Theorem cply1mul 20464

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 2823	. . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅)
4		cply1mul.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑃)
5	1, 2, 3, 4	coe1mul 20440	. . . . . . . . 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 1116	. . . . . . . 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 483	. . . . . . 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 483	. . . . . 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 7166	. . . . . . . . 9 ⊢ (𝑠 = 𝑛 → (0...𝑠) = (0...𝑛))
10		fvoveq1 7181	. . . . . . . . . 10 ⊢ (𝑠 = 𝑛 → ((coe1‘𝐺)‘(𝑠 − 𝑘)) = ((coe1‘𝐺)‘(𝑛 − 𝑘)))
11	10	oveq2d 7174	. . . . . . . . 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 5153	. . . . . . . 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 7174	. . . . . . 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 484	. . . . . 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 11907	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
16	15	adantl 484	. . . . . 6 ⊢ ((((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1‘𝐹)‘𝑐) = 0 ∧ ((coe1‘𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
17		ovexd 7193	. . . . . 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 6777	. . . . 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 3172	. . . . . . . . . 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 7166	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 0 → (𝑛 − 𝑘) = (𝑛 − 0))
21		nncn 11648	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
22	21	subid1d 10988	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ → (𝑛 − 0) = 𝑛)
23	22	adantr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (𝑛 − 0) = 𝑛)
24	20, 23	sylan9eqr 2880	. . . . . . . . . . . . . . . . . 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 2915	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → (𝑛 − 𝑘) ∈ ℕ)
27		fveqeq2 6681	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = (𝑛 − 𝑘) → (((coe1‘𝐺)‘𝑐) = 0 ↔ ((coe1‘𝐺)‘(𝑛 − 𝑘)) = 0 ))
28	27	rspcv 3620	. . . . . . . . . . . . . . . . 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 7166	. . . . . . . . . . . . . . . . . . . 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 13003	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0)
34	33	adantl 484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → 𝑘 ∈ ℕ0)
35	34	adantr 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → 𝑘 ∈ ℕ0)
36		eqid 2823	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (coe1‘𝐹) = (coe1‘𝐹)
37		eqid 2823	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Base‘𝑅) = (Base‘𝑅)
38	36, 4, 1, 37	coe1fvalcl 20382	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝑘 ∈ ℕ0) → ((coe1‘𝐹)‘𝑘) ∈ (Base‘𝑅))
39	32, 35, 38	syl2an 597	. . . . . . . . . . . . . . . . . . . . 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 19340	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘𝐹)‘𝑘) ∈ (Base‘𝑅)) → (((coe1‘𝐹)‘𝑘)(.r‘𝑅) 0 ) = 0 )
42	31, 39, 41	syl2anc 586	. . . . . . . . . . . . . . . . . . . 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 2880	. . . . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . 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 3026	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝑘 = 0 → 𝑘 ≠ 0)
53	52, 33	anim12ci 615	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → (𝑘 ∈ ℕ0 ∧ 𝑘 ≠ 0))
54		elnnne0 11914	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ ℕ ↔ (𝑘 ∈ ℕ0 ∧ 𝑘 ≠ 0))
55	53, 54	sylibr 236	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → 𝑘 ∈ ℕ)
56		fveqeq2 6681	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = 𝑘 → (((coe1‘𝐹)‘𝑐) = 0 ↔ ((coe1‘𝐹)‘𝑘) = 0 ))
57	56	rspcv 3620	. . . . . . . . . . . . . . . . . . . 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 7165	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2906	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝐺 ∈ 𝐵 ↔ 𝐺 ∈ (Base‘𝑃))
62	61	biimpi 218	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐺 ∈ 𝐵 → 𝐺 ∈ (Base‘𝑃))
63	62	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ (Base‘𝑃))
64	63	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → 𝐺 ∈ (Base‘𝑃))
65		fznn0sub 12942	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 ∈ (0...𝑛) → (𝑛 − 𝑘) ∈ ℕ0)
66		eqid 2823	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (coe1‘𝐺) = (coe1‘𝐺)
67		eqid 2823	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (Base‘𝑃) = (Base‘𝑃)
68	66, 67, 1, 37	coe1fvalcl 20382	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐺 ∈ (Base‘𝑃) ∧ (𝑛 − 𝑘) ∈ ℕ0) → ((coe1‘𝐺)‘(𝑛 − 𝑘)) ∈ (Base‘𝑅))
69	64, 65, 68	syl2an 597	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ 𝑘 ∈ (0...𝑛)) → ((coe1‘𝐺)‘(𝑛 − 𝑘)) ∈ (Base‘𝑅))
70	37, 3, 40	ringlz 19339	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘𝐺)‘(𝑛 − 𝑘)) ∈ (Base‘𝑅)) → ( 0 (.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 )
71	60, 69, 70	syl2anc 586	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ 𝑘 ∈ (0...𝑛)) → ( 0 (.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = 0 )
72	59, 71	sylan9eqr 2880	. . . . . . . . . . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . . 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 409	. . . . . . . . . . . . . 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 483	. . . . . . . . . . . 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 184	. . . . . . . . . 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	syl5bi 244	. . . . . . . . 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 409	. . . . . . . 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 458	. . . . . . 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 5163	. . . . . 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 7174	. . . . 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 19308	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
94		ovexd 7193	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (0...𝑛) ∈ V)
95	40	gsumz 18002	. . . . . . . . 9 ⊢ ((𝑅 ∈ Mnd ∧ (0...𝑛) ∈ V) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
96	93, 94, 95	syl2anc 586	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
97	96	adantr 483	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
98	97	adantr 483	. . . . . 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 483	. . . . 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 2862	. . . 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 3184	. . 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 6681	. . . 4 ⊢ (𝑐 = 𝑛 → (((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 ↔ ((coe1‘(𝐹 × 𝐺))‘𝑛) = 0 ))
103	102	cbvralvw 3451	. . 3 ⊢ (∀𝑐 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 ↔ ∀𝑛 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑛) = 0 )
104	101, 103	sylibr 236	. 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 415	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 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  ∀wral 3140  Vcvv 3496   ↦ cmpt 5148  ‘cfv 6357  (class class class)co 7158  0cc0 10539   − cmin 10872  ℕcn 11640  ℕ₀cn0 11900  ...cfz 12895  Basecbs 16485  ._rcmulr 16568  0_gc0g 16715   Σ_g cgsu 16716  Mndcmnd 17913  Ringcrg 19299  Poly₁cpl1 20347  coe₁cco1 20348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-iin 4924  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-se 5517  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-of 7411  df-ofr 7412  df-om 7583  df-1st 7691  df-2nd 7692  df-supp 7833  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-2o 8105  df-oadd 8108  df-er 8291  df-map 8410  df-pm 8411  df-ixp 8464  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-fsupp 8836  df-oi 8976  df-card 9370  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-z 11985  df-dec 12102  df-uz 12247  df-fz 12896  df-fzo 13037  df-seq 13373  df-hash 13694  df-struct 16487  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-ress 16493  df-plusg 16580  df-mulr 16581  df-sca 16583  df-vsca 16584  df-tset 16586  df-ple 16587  df-0g 16717  df-gsum 16718  df-mre 16859  df-mrc 16860  df-acs 16862  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-mhm 17958  df-submnd 17959  df-grp 18108  df-minusg 18109  df-mulg 18227  df-ghm 18358  df-cntz 18449  df-cmn 18910  df-abl 18911  df-mgp 19242  df-ur 19254  df-ring 19301  df-psr 20138  df-mpl 20140  df-opsr 20142  df-psr1 20350  df-ply1 20352  df-coe1 20353

This theorem is referenced by:  cpmatmcllem  21328