Theorem coe1tmmul 22161

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 = (0g‘𝑅)
coe1tm.k	⊢ 𝐾 = (Base‘𝑅)
coe1tm.p	⊢ 𝑃 = (Poly1‘𝑅)
coe1tm.x	⊢ 𝑋 = (var1‘𝑅)
coe1tm.m	⊢ · = ( ·𝑠 ‘𝑃)
coe1tm.n	⊢ 𝑁 = (mulGrp‘𝑃)
coe1tm.e	⊢ ↑ = (.g‘𝑁)
coe1tmmul.b	⊢ 𝐵 = (Base‘𝑃)
coe1tmmul.t	⊢ ∙ = (.r‘𝑃)
coe1tmmul.u	⊢ × = (.r‘𝑅)
coe1tmmul.a	⊢ (𝜑 → 𝐴 ∈ 𝐵)
coe1tmmul.r	⊢ (𝜑 → 𝑅 ∈ Ring)
coe1tmmul.c	⊢ (𝜑 → 𝐶 ∈ 𝐾)
coe1tmmul.d	⊢ (𝜑 → 𝐷 ∈ ℕ0)

Assertion

Ref	Expression
coe1tmmul	⊢ (𝜑 → (coe1‘((𝐶 · (𝐷 ↑ 𝑋)) ∙ 𝐴)) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, (𝐶 × ((coe1‘𝐴)‘(𝑥 − 𝐷))), 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.

Step	Hyp	Ref	Expression
1		coe1tmmul.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
2		coe1tmmul.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐾)
3		coe1tmmul.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ ℕ0)
4		coe1tm.k	. . . . 5 ⊢ 𝐾 = (Base‘𝑅)
5		coe1tm.p	. . . . 5 ⊢ 𝑃 = (Poly1‘𝑅)
6		coe1tm.x	. . . . 5 ⊢ 𝑋 = (var1‘𝑅)
7		coe1tm.m	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑃)
8		coe1tm.n	. . . . 5 ⊢ 𝑁 = (mulGrp‘𝑃)
9		coe1tm.e	. . . . 5 ⊢ ↑ = (.g‘𝑁)
10		coe1tmmul.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑃)
11	4, 5, 6, 7, 8, 9, 10	ply1tmcl 22156	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝐶 · (𝐷 ↑ 𝑋)) ∈ 𝐵)
12	1, 2, 3, 11	syl3anc 1373	. . 3 ⊢ (𝜑 → (𝐶 · (𝐷 ↑ 𝑋)) ∈ 𝐵)
13		coe1tmmul.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
14		coe1tmmul.t	. . . 4 ⊢ ∙ = (.r‘𝑃)
15		coe1tmmul.u	. . . 4 ⊢ × = (.r‘𝑅)
16	5, 14, 15, 10	coe1mul 22154	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 · (𝐷 ↑ 𝑋)) ∈ 𝐵 ∧ 𝐴 ∈ 𝐵) → (coe1‘((𝐶 · (𝐷 ↑ 𝑋)) ∙ 𝐴)) = (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) × ((coe1‘𝐴)‘(𝑥 − 𝑦)))))))
17	1, 12, 13, 16	syl3anc 1373	. 2 ⊢ (𝜑 → (coe1‘((𝐶 · (𝐷 ↑ 𝑋)) ∙ 𝐴)) = (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) × ((coe1‘𝐴)‘(𝑥 − 𝑦)))))))
18		eqeq2 2741	. . . 4 ⊢ ((𝐶 × ((coe1‘𝐴)‘(𝑥 − 𝐷))) = if(𝐷 ≤ 𝑥, (𝐶 × ((coe1‘𝐴)‘(𝑥 − 𝐷))), 0 ) → ((𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) × ((coe1‘𝐴)‘(𝑥 − 𝑦))))) = (𝐶 × ((coe1‘𝐴)‘(𝑥 − 𝐷))) ↔ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) × ((coe1‘𝐴)‘(𝑥 − 𝑦))))) = if(𝐷 ≤ 𝑥, (𝐶 × ((coe1‘𝐴)‘(𝑥 − 𝐷))), 0 )))
19		eqeq2 2741	. . . 4 ⊢ ( 0 = if(𝐷 ≤ 𝑥, (𝐶 × ((coe1‘𝐴)‘(𝑥 − 𝐷))), 0 ) → ((𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) × ((coe1‘𝐴)‘(𝑥 − 𝑦))))) = 0 ↔ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) × ((coe1‘𝐴)‘(𝑥 − 𝑦))))) = if(𝐷 ≤ 𝑥, (𝐶 × ((coe1‘𝐴)‘(𝑥 − 𝐷))), 0 )))
20		coe1tm.z	. . . . . 6 ⊢ 0 = (0g‘𝑅)
21	1	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → 𝑅 ∈ Ring)
22		ringmnd 20128	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
23	21, 22	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → 𝑅 ∈ Mnd)
24		ovexd 7384	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → (0...𝑥) ∈ V)
25	3	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → 𝐷 ∈ ℕ0)
26		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → 𝐷 ≤ 𝑥)
27		fznn0 13522	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ0 → (𝐷 ∈ (0...𝑥) ↔ (𝐷 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)))
28	27	ad2antlr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → (𝐷 ∈ (0...𝑥) ↔ (𝐷 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)))
29	25, 26, 28	mpbir2and 713	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → 𝐷 ∈ (0...𝑥))
30	1	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝑦 ∈ (0...𝑥)) → 𝑅 ∈ Ring)
31		eqid 2729	. . . . . . . . . . . . 13 ⊢ (coe1‘(𝐶 · (𝐷 ↑ 𝑋))) = (coe1‘(𝐶 · (𝐷 ↑ 𝑋)))
32	31, 10, 5, 4	coe1f 22094	. . . . . . . . . . . 12 ⊢ ((𝐶 · (𝐷 ↑ 𝑋)) ∈ 𝐵 → (coe1‘(𝐶 · (𝐷 ↑ 𝑋))):ℕ0⟶𝐾)
33	12, 32	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (coe1‘(𝐶 · (𝐷 ↑ 𝑋))):ℕ0⟶𝐾)
34	33	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → (coe1‘(𝐶 · (𝐷 ↑ 𝑋))):ℕ0⟶𝐾)
35		elfznn0 13523	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℕ0)
36		ffvelcdm 7015	. . . . . . . . . 10 ⊢ (((coe1‘(𝐶 · (𝐷 ↑ 𝑋))):ℕ0⟶𝐾 ∧ 𝑦 ∈ ℕ0) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) ∈ 𝐾)
37	34, 35, 36	syl2an 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝑦 ∈ (0...𝑥)) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) ∈ 𝐾)
38		eqid 2729	. . . . . . . . . . . . 13 ⊢ (coe1‘𝐴) = (coe1‘𝐴)
39	38, 10, 5, 4	coe1f 22094	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝐵 → (coe1‘𝐴):ℕ0⟶𝐾)
40	13, 39	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (coe1‘𝐴):ℕ0⟶𝐾)
41	40	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → (coe1‘𝐴):ℕ0⟶𝐾)
42		fznn0sub 13459	. . . . . . . . . 10 ⊢ (𝑦 ∈ (0...𝑥) → (𝑥 − 𝑦) ∈ ℕ0)
43		ffvelcdm 7015	. . . . . . . . . 10 ⊢ (((coe1‘𝐴):ℕ0⟶𝐾 ∧ (𝑥 − 𝑦) ∈ ℕ0) → ((coe1‘𝐴)‘(𝑥 − 𝑦)) ∈ 𝐾)
44	41, 42, 43	syl2an 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝑦 ∈ (0...𝑥)) → ((coe1‘𝐴)‘(𝑥 − 𝑦)) ∈ 𝐾)
45	4, 15	ringcl 20135	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) ∈ 𝐾 ∧ ((coe1‘𝐴)‘(𝑥 − 𝑦)) ∈ 𝐾) → (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) × ((coe1‘𝐴)‘(𝑥 − 𝑦))) ∈ 𝐾)
46	30, 37, 44, 45	syl3anc 1373	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) × ((coe1‘𝐴)‘(𝑥 − 𝑦))) ∈ 𝐾)
47	46	fmpttd 7049	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) × ((coe1‘𝐴)‘(𝑥 − 𝑦)))):(0...𝑥)⟶𝐾)
48	47	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) × ((coe1‘𝐴)‘(𝑥 − 𝑦)))):(0...𝑥)⟶𝐾)
49	1	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) ∧ 𝑦 ∈ ((0...𝑥) ∖ {𝐷})) → 𝑅 ∈ Ring)
50	2	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) ∧ 𝑦 ∈ ((0...𝑥) ∖ {𝐷})) → 𝐶 ∈ 𝐾)
51	3	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) ∧ 𝑦 ∈ ((0...𝑥) ∖ {𝐷})) → 𝐷 ∈ ℕ0)
52		eldifi 4082	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((0...𝑥) ∖ {𝐷}) → 𝑦 ∈ (0...𝑥))
53	52, 35	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ((0...𝑥) ∖ {𝐷}) → 𝑦 ∈ ℕ0)
54	53	adantl 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) ∧ 𝑦 ∈ ((0...𝑥) ∖ {𝐷})) → 𝑦 ∈ ℕ0)
55		eldifsni 4741	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((0...𝑥) ∖ {𝐷}) → 𝑦 ≠ 𝐷)
56	55	necomd 2980	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ((0...𝑥) ∖ {𝐷}) → 𝐷 ≠ 𝑦)
57	56	adantl 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) ∧ 𝑦 ∈ ((0...𝑥) ∖ {𝐷})) → 𝐷 ≠ 𝑦)
58	20, 4, 5, 6, 7, 8, 9, 49, 50, 51, 54, 57	coe1tmfv2 22159	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) ∧ 𝑦 ∈ ((0...𝑥) ∖ {𝐷})) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) = 0 )
59	58	oveq1d 7364	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) ∧ 𝑦 ∈ ((0...𝑥) ∖ {𝐷})) → (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) × ((coe1‘𝐴)‘(𝑥 − 𝑦))) = ( 0 × ((coe1‘𝐴)‘(𝑥 − 𝑦))))
60	4, 15, 20	ringlz 20178	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘𝐴)‘(𝑥 − 𝑦)) ∈ 𝐾) → ( 0 × ((coe1‘𝐴)‘(𝑥 − 𝑦))) = 0 )
61	30, 44, 60	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝑦 ∈ (0...𝑥)) → ( 0 × ((coe1‘𝐴)‘(𝑥 − 𝑦))) = 0 )
62	52, 61	sylan2 593	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝑦 ∈ ((0...𝑥) ∖ {𝐷})) → ( 0 × ((coe1‘𝐴)‘(𝑥 − 𝑦))) = 0 )
63	62	adantlr 715	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) ∧ 𝑦 ∈ ((0...𝑥) ∖ {𝐷})) → ( 0 × ((coe1‘𝐴)‘(𝑥 − 𝑦))) = 0 )
64	59, 63	eqtrd 2764	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) ∧ 𝑦 ∈ ((0...𝑥) ∖ {𝐷})) → (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) × ((coe1‘𝐴)‘(𝑥 − 𝑦))) = 0 )
65	64, 24	suppss2 8133	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) × ((coe1‘𝐴)‘(𝑥 − 𝑦)))) supp 0 ) ⊆ {𝐷})
66	4, 20, 23, 24, 29, 48, 65	gsumpt 19841	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) × ((coe1‘𝐴)‘(𝑥 − 𝑦))))) = ((𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) × ((coe1‘𝐴)‘(𝑥 − 𝑦))))‘𝐷))
67		fveq2 6822	. . . . . . . . 9 ⊢ (𝑦 = 𝐷 → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) = ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝐷))
68		oveq2 7357	. . . . . . . . . 10 ⊢ (𝑦 = 𝐷 → (𝑥 − 𝑦) = (𝑥 − 𝐷))
69	68	fveq2d 6826	. . . . . . . . 9 ⊢ (𝑦 = 𝐷 → ((coe1‘𝐴)‘(𝑥 − 𝑦)) = ((coe1‘𝐴)‘(𝑥 − 𝐷)))
70	67, 69	oveq12d 7367	. . . . . . . 8 ⊢ (𝑦 = 𝐷 → (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) × ((coe1‘𝐴)‘(𝑥 − 𝑦))) = (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝐷) × ((coe1‘𝐴)‘(𝑥 − 𝐷))))
71		eqid 2729	. . . . . . . 8 ⊢ (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) × ((coe1‘𝐴)‘(𝑥 − 𝑦)))) = (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) × ((coe1‘𝐴)‘(𝑥 − 𝑦))))
72		ovex 7382	. . . . . . . 8 ⊢ (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝐷) × ((coe1‘𝐴)‘(𝑥 − 𝐷))) ∈ V
73	70, 71, 72	fvmpt 6930	. . . . . . 7 ⊢ (𝐷 ∈ (0...𝑥) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) × ((coe1‘𝐴)‘(𝑥 − 𝑦))))‘𝐷) = (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝐷) × ((coe1‘𝐴)‘(𝑥 − 𝐷))))
74	29, 73	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) × ((coe1‘𝐴)‘(𝑥 − 𝑦))))‘𝐷) = (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝐷) × ((coe1‘𝐴)‘(𝑥 − 𝐷))))
75	20, 4, 5, 6, 7, 8, 9	coe1tmfv1 22158	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝐷) = 𝐶)
76	1, 2, 3, 75	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝐷) = 𝐶)
77	76	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝐷) = 𝐶)
78	77	oveq1d 7364	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝐷) × ((coe1‘𝐴)‘(𝑥 − 𝐷))) = (𝐶 × ((coe1‘𝐴)‘(𝑥 − 𝐷))))
79	74, 78	eqtrd 2764	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) × ((coe1‘𝐴)‘(𝑥 − 𝑦))))‘𝐷) = (𝐶 × ((coe1‘𝐴)‘(𝑥 − 𝐷))))
80	66, 79	eqtrd 2764	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) × ((coe1‘𝐴)‘(𝑥 − 𝑦))))) = (𝐶 × ((coe1‘𝐴)‘(𝑥 − 𝐷))))
81	1	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) ∧ 𝑦 ∈ (0...𝑥)) → 𝑅 ∈ Ring)
82	2	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) ∧ 𝑦 ∈ (0...𝑥)) → 𝐶 ∈ 𝐾)
83	3	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) ∧ 𝑦 ∈ (0...𝑥)) → 𝐷 ∈ ℕ0)
84	35	adantl 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 ∈ ℕ0)
85		elfzle2 13431	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0...𝑥) → 𝑦 ≤ 𝑥)
86	85	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 ≤ 𝑥)
87		breq1 5095	. . . . . . . . . . . . . 14 ⊢ (𝐷 = 𝑦 → (𝐷 ≤ 𝑥 ↔ 𝑦 ≤ 𝑥))
88	86, 87	syl5ibrcom 247	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝑦 ∈ (0...𝑥)) → (𝐷 = 𝑦 → 𝐷 ≤ 𝑥))
89	88	necon3bd 2939	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝑦 ∈ (0...𝑥)) → (¬ 𝐷 ≤ 𝑥 → 𝐷 ≠ 𝑦))
90	89	imp 406	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝑦 ∈ (0...𝑥)) ∧ ¬ 𝐷 ≤ 𝑥) → 𝐷 ≠ 𝑦)
91	90	an32s 652	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) ∧ 𝑦 ∈ (0...𝑥)) → 𝐷 ≠ 𝑦)
92	20, 4, 5, 6, 7, 8, 9, 81, 82, 83, 84, 91	coe1tmfv2 22159	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) ∧ 𝑦 ∈ (0...𝑥)) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) = 0 )
93	92	oveq1d 7364	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) × ((coe1‘𝐴)‘(𝑥 − 𝑦))) = ( 0 × ((coe1‘𝐴)‘(𝑥 − 𝑦))))
94	61	adantlr 715	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) ∧ 𝑦 ∈ (0...𝑥)) → ( 0 × ((coe1‘𝐴)‘(𝑥 − 𝑦))) = 0 )
95	93, 94	eqtrd 2764	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) × ((coe1‘𝐴)‘(𝑥 − 𝑦))) = 0 )
96	95	mpteq2dva 5185	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) → (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) × ((coe1‘𝐴)‘(𝑥 − 𝑦)))) = (𝑦 ∈ (0...𝑥) ↦ 0 ))
97	96	oveq2d 7365	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) × ((coe1‘𝐴)‘(𝑥 − 𝑦))))) = (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )))
98	1, 22	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Mnd)
99	98	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) → 𝑅 ∈ Mnd)
100		ovexd 7384	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) → (0...𝑥) ∈ V)
101	20	gsumz 18710	. . . . . 6 ⊢ ((𝑅 ∈ Mnd ∧ (0...𝑥) ∈ V) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
102	99, 100, 101	syl2anc 584	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
103	97, 102	eqtrd 2764	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) × ((coe1‘𝐴)‘(𝑥 − 𝑦))))) = 0 )
104	18, 19, 80, 103	ifbothda 4515	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) × ((coe1‘𝐴)‘(𝑥 − 𝑦))))) = if(𝐷 ≤ 𝑥, (𝐶 × ((coe1‘𝐴)‘(𝑥 − 𝐷))), 0 ))
105	104	mpteq2dva 5185	. 2 ⊢ (𝜑 → (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝑦) × ((coe1‘𝐴)‘(𝑥 − 𝑦)))))) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, (𝐶 × ((coe1‘𝐴)‘(𝑥 − 𝐷))), 0 )))
106	17, 105	eqtrd 2764	1 ⊢ (𝜑 → (coe1‘((𝐶 · (𝐷 ↑ 𝑋)) ∙ 𝐴)) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, (𝐶 × ((coe1‘𝐴)‘(𝑥 − 𝐷))), 0 )))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  Vcvv 3436   ∖ cdif 3900  ifcif 4476  {csn 4577   class class class wbr 5092   ↦ cmpt 5173  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349  0cc0 11009   ≤ cle 11150   − cmin 11347  ℕ₀cn0 12384  ...cfz 13410  Basecbs 17120  ._rcmulr 17162   ·_𝑠 cvsca 17165  0_gc0g 17343   Σ_g cgsu 17344  Mndcmnd 18608  ._gcmg 18946  mulGrpcmgp 20025  Ringcrg 20118  var₁cv1 22058  Poly₁cpl1 22059  coe₁cco1 22060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613  df-ofr 7614  df-om 7800  df-1st 7924  df-2nd 7925  df-supp 8094  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-er 8625  df-map 8755  df-pm 8756  df-ixp 8825  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-fsupp 9252  df-sup 9332  df-oi 9402  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-z 12472  df-dec 12592  df-uz 12736  df-fz 13411  df-fzo 13558  df-seq 13909  df-hash 14238  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-sca 17177  df-vsca 17178  df-ip 17179  df-tset 17180  df-ple 17181  df-ds 17183  df-hom 17185  df-cco 17186  df-0g 17345  df-gsum 17346  df-prds 17351  df-pws 17353  df-mre 17488  df-mrc 17489  df-acs 17491  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-mhm 18657  df-submnd 18658  df-grp 18815  df-minusg 18816  df-sbg 18817  df-mulg 18947  df-subg 19002  df-ghm 19092  df-cntz 19196  df-cmn 19661  df-abl 19662  df-mgp 20026  df-rng 20038  df-ur 20067  df-ring 20120  df-subrng 20431  df-subrg 20455  df-lmod 20765  df-lss 20835  df-psr 21816  df-mvr 21817  df-mpl 21818  df-opsr 21820  df-psr1 22062  df-vr1 22063  df-ply1 22064  df-coe1 22065

This theorem is referenced by:  coe1pwmul  22163  coe1sclmul  22166