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Theorem cply1mul 19426
Description: The product of two constant polynomials is a constant polynomial. (Contributed by AV, 18-Nov-2019.)
Hypotheses
Ref Expression
cply1mul.p 𝑃 = (Poly1𝑅)
cply1mul.b 𝐵 = (Base‘𝑃)
cply1mul.0 0 = (0g𝑅)
cply1mul.m × = (.r𝑃)
Assertion
Ref Expression
cply1mul ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘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.
StepHypRef Expression
1 cply1mul.p . . . . . . . . . 10 𝑃 = (Poly1𝑅)
2 cply1mul.m . . . . . . . . . 10 × = (.r𝑃)
3 eqid 2604 . . . . . . . . . 10 (.r𝑅) = (.r𝑅)
4 cply1mul.b . . . . . . . . . 10 𝐵 = (Base‘𝑃)
51, 2, 3, 4coe1mul 19402 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (coe1‘(𝐹 × 𝐺)) = (𝑠 ∈ ℕ0 ↦ (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑠𝑘)))))))
653expb 1257 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (coe1‘(𝐹 × 𝐺)) = (𝑠 ∈ ℕ0 ↦ (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑠𝑘)))))))
76adantr 479 . . . . . . 7 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) → (coe1‘(𝐹 × 𝐺)) = (𝑠 ∈ ℕ0 ↦ (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑠𝑘)))))))
87adantr 479 . . . . . 6 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → (coe1‘(𝐹 × 𝐺)) = (𝑠 ∈ ℕ0 ↦ (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑠𝑘)))))))
9 oveq2 6530 . . . . . . . . 9 (𝑠 = 𝑛 → (0...𝑠) = (0...𝑛))
10 oveq1 6529 . . . . . . . . . . 11 (𝑠 = 𝑛 → (𝑠𝑘) = (𝑛𝑘))
1110fveq2d 6087 . . . . . . . . . 10 (𝑠 = 𝑛 → ((coe1𝐺)‘(𝑠𝑘)) = ((coe1𝐺)‘(𝑛𝑘)))
1211oveq2d 6538 . . . . . . . . 9 (𝑠 = 𝑛 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑠𝑘))) = (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))))
139, 12mpteq12dv 4652 . . . . . . . 8 (𝑠 = 𝑛 → (𝑘 ∈ (0...𝑠) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑠𝑘)))) = (𝑘 ∈ (0...𝑛) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘)))))
1413oveq2d 6538 . . . . . . 7 (𝑠 = 𝑛 → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑠𝑘))))) = (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))))))
1514adantl 480 . . . . . 6 (((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) ∧ 𝑠 = 𝑛) → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑠𝑘))))) = (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))))))
16 nnnn0 11141 . . . . . . 7 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
1716adantl 480 . . . . . 6 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
18 ovex 6550 . . . . . . 7 (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))))) ∈ V
1918a1i 11 . . . . . 6 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))))) ∈ V)
208, 15, 17, 19fvmptd 6177 . . . . 5 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → ((coe1‘(𝐹 × 𝐺))‘𝑛) = (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))))))
21 r19.26 3040 . . . . . . . . . 10 (∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 ) ↔ (∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 ∧ ∀𝑐 ∈ ℕ ((coe1𝐺)‘𝑐) = 0 ))
22 oveq2 6530 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 0 → (𝑛𝑘) = (𝑛 − 0))
23 nncn 10870 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
2423subid1d 10227 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ℕ → (𝑛 − 0) = 𝑛)
2524adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (𝑛 − 0) = 𝑛)
2622, 25sylan9eqr 2660 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → (𝑛𝑘) = 𝑛)
27 simpll 785 . . . . . . . . . . . . . . . . . . 19 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → 𝑛 ∈ ℕ)
2826, 27eqeltrd 2682 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → (𝑛𝑘) ∈ ℕ)
29 fveq2 6083 . . . . . . . . . . . . . . . . . . . 20 (𝑐 = (𝑛𝑘) → ((coe1𝐺)‘𝑐) = ((coe1𝐺)‘(𝑛𝑘)))
3029eqeq1d 2606 . . . . . . . . . . . . . . . . . . 19 (𝑐 = (𝑛𝑘) → (((coe1𝐺)‘𝑐) = 0 ↔ ((coe1𝐺)‘(𝑛𝑘)) = 0 ))
3130rspcv 3272 . . . . . . . . . . . . . . . . . 18 ((𝑛𝑘) ∈ ℕ → (∀𝑐 ∈ ℕ ((coe1𝐺)‘𝑐) = 0 → ((coe1𝐺)‘(𝑛𝑘)) = 0 ))
3228, 31syl 17 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → (∀𝑐 ∈ ℕ ((coe1𝐺)‘𝑐) = 0 → ((coe1𝐺)‘(𝑛𝑘)) = 0 ))
33 oveq2 6530 . . . . . . . . . . . . . . . . . . . . 21 (((coe1𝐺)‘(𝑛𝑘)) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = (((coe1𝐹)‘𝑘)(.r𝑅) 0 ))
34 simpll 785 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) → 𝑅 ∈ Ring)
35 simpl 471 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹𝐵𝐺𝐵) → 𝐹𝐵)
3635adantl 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → 𝐹𝐵)
37 elfznn0 12252 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0)
3837adantl 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → 𝑘 ∈ ℕ0)
3938adantr 479 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → 𝑘 ∈ ℕ0)
40 eqid 2604 . . . . . . . . . . . . . . . . . . . . . . . 24 (coe1𝐹) = (coe1𝐹)
41 eqid 2604 . . . . . . . . . . . . . . . . . . . . . . . 24 (Base‘𝑅) = (Base‘𝑅)
4240, 4, 1, 41coe1fvalcl 19344 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹𝐵𝑘 ∈ ℕ0) → ((coe1𝐹)‘𝑘) ∈ (Base‘𝑅))
4336, 39, 42syl2an 492 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) → ((coe1𝐹)‘𝑘) ∈ (Base‘𝑅))
44 cply1mul.0 . . . . . . . . . . . . . . . . . . . . . . 23 0 = (0g𝑅)
4541, 3, 44ringrz 18352 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ Ring ∧ ((coe1𝐹)‘𝑘) ∈ (Base‘𝑅)) → (((coe1𝐹)‘𝑘)(.r𝑅) 0 ) = 0 )
4634, 43, 45syl2anc 690 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) → (((coe1𝐹)‘𝑘)(.r𝑅) 0 ) = 0 )
4733, 46sylan9eqr 2660 . . . . . . . . . . . . . . . . . . . 20 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) ∧ ((coe1𝐺)‘(𝑛𝑘)) = 0 ) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )
4847ex 448 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) → (((coe1𝐺)‘(𝑛𝑘)) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))
4948expcom 449 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (((coe1𝐺)‘(𝑛𝑘)) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )))
5049com23 83 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → (((coe1𝐺)‘(𝑛𝑘)) = 0 → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )))
5132, 50syld 45 . . . . . . . . . . . . . . . 16 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → (∀𝑐 ∈ ℕ ((coe1𝐺)‘𝑐) = 0 → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )))
5251com12 32 . . . . . . . . . . . . . . 15 (∀𝑐 ∈ ℕ ((coe1𝐺)‘𝑐) = 0 → (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )))
5352expd 450 . . . . . . . . . . . . . 14 (∀𝑐 ∈ ℕ ((coe1𝐺)‘𝑐) = 0 → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (𝑘 = 0 → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
5453com24 92 . . . . . . . . . . . . 13 (∀𝑐 ∈ ℕ ((coe1𝐺)‘𝑐) = 0 → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (𝑘 = 0 → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
5554adantl 480 . . . . . . . . . . . 12 ((∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 ∧ ∀𝑐 ∈ ℕ ((coe1𝐺)‘𝑐) = 0 ) → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (𝑘 = 0 → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
5655com13 85 . . . . . . . . . . 11 (𝑘 = 0 → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → ((∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 ∧ ∀𝑐 ∈ ℕ ((coe1𝐺)‘𝑐) = 0 ) → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
57 df-ne 2776 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ≠ 0 ↔ ¬ 𝑘 = 0)
5857biimpri 216 . . . . . . . . . . . . . . . . . . . . . 22 𝑘 = 0 → 𝑘 ≠ 0)
5958, 37anim12ci 588 . . . . . . . . . . . . . . . . . . . . 21 ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → (𝑘 ∈ ℕ0𝑘 ≠ 0))
60 elnnne0 11148 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ ℕ ↔ (𝑘 ∈ ℕ0𝑘 ≠ 0))
6159, 60sylibr 222 . . . . . . . . . . . . . . . . . . . 20 ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → 𝑘 ∈ ℕ)
62 fveq2 6083 . . . . . . . . . . . . . . . . . . . . . 22 (𝑐 = 𝑘 → ((coe1𝐹)‘𝑐) = ((coe1𝐹)‘𝑘))
6362eqeq1d 2606 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = 𝑘 → (((coe1𝐹)‘𝑐) = 0 ↔ ((coe1𝐹)‘𝑘) = 0 ))
6463rspcv 3272 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ ℕ → (∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 → ((coe1𝐹)‘𝑘) = 0 ))
6561, 64syl 17 . . . . . . . . . . . . . . . . . . 19 ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → (∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 → ((coe1𝐹)‘𝑘) = 0 ))
66 oveq1 6529 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((coe1𝐹)‘𝑘) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = ( 0 (.r𝑅)((coe1𝐺)‘(𝑛𝑘))))
67 simpll 785 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ 𝑘 ∈ (0...𝑛)) → 𝑅 ∈ Ring)
684eleq2i 2674 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐺𝐵𝐺 ∈ (Base‘𝑃))
6968biimpi 204 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐺𝐵𝐺 ∈ (Base‘𝑃))
7069adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐹𝐵𝐺𝐵) → 𝐺 ∈ (Base‘𝑃))
7170adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → 𝐺 ∈ (Base‘𝑃))
72 fznn0sub 12194 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 ∈ (0...𝑛) → (𝑛𝑘) ∈ ℕ0)
73 eqid 2604 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (coe1𝐺) = (coe1𝐺)
74 eqid 2604 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (Base‘𝑃) = (Base‘𝑃)
7573, 74, 1, 41coe1fvalcl 19344 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐺 ∈ (Base‘𝑃) ∧ (𝑛𝑘) ∈ ℕ0) → ((coe1𝐺)‘(𝑛𝑘)) ∈ (Base‘𝑅))
7671, 72, 75syl2an 492 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ 𝑘 ∈ (0...𝑛)) → ((coe1𝐺)‘(𝑛𝑘)) ∈ (Base‘𝑅))
7741, 3, 44ringlz 18351 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑅 ∈ Ring ∧ ((coe1𝐺)‘(𝑛𝑘)) ∈ (Base‘𝑅)) → ( 0 (.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )
7867, 76, 77syl2anc 690 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ 𝑘 ∈ (0...𝑛)) → ( 0 (.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )
7966, 78sylan9eqr 2660 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ 𝑘 ∈ (0...𝑛)) ∧ ((coe1𝐹)‘𝑘) = 0 ) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )
8079ex 448 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))
8180ex 448 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (𝑘 ∈ (0...𝑛) → (((coe1𝐹)‘𝑘) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )))
8281com23 83 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (((coe1𝐹)‘𝑘) = 0 → (𝑘 ∈ (0...𝑛) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )))
8382a1dd 47 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (((coe1𝐹)‘𝑘) = 0 → (𝑛 ∈ ℕ → (𝑘 ∈ (0...𝑛) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
8483com14 93 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (0...𝑛) → (((coe1𝐹)‘𝑘) = 0 → (𝑛 ∈ ℕ → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
8584adantl 480 . . . . . . . . . . . . . . . . . . 19 ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘) = 0 → (𝑛 ∈ ℕ → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
8665, 85syld 45 . . . . . . . . . . . . . . . . . 18 ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → (∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 → (𝑛 ∈ ℕ → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
8786com24 92 . . . . . . . . . . . . . . . . 17 ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (𝑛 ∈ ℕ → (∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
8887ex 448 . . . . . . . . . . . . . . . 16 𝑘 = 0 → (𝑘 ∈ (0...𝑛) → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (𝑛 ∈ ℕ → (∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )))))
8988com14 93 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → (𝑘 ∈ (0...𝑛) → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (¬ 𝑘 = 0 → (∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )))))
9089imp 443 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (¬ 𝑘 = 0 → (∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
9190com14 93 . . . . . . . . . . . . 13 (∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (¬ 𝑘 = 0 → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
9291adantr 479 . . . . . . . . . . . 12 ((∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 ∧ ∀𝑐 ∈ ℕ ((coe1𝐺)‘𝑐) = 0 ) → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (¬ 𝑘 = 0 → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
9392com13 85 . . . . . . . . . . 11 𝑘 = 0 → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → ((∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 ∧ ∀𝑐 ∈ ℕ ((coe1𝐺)‘𝑐) = 0 ) → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
9456, 93pm2.61i 174 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → ((∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 ∧ ∀𝑐 ∈ ℕ ((coe1𝐺)‘𝑐) = 0 ) → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )))
9521, 94syl5bi 230 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 ) → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )))
9695imp 443 . . . . . . . 8 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))
9796impl 647 . . . . . . 7 (((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )
9897mpteq2dva 4661 . . . . . 6 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ (0...𝑛) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘)))) = (𝑘 ∈ (0...𝑛) ↦ 0 ))
9998oveq2d 6538 . . . . 5 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))))) = (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )))
100 ringmnd 18320 . . . . . . . . 9 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
101 ovex 6550 . . . . . . . . . 10 (0...𝑛) ∈ V
102101a1i 11 . . . . . . . . 9 (𝑅 ∈ Ring → (0...𝑛) ∈ V)
10344gsumz 17138 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ (0...𝑛) ∈ V) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
104100, 102, 103syl2anc 690 . . . . . . . 8 (𝑅 ∈ Ring → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
105104adantr 479 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
106105adantr 479 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
107106adantr 479 . . . . 5 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
10820, 99, 1073eqtrd 2642 . . . 4 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → ((coe1‘(𝐹 × 𝐺))‘𝑛) = 0 )
109108ralrimiva 2943 . . 3 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) → ∀𝑛 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑛) = 0 )
110 fveq2 6083 . . . . 5 (𝑐 = 𝑛 → ((coe1‘(𝐹 × 𝐺))‘𝑐) = ((coe1‘(𝐹 × 𝐺))‘𝑛))
111110eqeq1d 2606 . . . 4 (𝑐 = 𝑛 → (((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 ↔ ((coe1‘(𝐹 × 𝐺))‘𝑛) = 0 ))
112111cbvralv 3141 . . 3 (∀𝑐 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 ↔ ∀𝑛 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑛) = 0 )
113109, 112sylibr 222 . 2 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) → ∀𝑐 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 )
114113ex 448 1 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 ) → ∀𝑐 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wcel 1975  wne 2774  wral 2890  Vcvv 3167  cmpt 4632  cfv 5785  (class class class)co 6522  0cc0 9787  cmin 10112  cn 10862  0cn0 11134  ...cfz 12147  Basecbs 15636  .rcmulr 15710  0gc0g 15864   Σg cgsu 15865  Mndcmnd 17058  Ringcrg 18311  Poly1cpl1 19309  coe1cco1 19310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819  ax-inf2 8393  ax-cnex 9843  ax-resscn 9844  ax-1cn 9845  ax-icn 9846  ax-addcl 9847  ax-addrcl 9848  ax-mulcl 9849  ax-mulrcl 9850  ax-mulcom 9851  ax-addass 9852  ax-mulass 9853  ax-distr 9854  ax-i2m1 9855  ax-1ne0 9856  ax-1rid 9857  ax-rnegex 9858  ax-rrecex 9859  ax-cnre 9860  ax-pre-lttri 9861  ax-pre-lttrn 9862  ax-pre-ltadd 9863  ax-pre-mulgt0 9864
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-nel 2777  df-ral 2895  df-rex 2896  df-reu 2897  df-rmo 2898  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-pss 3550  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-tp 4124  df-op 4126  df-uni 4362  df-int 4400  df-iun 4446  df-iin 4447  df-br 4573  df-opab 4633  df-mpt 4634  df-tr 4670  df-eprel 4934  df-id 4938  df-po 4944  df-so 4945  df-fr 4982  df-se 4983  df-we 4984  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-pred 5578  df-ord 5624  df-on 5625  df-lim 5626  df-suc 5627  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-isom 5794  df-riota 6484  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-of 6767  df-ofr 6768  df-om 6930  df-1st 7031  df-2nd 7032  df-supp 7155  df-wrecs 7266  df-recs 7327  df-rdg 7365  df-1o 7419  df-2o 7420  df-oadd 7423  df-er 7601  df-map 7718  df-pm 7719  df-ixp 7767  df-en 7814  df-dom 7815  df-sdom 7816  df-fin 7817  df-fsupp 8131  df-oi 8270  df-card 8620  df-pnf 9927  df-mnf 9928  df-xr 9929  df-ltxr 9930  df-le 9931  df-sub 10114  df-neg 10115  df-nn 10863  df-2 10921  df-3 10922  df-4 10923  df-5 10924  df-6 10925  df-7 10926  df-8 10927  df-9 10928  df-n0 11135  df-z 11206  df-dec 11321  df-uz 11515  df-fz 12148  df-fzo 12285  df-seq 12614  df-hash 12930  df-struct 15638  df-ndx 15639  df-slot 15640  df-base 15641  df-sets 15642  df-ress 15643  df-plusg 15722  df-mulr 15723  df-sca 15725  df-vsca 15726  df-tset 15728  df-ple 15729  df-0g 15866  df-gsum 15867  df-mre 16010  df-mrc 16011  df-acs 16013  df-mgm 17006  df-sgrp 17048  df-mnd 17059  df-mhm 17099  df-submnd 17100  df-grp 17189  df-minusg 17190  df-mulg 17305  df-ghm 17422  df-cntz 17514  df-cmn 17959  df-abl 17960  df-mgp 18254  df-ur 18266  df-ring 18313  df-psr 19118  df-mpl 19120  df-opsr 19122  df-psr1 19312  df-ply1 19314  df-coe1 19315
This theorem is referenced by:  cpmatmcllem  20279
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