦))
12520, 5, 6, 7, 8, 9, 10, 106, 107, 108, 109, 124coe1tmfv2 19412 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)) = 0 )
126125oveq2d 6543 . . . . . . . . 9 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = (((coe1𝐴)‘𝑦) × 0 ))
12745ad2antrr 757 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (coe1𝐴):ℕ0𝐾)
128127, 114ffvelrnd 6253 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → ((coe1𝐴)‘𝑦) ∈ 𝐾)
129106, 128, 82syl2anc 690 . . . . . . . . 9 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (((coe1𝐴)‘𝑦) × 0 ) = 0 )
130126, 129eqtrd 2643 . . . . . . . 8 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = 0 )
131130anassrs 677 . . . . . . 7 ((((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = 0 )
132131mpteq2dva 4666 . . . . . 6 (((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) → (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))) = (𝑦 ∈ (0...𝑥) ↦ 0 ))
133132oveq2d 6543 . . . . 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 17143 . . . . . . 7 ((𝑅 ∈ Mnd ∧ (0...𝑥) ∈ V) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
136134, 24, 135sylancl 692 . . . . . 6 (𝜑 → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
137136ad2antrr 757 . . . . 5 (((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
138133, 137eqtrd 2643 . . . 4 (((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = 0 )
13918, 19, 105, 138ifbothda 4072 . . 3 ((𝜑𝑥 ∈ ℕ0) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 ))
140139mpteq2dva 4666 . 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 2643 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 194  wa 382   = wceq 1474  wcel 1976  wne 2779  Vcvv 3172  cdif 3536  ifcif 4035  {csn 4124   class class class wbr 4577  cmpt 4637  wf 5786  cfv 5790  (class class class)co 6527  cc 9790  cr 9791  0cc0 9792   < clt 9930  cle 9931  cmin 10117  0cn0 11139  ...cfz 12152  Basecbs 15641  .rcmulr 15715   ·𝑠 cvsca 15718  0gc0g 15869   Σg cgsu 15870  Mndcmnd 17063  .gcmg 17309  mulGrpcmgp 18258  Ringcrg 18316  var1cv1 19313  Poly1cpl1 19314  coe1cco1 19315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-inf2 8398  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-of 6772  df-ofr 6773  df-om 6935  df-1st 7036  df-2nd 7037  df-supp 7160  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-2o 7425  df-oadd 7428  df-er 7606  df-map 7723  df-pm 7724  df-ixp 7772  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-fsupp 8136  df-oi 8275  df-card 8625  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-3 10927  df-4 10928  df-5 10929  df-6 10930  df-7 10931  df-8 10932  df-9 10933  df-n0 11140  df-z 11211  df-dec 11326  df-uz 11520  df-fz 12153  df-fzo 12290  df-seq 12619  df-hash 12935  df-struct 15643  df-ndx 15644  df-slot 15645  df-base 15646  df-sets 15647  df-ress 15648  df-plusg 15727  df-mulr 15728  df-sca 15730  df-vsca 15731  df-tset 15733  df-ple 15734  df-0g 15871  df-gsum 15872  df-mre 16015  df-mrc 16016  df-acs 16018  df-mgm 17011  df-sgrp 17053  df-mnd 17064  df-mhm 17104  df-submnd 17105  df-grp 17194  df-minusg 17195  df-sbg 17196  df-mulg 17310  df-subg 17360  df-ghm 17427  df-cntz 17519  df-cmn 17964  df-abl 17965  df-mgp 18259  df-ur 18271  df-ring 18318  df-subrg 18547  df-lmod 18634  df-lss 18700  df-psr 19123  df-mvr 19124  df-mpl 19125  df-opsr 19127  df-psr1 19317  df-vr1 19318  df-ply1 19319  df-coe1 19320
This theorem is referenced by:  coe1tmmul2fv  19415  coe1sclmul2  19421
  Copyright terms: Public domain W3C validator