Theorem cply1mul 21665

Description: The product of two constant polynomials is a constant polynomial. (Contributed by AV, 18-Nov-2019.)

Hypotheses

Ref	Expression
cply1mul.p	⊢ 𝑃 = (Poly1‘𝑅)
cply1mul.b	⊢ 𝐵 = (Base‘𝑃)
cply1mul.0	⊢ 0 = (0g‘𝑅)
cply1mul.m	⊢ × = (.r‘𝑃)

Assertion

Ref	Expression
cply1mul	⊢ ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (∀𝑐 ∈ ℕ (((coe1‘𝐹)‘𝑐) = 0 ∧ ((coe1‘𝐺)‘𝑐) = 0 ) → ∀𝑐 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 ))

Distinct variable groups:   𝐹,𝑐   𝐺,𝑐   × ,𝑐   0 ,𝑐

Allowed substitution hints:   𝐵(𝑐)   𝑃(𝑐)   𝑅(𝑐)

Proof of Theorem cply1mul

Dummy variables 𝑘 𝑛 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cply1mul.p	. . . . . . . . . 10 ⊢ 𝑃 = (Poly1‘𝑅)
2		cply1mul.m	. . . . . . . . . 10 ⊢ × = (.r‘𝑃)
3		eqid 2736	. . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅)
4		cply1mul.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑃)
5	1, 2, 3, 4	coe1mul 21641	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘(𝐹 × 𝐺)) = (𝑠 ∈ ℕ0 ↦ (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑠 − 𝑘)))))))
6	5	3expb 1120	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (coe1‘(𝐹 × 𝐺)) = (𝑠 ∈ ℕ0 ↦ (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑠 − 𝑘)))))))
7	6	adantr 481	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1‘𝐹)‘𝑐) = 0 ∧ ((coe1‘𝐺)‘𝑐) = 0 )) → (coe1‘(𝐹 × 𝐺)) = (𝑠 ∈ ℕ0 ↦ (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑠 − 𝑘)))))))
8	7	adantr 481	. . . . . 6 ⊢ ((((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1‘𝐹)‘𝑐) = 0 ∧ ((coe1‘𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → (coe1‘(𝐹 × 𝐺)) = (𝑠 ∈ ℕ0 ↦ (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑠 − 𝑘)))))))
9		oveq2 7365	. . . . . . . . 9 ⊢ (𝑠 = 𝑛 → (0...𝑠) = (0...𝑛))
10		fvoveq1 7380	. . . . . . . . . 10 ⊢ (𝑠 = 𝑛 → ((coe1‘𝐺)‘(𝑠 − 𝑘)) = ((coe1‘𝐺)‘(𝑛 − 𝑘)))
11	10	oveq2d 7373	. . . . . . . . 9 ⊢ (𝑠 = 𝑛 → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑠 − 𝑘))) = (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))))
12	9, 11	mpteq12dv 5196	. . . . . . . 8 ⊢ (𝑠 = 𝑛 → (𝑘 ∈ (0...𝑠) ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑠 − 𝑘)))) = (𝑘 ∈ (0...𝑛) ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘)))))
13	12	oveq2d 7373	. . . . . . 7 ⊢ (𝑠 = 𝑛 → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑠 − 𝑘))))) = (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))))))
14	13	adantl 482	. . . . . 6 ⊢ (((((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1‘𝐹)‘𝑐) = 0 ∧ ((coe1‘𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) ∧ 𝑠 = 𝑛) → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑠 − 𝑘))))) = (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))))))
15		nnnn0 12420	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
16	15	adantl 482	. . . . . 6 ⊢ ((((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1‘𝐹)‘𝑐) = 0 ∧ ((coe1‘𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
17		ovexd 7392	. . . . . 6 ⊢ ((((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1‘𝐹)‘𝑐) = 0 ∧ ((coe1‘𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))))) ∈ V)
18	8, 14, 16, 17	fvmptd 6955	. . . . 5 ⊢ ((((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1‘𝐹)‘𝑐) = 0 ∧ ((coe1‘𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → ((coe1‘(𝐹 × 𝐺))‘𝑛) = (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))))))
19		r19.26 3114	. . . . . . . . . 10 ⊢ (∀𝑐 ∈ ℕ (((coe1‘𝐹)‘𝑐) = 0 ∧ ((coe1‘𝐺)‘𝑐) = 0 ) ↔ (∀𝑐 ∈ ℕ ((coe1‘𝐹)‘𝑐) = 0 ∧ ∀𝑐 ∈ ℕ ((coe1‘𝐺)‘𝑐) = 0 ))
20		oveq2 7365	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 0 → (𝑛 − 𝑘) = (𝑛 − 0))
21		nncn 12161	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
22	21	subid1d 11501	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ → (𝑛 − 0) = 𝑛)
23	22	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (𝑛 − 0) = 𝑛)
24	20, 23	sylan9eqr 2798	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → (𝑛 − 𝑘) = 𝑛)
25		simpll 765	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → 𝑛 ∈ ℕ)
26	24, 25	eqeltrd 2838	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → (𝑛 − 𝑘) ∈ ℕ)
27		fveqeq2 6851	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = (𝑛 − 𝑘) → (((coe1‘𝐺)‘𝑐) = 0 ↔ ((coe1‘𝐺)‘(𝑛 − 𝑘)) = 0 ))
28	27	rspcv 3577	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 − 𝑘) ∈ ℕ → (∀𝑐 ∈ ℕ ((coe1‘𝐺)‘𝑐) = 0 → ((coe1‘𝐺)‘(𝑛 − 𝑘)) = 0 ))
29	26, 28	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → (∀𝑐 ∈ ℕ ((coe1‘𝐺)‘𝑐) = 0 → ((coe1‘𝐺)‘(𝑛 − 𝑘)) = 0 ))
30		oveq2 7365	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((coe1‘𝐺)‘(𝑛 − 𝑘)) = 0 → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = (((coe1‘𝐹)‘𝑘)(.r‘𝑅) 0 ))
31		simpll 765	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) → 𝑅 ∈ Ring)
32		simprl 769	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → 𝐹 ∈ 𝐵)
33		elfznn0 13534	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0)
34	33	adantl 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → 𝑘 ∈ ℕ0)
35	34	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → 𝑘 ∈ ℕ0)
36		eqid 2736	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (coe1‘𝐹) = (coe1‘𝐹)
37		eqid 2736	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Base‘𝑅) = (Base‘𝑅)
38	36, 4, 1, 37	coe1fvalcl 21583	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐹)‘𝑘) ∈ (Base‘𝑅))
39	32, 35, 38	syl2an 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) → ((coe1‘𝐹)‘𝑘) ∈ (Base‘𝑅))
40		cply1mul.0	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 = (0g‘𝑅)
41	37, 3, 40	ringrz 20012	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘𝐹)‘𝑘) ∈ (Base‘𝑅)) → (((coe1‘𝐹)‘𝑘)(.r‘𝑅) 0 ) = 0 )
42	31, 39, 41	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) → (((coe1‘𝐹)‘𝑘)(.r‘𝑅) 0 ) = 0 )
43	30, 42	sylan9eqr 2798	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) ∧ ((coe1‘𝐺)‘(𝑛 − 𝑘)) = 0 ) → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 )
44	43	ex 413	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) → (((coe1‘𝐺)‘(𝑛 − 𝑘)) = 0 → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 ))
45	44	expcom 414	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (((coe1‘𝐺)‘(𝑛 − 𝑘)) = 0 → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 )))
46	45	com23 86	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → (((coe1‘𝐺)‘(𝑛 − 𝑘)) = 0 → ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 )))
47	29, 46	syldc 48	. . . . . . . . . . . . . . 15 ⊢ (∀𝑐 ∈ ℕ ((coe1‘𝐺)‘𝑐) = 0 → (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 )))
48	47	expd 416	. . . . . . . . . . . . . 14 ⊢ (∀𝑐 ∈ ℕ ((coe1‘𝐺)‘𝑐) = 0 → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (𝑘 = 0 → ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 ))))
49	48	com24 95	. . . . . . . . . . . . 13 ⊢ (∀𝑐 ∈ ℕ ((coe1‘𝐺)‘𝑐) = 0 → ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (𝑘 = 0 → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 ))))
50	49	adantl 482	. . . . . . . . . . . 12 ⊢ ((∀𝑐 ∈ ℕ ((coe1‘𝐹)‘𝑐) = 0 ∧ ∀𝑐 ∈ ℕ ((coe1‘𝐺)‘𝑐) = 0 ) → ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (𝑘 = 0 → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 ))))
51	50	com13 88	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → ((∀𝑐 ∈ ℕ ((coe1‘𝐹)‘𝑐) = 0 ∧ ∀𝑐 ∈ ℕ ((coe1‘𝐺)‘𝑐) = 0 ) → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 ))))
52		neqne 2951	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝑘 = 0 → 𝑘 ≠ 0)
53	52, 33	anim12ci 614	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → (𝑘 ∈ ℕ0 ∧ 𝑘 ≠ 0))
54		elnnne0 12427	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ ℕ ↔ (𝑘 ∈ ℕ0 ∧ 𝑘 ≠ 0))
55	53, 54	sylibr 233	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → 𝑘 ∈ ℕ)
56		fveqeq2 6851	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = 𝑘 → (((coe1‘𝐹)‘𝑐) = 0 ↔ ((coe1‘𝐹)‘𝑘) = 0 ))
57	56	rspcv 3577	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ ℕ → (∀𝑐 ∈ ℕ ((coe1‘𝐹)‘𝑐) = 0 → ((coe1‘𝐹)‘𝑘) = 0 ))
58	55, 57	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → (∀𝑐 ∈ ℕ ((coe1‘𝐹)‘𝑐) = 0 → ((coe1‘𝐹)‘𝑘) = 0 ))
59		oveq1 7364	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((coe1‘𝐹)‘𝑘) = 0 → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = ( 0 (.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))))
60		simpll 765	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ 𝑘 ∈ (0...𝑛)) → 𝑅 ∈ Ring)
61	4	eleq2i 2829	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝐺 ∈ 𝐵 ↔ 𝐺 ∈ (Base‘𝑃))
62	61	biimpi 215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐺 ∈ 𝐵 → 𝐺 ∈ (Base‘𝑃))
63	62	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ (Base‘𝑃))
64	63	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → 𝐺 ∈ (Base‘𝑃))
65		fznn0sub 13473	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 ∈ (0...𝑛) → (𝑛 − 𝑘) ∈ ℕ0)
66		eqid 2736	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (coe1‘𝐺) = (coe1‘𝐺)
67		eqid 2736	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (Base‘𝑃) = (Base‘𝑃)
68	66, 67, 1, 37	coe1fvalcl 21583	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐺 ∈ (Base‘𝑃) ∧ (𝑛 − 𝑘) ∈ ℕ0) → ((coe1‘𝐺)‘(𝑛 − 𝑘)) ∈ (Base‘𝑅))
69	64, 65, 68	syl2an 596	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ 𝑘 ∈ (0...𝑛)) → ((coe1‘𝐺)‘(𝑛 − 𝑘)) ∈ (Base‘𝑅))
70	37, 3, 40	ringlz 20011	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘𝐺)‘(𝑛 − 𝑘)) ∈ (Base‘𝑅)) → ( 0 (.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 )
71	60, 69, 70	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ 𝑘 ∈ (0...𝑛)) → ( 0 (.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 )
72	59, 71	sylan9eqr 2798	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ 𝑘 ∈ (0...𝑛)) ∧ ((coe1‘𝐹)‘𝑘) = 0 ) → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 )
73	72	ex 413	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ 𝑘 ∈ (0...𝑛)) → (((coe1‘𝐹)‘𝑘) = 0 → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 ))
74	73	ex 413	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (𝑘 ∈ (0...𝑛) → (((coe1‘𝐹)‘𝑘) = 0 → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 )))
75	74	com23 86	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (((coe1‘𝐹)‘𝑘) = 0 → (𝑘 ∈ (0...𝑛) → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 )))
76	75	a1dd 50	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (((coe1‘𝐹)‘𝑘) = 0 → (𝑛 ∈ ℕ → (𝑘 ∈ (0...𝑛) → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 ))))
77	76	com14 96	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (0...𝑛) → (((coe1‘𝐹)‘𝑘) = 0 → (𝑛 ∈ ℕ → ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 ))))
78	77	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → (((coe1‘𝐹)‘𝑘) = 0 → (𝑛 ∈ ℕ → ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 ))))
79	58, 78	syld 47	. . . . . . . . . . . . . . . . . 18 ⊢ ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → (∀𝑐 ∈ ℕ ((coe1‘𝐹)‘𝑐) = 0 → (𝑛 ∈ ℕ → ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 ))))
80	79	com24 95	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (𝑛 ∈ ℕ → (∀𝑐 ∈ ℕ ((coe1‘𝐹)‘𝑐) = 0 → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 ))))
81	80	ex 413	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑘 = 0 → (𝑘 ∈ (0...𝑛) → ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (𝑛 ∈ ℕ → (∀𝑐 ∈ ℕ ((coe1‘𝐹)‘𝑐) = 0 → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 )))))
82	81	com14 96	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → (𝑘 ∈ (0...𝑛) → ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (¬ 𝑘 = 0 → (∀𝑐 ∈ ℕ ((coe1‘𝐹)‘𝑐) = 0 → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 )))))
83	82	imp 407	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (¬ 𝑘 = 0 → (∀𝑐 ∈ ℕ ((coe1‘𝐹)‘𝑐) = 0 → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 ))))
84	83	com14 96	. . . . . . . . . . . . 13 ⊢ (∀𝑐 ∈ ℕ ((coe1‘𝐹)‘𝑐) = 0 → ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (¬ 𝑘 = 0 → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 ))))
85	84	adantr 481	. . . . . . . . . . . 12 ⊢ ((∀𝑐 ∈ ℕ ((coe1‘𝐹)‘𝑐) = 0 ∧ ∀𝑐 ∈ ℕ ((coe1‘𝐺)‘𝑐) = 0 ) → ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (¬ 𝑘 = 0 → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 ))))
86	85	com13 88	. . . . . . . . . . 11 ⊢ (¬ 𝑘 = 0 → ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → ((∀𝑐 ∈ ℕ ((coe1‘𝐹)‘𝑐) = 0 ∧ ∀𝑐 ∈ ℕ ((coe1‘𝐺)‘𝑐) = 0 ) → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 ))))
87	51, 86	pm2.61i 182	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → ((∀𝑐 ∈ ℕ ((coe1‘𝐹)‘𝑐) = 0 ∧ ∀𝑐 ∈ ℕ ((coe1‘𝐺)‘𝑐) = 0 ) → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 )))
88	19, 87	biimtrid 241	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (∀𝑐 ∈ ℕ (((coe1‘𝐹)‘𝑐) = 0 ∧ ((coe1‘𝐺)‘𝑐) = 0 ) → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 )))
89	88	imp 407	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1‘𝐹)‘𝑐) = 0 ∧ ((coe1‘𝐺)‘𝑐) = 0 )) → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 ))
90	89	impl 456	. . . . . . 7 ⊢ (((((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1‘𝐹)‘𝑐) = 0 ∧ ((coe1‘𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑛)) → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 )
91	90	mpteq2dva 5205	. . . . . 6 ⊢ ((((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1‘𝐹)‘𝑐) = 0 ∧ ((coe1‘𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ (0...𝑛) ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘)))) = (𝑘 ∈ (0...𝑛) ↦ 0 ))
92	91	oveq2d 7373	. . . . 5 ⊢ ((((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1‘𝐹)‘𝑐) = 0 ∧ ((coe1‘𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))))) = (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )))
93		ringmnd 19974	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
94		ovexd 7392	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (0...𝑛) ∈ V)
95	40	gsumz 18646	. . . . . . . . 9 ⊢ ((𝑅 ∈ Mnd ∧ (0...𝑛) ∈ V) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
96	93, 94, 95	syl2anc 584	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
97	96	adantr 481	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
98	97	adantr 481	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1‘𝐹)‘𝑐) = 0 ∧ ((coe1‘𝐺)‘𝑐) = 0 )) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
99	98	adantr 481	. . . . 5 ⊢ ((((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1‘𝐹)‘𝑐) = 0 ∧ ((coe1‘𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
100	18, 92, 99	3eqtrd 2780	. . . 4 ⊢ ((((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1‘𝐹)‘𝑐) = 0 ∧ ((coe1‘𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → ((coe1‘(𝐹 × 𝐺))‘𝑛) = 0 )
101	100	ralrimiva 3143	. . 3 ⊢ (((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1‘𝐹)‘𝑐) = 0 ∧ ((coe1‘𝐺)‘𝑐) = 0 )) → ∀𝑛 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑛) = 0 )
102		fveqeq2 6851	. . . 4 ⊢ (𝑐 = 𝑛 → (((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 ↔ ((coe1‘(𝐹 × 𝐺))‘𝑛) = 0 ))
103	102	cbvralvw 3225	. . 3 ⊢ (∀𝑐 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 ↔ ∀𝑛 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑛) = 0 )
104	101, 103	sylibr 233	. 2 ⊢ (((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1‘𝐹)‘𝑐) = 0 ∧ ((coe1‘𝐺)‘𝑐) = 0 )) → ∀𝑐 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 )
105	104	ex 413	1 ⊢ ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (∀𝑐 ∈ ℕ (((coe1‘𝐹)‘𝑐) = 0 ∧ ((coe1‘𝐺)‘𝑐) = 0 ) → ∀𝑐 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  Vcvv 3445   ↦ cmpt 5188  ‘cfv 6496  (class class class)co 7357  0cc0 11051   − cmin 11385  ℕcn 12153  ℕ₀cn0 12413  ...cfz 13424  Basecbs 17083  ._rcmulr 17134  0_gc0g 17321   Σ_g cgsu 17322  Mndcmnd 18556  Ringcrg 19964  Poly₁cpl1 21548  coe₁cco1 21549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-hom 17157  df-cco 17158  df-0g 17323  df-gsum 17324  df-prds 17329  df-pws 17331  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-mulg 18873  df-ghm 19006  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-psr 21311  df-mpl 21313  df-opsr 21315  df-psr1 21551  df-ply1 21553  df-coe1 21554

This theorem is referenced by:  cpmatmcllem  22067