Theorem cply1mul 22231

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 2733	. . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅)
4		cply1mul.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑃)
5	1, 2, 3, 4	coe1mul 22203	. . . . . . . . 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 7363	. . . . . . . . 9 ⊢ (𝑠 = 𝑛 → (0...𝑠) = (0...𝑛))
10		fvoveq1 7378	. . . . . . . . . 10 ⊢ (𝑠 = 𝑛 → ((coe1‘𝐺)‘(𝑠 − 𝑘)) = ((coe1‘𝐺)‘(𝑛 − 𝑘)))
11	10	oveq2d 7371	. . . . . . . . 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 5182	. . . . . . . 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 7371	. . . . . . 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 12399	. . . . . . 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 7390	. . . . . 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 6945	. . . . 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 3093	. . . . . . . . . 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 7363	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 0 → (𝑛 − 𝑘) = (𝑛 − 0))
21		nncn 12144	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
22	21	subid1d 11472	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ → (𝑛 − 0) = 𝑛)
23	22	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (𝑛 − 0) = 𝑛)
24	20, 23	sylan9eqr 2790	. . . . . . . . . . . . . . . . . 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 2833	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → (𝑛 − 𝑘) ∈ ℕ)
27		fveqeq2 6840	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = (𝑛 − 𝑘) → (((coe1‘𝐺)‘𝑐) = 0 ↔ ((coe1‘𝐺)‘(𝑛 − 𝑘)) = 0 ))
28	27	rspcv 3569	. . . . . . . . . . . . . . . . 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 7363	. . . . . . . . . . . . . . . . . . . 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 13527	. . . . . . . . . . . . . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (coe1‘𝐹) = (coe1‘𝐹)
37		eqid 2733	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Base‘𝑅) = (Base‘𝑅)
38	36, 4, 1, 37	coe1fvalcl 22144	. . . . . . . . . . . . . . . . . . . . . 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 20220	. . . . . . . . . . . . . . . . . . . . 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 2790	. . . . . . . . . . . . . . . . . . 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 2937	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝑘 = 0 → 𝑘 ≠ 0)
53	52, 33	anim12ci 614	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → (𝑘 ∈ ℕ0 ∧ 𝑘 ≠ 0))
54		elnnne0 12406	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ ℕ ↔ (𝑘 ∈ ℕ0 ∧ 𝑘 ≠ 0))
55	53, 54	sylibr 234	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → 𝑘 ∈ ℕ)
56		fveqeq2 6840	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = 𝑘 → (((coe1‘𝐹)‘𝑐) = 0 ↔ ((coe1‘𝐹)‘𝑘) = 0 ))
57	56	rspcv 3569	. . . . . . . . . . . . . . . . . . . 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 7362	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2825	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13463	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 ∈ (0...𝑛) → (𝑛 − 𝑘) ∈ ℕ0)
66		eqid 2733	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (coe1‘𝐺) = (coe1‘𝐺)
67		eqid 2733	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (Base‘𝑃) = (Base‘𝑃)
68	66, 67, 1, 37	coe1fvalcl 22144	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 20219	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2790	. . . . . . . . . . . . . . . . . . . . . . . . 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 5188	. . . . . 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 7371	. . . . 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 20169	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
94		ovexd 7390	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (0...𝑛) ∈ V)
95	40	gsumz 18752	. . . . . . . . 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 2772	. . . 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 3125	. . 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 6840	. . . 4 ⊢ (𝑐 = 𝑛 → (((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 ↔ ((coe1‘(𝐹 × 𝐺))‘𝑛) = 0 ))
103	102	cbvralvw 3211	. . 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 1541   ∈ wcel 2113   ≠ wne 2929  ∀wral 3048  Vcvv 3437   ↦ cmpt 5176  ‘cfv 6489  (class class class)co 7355  0cc0 11017   − cmin 11355  ℕcn 12136  ℕ₀cn0 12392  ...cfz 13414  Basecbs 17127  ._rcmulr 17169  0_gc0g 17350   Σ_g cgsu 17351  Mndcmnd 18650  Ringcrg 20159  Poly₁cpl1 22108  coe₁cco1 22109

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-cnex 11073  ax-resscn 11074  ax-1cn 11075  ax-icn 11076  ax-addcl 11077  ax-addrcl 11078  ax-mulcl 11079  ax-mulrcl 11080  ax-mulcom 11081  ax-addass 11082  ax-mulass 11083  ax-distr 11084  ax-i2m1 11085  ax-1ne0 11086  ax-1rid 11087  ax-rnegex 11088  ax-rrecex 11089  ax-cnre 11090  ax-pre-lttri 11091  ax-pre-lttrn 11092  ax-pre-ltadd 11093  ax-pre-mulgt0 11094

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-iin 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-isom 6498  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-of 7619  df-ofr 7620  df-om 7806  df-1st 7930  df-2nd 7931  df-supp 8100  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-2o 8395  df-er 8631  df-map 8761  df-pm 8762  df-ixp 8832  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-fsupp 9257  df-sup 9337  df-oi 9407  df-card 9843  df-pnf 11159  df-mnf 11160  df-xr 11161  df-ltxr 11162  df-le 11163  df-sub 11357  df-neg 11358  df-nn 12137  df-2 12199  df-3 12200  df-4 12201  df-5 12202  df-6 12203  df-7 12204  df-8 12205  df-9 12206  df-n0 12393  df-z 12480  df-dec 12599  df-uz 12743  df-fz 13415  df-fzo 13562  df-seq 13916  df-hash 14245  df-struct 17065  df-sets 17082  df-slot 17100  df-ndx 17112  df-base 17128  df-ress 17149  df-plusg 17181  df-mulr 17182  df-sca 17184  df-vsca 17185  df-ip 17186  df-tset 17187  df-ple 17188  df-ds 17190  df-hom 17192  df-cco 17193  df-0g 17352  df-gsum 17353  df-prds 17358  df-pws 17360  df-mre 17496  df-mrc 17497  df-acs 17499  df-mgm 18556  df-sgrp 18635  df-mnd 18651  df-mhm 18699  df-submnd 18700  df-grp 18857  df-minusg 18858  df-mulg 18989  df-ghm 19133  df-cntz 19237  df-cmn 19702  df-abl 19703  df-mgp 20067  df-rng 20079  df-ur 20108  df-ring 20161  df-psr 21856  df-mpl 21858  df-opsr 21860  df-psr1 22111  df-ply1 22113  df-coe1 22114

This theorem is referenced by:  cpmatmcllem  22653