Theorem cply1mul 22234

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 2735	. . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅)
4		cply1mul.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑃)
5	1, 2, 3, 4	coe1mul 22207	. . . . . . . . 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 480	. . . . . . 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 480	. . . . . 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 7413	. . . . . . . . 9 ⊢ (𝑠 = 𝑛 → (0...𝑠) = (0...𝑛))
10		fvoveq1 7428	. . . . . . . . . 10 ⊢ (𝑠 = 𝑛 → ((coe1‘𝐺)‘(𝑠 − 𝑘)) = ((coe1‘𝐺)‘(𝑛 − 𝑘)))
11	10	oveq2d 7421	. . . . . . . . 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 5207	. . . . . . . 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 7421	. . . . . . 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 481	. . . . . 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 12508	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
16	15	adantl 481	. . . . . 6 ⊢ ((((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1‘𝐹)‘𝑐) = 0 ∧ ((coe1‘𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
17		ovexd 7440	. . . . . 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 6993	. . . . 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 3098	. . . . . . . . . 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 7413	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 0 → (𝑛 − 𝑘) = (𝑛 − 0))
21		nncn 12248	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
22	21	subid1d 11583	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ → (𝑛 − 0) = 𝑛)
23	22	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (𝑛 − 0) = 𝑛)
24	20, 23	sylan9eqr 2792	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → (𝑛 − 𝑘) = 𝑛)
25		simpll 766	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → 𝑛 ∈ ℕ)
26	24, 25	eqeltrd 2834	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → (𝑛 − 𝑘) ∈ ℕ)
27		fveqeq2 6885	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = (𝑛 − 𝑘) → (((coe1‘𝐺)‘𝑐) = 0 ↔ ((coe1‘𝐺)‘(𝑛 − 𝑘)) = 0 ))
28	27	rspcv 3597	. . . . . . . . . . . . . . . . 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 7413	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((coe1‘𝐺)‘(𝑛 − 𝑘)) = 0 → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = (((coe1‘𝐹)‘𝑘)(.r‘𝑅) 0 ))
31		simpll 766	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) → 𝑅 ∈ Ring)
32		simprl 770	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → 𝐹 ∈ 𝐵)
33		elfznn0 13637	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0)
34	33	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → 𝑘 ∈ ℕ0)
35	34	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → 𝑘 ∈ ℕ0)
36		eqid 2735	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (coe1‘𝐹) = (coe1‘𝐹)
37		eqid 2735	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Base‘𝑅) = (Base‘𝑅)
38	36, 4, 1, 37	coe1fvalcl 22148	. . . . . . . . . . . . . . . . . . . . . 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 20254	. . . . . . . . . . . . . . . . . . . . 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 2792	. . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . 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 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 ))))
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 2940	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝑘 = 0 → 𝑘 ≠ 0)
53	52, 33	anim12ci 614	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → (𝑘 ∈ ℕ0 ∧ 𝑘 ≠ 0))
54		elnnne0 12515	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ ℕ ↔ (𝑘 ∈ ℕ0 ∧ 𝑘 ≠ 0))
55	53, 54	sylibr 234	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → 𝑘 ∈ ℕ)
56		fveqeq2 6885	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = 𝑘 → (((coe1‘𝐹)‘𝑐) = 0 ↔ ((coe1‘𝐹)‘𝑘) = 0 ))
57	56	rspcv 3597	. . . . . . . . . . . . . . . . . . . 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 7412	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((coe1‘𝐹)‘𝑘) = 0 → (((coe1‘𝐹)‘𝑘)(.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))) = ( 0 (.r‘𝑅)((coe1‘𝐺)‘(𝑛 − 𝑘))))
60		simpll 766	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) ∧ 𝑘 ∈ (0...𝑛)) → 𝑅 ∈ Ring)
61	4	eleq2i 2826	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝐺 ∈ 𝐵 ↔ 𝐺 ∈ (Base‘𝑃))
62	61	biimpi 216	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐺 ∈ 𝐵 → 𝐺 ∈ (Base‘𝑃))
63	62	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ (Base‘𝑃))
64	63	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → 𝐺 ∈ (Base‘𝑃))
65		fznn0sub 13573	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 ∈ (0...𝑛) → (𝑛 − 𝑘) ∈ ℕ0)
66		eqid 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (coe1‘𝐺) = (coe1‘𝐺)
67		eqid 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (Base‘𝑃) = (Base‘𝑃)
68	66, 67, 1, 37	coe1fvalcl 22148	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 20253	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2792	. . . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 242	. . . . . . . . 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 406	. . . . . . . 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 455	. . . . . . 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 5214	. . . . . 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 7421	. . . . 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 20203	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
94		ovexd 7440	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (0...𝑛) ∈ V)
95	40	gsumz 18814	. . . . . . . . 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 480	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵)) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
98	97	adantr 480	. . . . . 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 480	. . . . 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 2774	. . . 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 3132	. . 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 6885	. . . 4 ⊢ (𝑐 = 𝑛 → (((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 ↔ ((coe1‘(𝐹 × 𝐺))‘𝑛) = 0 ))
103	102	cbvralvw 3220	. . 3 ⊢ (∀𝑐 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 ↔ ∀𝑛 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑛) = 0 )
104	101, 103	sylibr 234	. 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 412	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 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  Vcvv 3459   ↦ cmpt 5201  ‘cfv 6531  (class class class)co 7405  0cc0 11129   − cmin 11466  ℕcn 12240  ℕ₀cn0 12501  ...cfz 13524  Basecbs 17228  ._rcmulr 17272  0_gc0g 17453   Σ_g cgsu 17454  Mndcmnd 18712  Ringcrg 20193  Poly₁cpl1 22112  coe₁cco1 22113

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7671  df-ofr 7672  df-om 7862  df-1st 7988  df-2nd 7989  df-supp 8160  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8719  df-map 8842  df-pm 8843  df-ixp 8912  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-fsupp 9374  df-sup 9454  df-oi 9524  df-card 9953  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12502  df-z 12589  df-dec 12709  df-uz 12853  df-fz 13525  df-fzo 13672  df-seq 14020  df-hash 14349  df-struct 17166  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17252  df-plusg 17284  df-mulr 17285  df-sca 17287  df-vsca 17288  df-ip 17289  df-tset 17290  df-ple 17291  df-ds 17293  df-hom 17295  df-cco 17296  df-0g 17455  df-gsum 17456  df-prds 17461  df-pws 17463  df-mre 17598  df-mrc 17599  df-acs 17601  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-mhm 18761  df-submnd 18762  df-grp 18919  df-minusg 18920  df-mulg 19051  df-ghm 19196  df-cntz 19300  df-cmn 19763  df-abl 19764  df-mgp 20101  df-rng 20113  df-ur 20142  df-ring 20195  df-psr 21869  df-mpl 21871  df-opsr 21873  df-psr1 22115  df-ply1 22117  df-coe1 22118

This theorem is referenced by:  cpmatmcllem  22656