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Theorem cply1mul 21681
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 2733 . . . . . . . . . 10 (.r𝑅) = (.r𝑅)
4 cply1mul.b . . . . . . . . . 10 𝐵 = (Base‘𝑃)
51, 2, 3, 4coe1mul 21657 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (coe1‘(𝐹 × 𝐺)) = (𝑠 ∈ ℕ0 ↦ (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑠𝑘)))))))
653expb 1121 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (coe1‘(𝐹 × 𝐺)) = (𝑠 ∈ ℕ0 ↦ (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑠𝑘)))))))
76adantr 482 . . . . . . 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 482 . . . . . 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 7366 . . . . . . . . 9 (𝑠 = 𝑛 → (0...𝑠) = (0...𝑛))
10 fvoveq1 7381 . . . . . . . . . 10 (𝑠 = 𝑛 → ((coe1𝐺)‘(𝑠𝑘)) = ((coe1𝐺)‘(𝑛𝑘)))
1110oveq2d 7374 . . . . . . . . 9 (𝑠 = 𝑛 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑠𝑘))) = (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))))
129, 11mpteq12dv 5197 . . . . . . . 8 (𝑠 = 𝑛 → (𝑘 ∈ (0...𝑠) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑠𝑘)))) = (𝑘 ∈ (0...𝑛) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘)))))
1312oveq2d 7374 . . . . . . 7 (𝑠 = 𝑛 → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑠𝑘))))) = (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))))))
1413adantl 483 . . . . . 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))))))
15 nnnn0 12425 . . . . . . 7 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
1615adantl 483 . . . . . 6 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
17 ovexd 7393 . . . . . 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)
188, 14, 16, 17fvmptd 6956 . . . . 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))))))
19 r19.26 3111 . . . . . . . . . 10 (∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 ) ↔ (∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 ∧ ∀𝑐 ∈ ℕ ((coe1𝐺)‘𝑐) = 0 ))
20 oveq2 7366 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (𝑛𝑘) = (𝑛 − 0))
21 nncn 12166 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
2221subid1d 11506 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ → (𝑛 − 0) = 𝑛)
2322adantr 482 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (𝑛 − 0) = 𝑛)
2420, 23sylan9eqr 2795 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → (𝑛𝑘) = 𝑛)
25 simpll 766 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → 𝑛 ∈ ℕ)
2624, 25eqeltrd 2834 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → (𝑛𝑘) ∈ ℕ)
27 fveqeq2 6852 . . . . . . . . . . . . . . . . . 18 (𝑐 = (𝑛𝑘) → (((coe1𝐺)‘𝑐) = 0 ↔ ((coe1𝐺)‘(𝑛𝑘)) = 0 ))
2827rspcv 3576 . . . . . . . . . . . . . . . . 17 ((𝑛𝑘) ∈ ℕ → (∀𝑐 ∈ ℕ ((coe1𝐺)‘𝑐) = 0 → ((coe1𝐺)‘(𝑛𝑘)) = 0 ))
2926, 28syl 17 . . . . . . . . . . . . . . . 16 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → (∀𝑐 ∈ ℕ ((coe1𝐺)‘𝑐) = 0 → ((coe1𝐺)‘(𝑛𝑘)) = 0 ))
30 oveq2 7366 . . . . . . . . . . . . . . . . . . . 20 (((coe1𝐺)‘(𝑛𝑘)) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = (((coe1𝐹)‘𝑘)(.r𝑅) 0 ))
31 simpll 766 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) → 𝑅 ∈ Ring)
32 simprl 770 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → 𝐹𝐵)
33 elfznn0 13540 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0)
3433adantl 483 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → 𝑘 ∈ ℕ0)
3534adantr 482 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → 𝑘 ∈ ℕ0)
36 eqid 2733 . . . . . . . . . . . . . . . . . . . . . . 23 (coe1𝐹) = (coe1𝐹)
37 eqid 2733 . . . . . . . . . . . . . . . . . . . . . . 23 (Base‘𝑅) = (Base‘𝑅)
3836, 4, 1, 37coe1fvalcl 21599 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹𝐵𝑘 ∈ ℕ0) → ((coe1𝐹)‘𝑘) ∈ (Base‘𝑅))
3932, 35, 38syl2an 597 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) → ((coe1𝐹)‘𝑘) ∈ (Base‘𝑅))
40 cply1mul.0 . . . . . . . . . . . . . . . . . . . . . 22 0 = (0g𝑅)
4137, 3, 40ringrz 20017 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Ring ∧ ((coe1𝐹)‘𝑘) ∈ (Base‘𝑅)) → (((coe1𝐹)‘𝑘)(.r𝑅) 0 ) = 0 )
4231, 39, 41syl2anc 585 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) → (((coe1𝐹)‘𝑘)(.r𝑅) 0 ) = 0 )
4330, 42sylan9eqr 2795 . . . . . . . . . . . . . . . . . . 19 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) ∧ ((coe1𝐺)‘(𝑛𝑘)) = 0 ) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )
4443ex 414 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) → (((coe1𝐺)‘(𝑛𝑘)) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))
4544expcom 415 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (((coe1𝐺)‘(𝑛𝑘)) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )))
4645com23 86 . . . . . . . . . . . . . . . 16 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → (((coe1𝐺)‘(𝑛𝑘)) = 0 → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )))
4729, 46syldc 48 . . . . . . . . . . . . . . 15 (∀𝑐 ∈ ℕ ((coe1𝐺)‘𝑐) = 0 → (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )))
4847expd 417 . . . . . . . . . . . . . 14 (∀𝑐 ∈ ℕ ((coe1𝐺)‘𝑐) = 0 → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (𝑘 = 0 → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
4948com24 95 . . . . . . . . . . . . 13 (∀𝑐 ∈ ℕ ((coe1𝐺)‘𝑐) = 0 → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (𝑘 = 0 → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
5049adantl 483 . . . . . . . . . . . 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 ))))
5150com13 88 . . . . . . . . . . 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 ))))
52 neqne 2948 . . . . . . . . . . . . . . . . . . . . . 22 𝑘 = 0 → 𝑘 ≠ 0)
5352, 33anim12ci 615 . . . . . . . . . . . . . . . . . . . . 21 ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → (𝑘 ∈ ℕ0𝑘 ≠ 0))
54 elnnne0 12432 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ ℕ ↔ (𝑘 ∈ ℕ0𝑘 ≠ 0))
5553, 54sylibr 233 . . . . . . . . . . . . . . . . . . . 20 ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → 𝑘 ∈ ℕ)
56 fveqeq2 6852 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = 𝑘 → (((coe1𝐹)‘𝑐) = 0 ↔ ((coe1𝐹)‘𝑘) = 0 ))
5756rspcv 3576 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ ℕ → (∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 → ((coe1𝐹)‘𝑘) = 0 ))
5855, 57syl 17 . . . . . . . . . . . . . . . . . . 19 ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → (∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 → ((coe1𝐹)‘𝑘) = 0 ))
59 oveq1 7365 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((coe1𝐹)‘𝑘) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = ( 0 (.r𝑅)((coe1𝐺)‘(𝑛𝑘))))
60 simpll 766 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ 𝑘 ∈ (0...𝑛)) → 𝑅 ∈ Ring)
614eleq2i 2826 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐺𝐵𝐺 ∈ (Base‘𝑃))
6261biimpi 215 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐺𝐵𝐺 ∈ (Base‘𝑃))
6362adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐹𝐵𝐺𝐵) → 𝐺 ∈ (Base‘𝑃))
6463adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → 𝐺 ∈ (Base‘𝑃))
65 fznn0sub 13479 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 ∈ (0...𝑛) → (𝑛𝑘) ∈ ℕ0)
66 eqid 2733 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (coe1𝐺) = (coe1𝐺)
67 eqid 2733 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (Base‘𝑃) = (Base‘𝑃)
6866, 67, 1, 37coe1fvalcl 21599 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐺 ∈ (Base‘𝑃) ∧ (𝑛𝑘) ∈ ℕ0) → ((coe1𝐺)‘(𝑛𝑘)) ∈ (Base‘𝑅))
6964, 65, 68syl2an 597 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ 𝑘 ∈ (0...𝑛)) → ((coe1𝐺)‘(𝑛𝑘)) ∈ (Base‘𝑅))
7037, 3, 40ringlz 20016 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑅 ∈ Ring ∧ ((coe1𝐺)‘(𝑛𝑘)) ∈ (Base‘𝑅)) → ( 0 (.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )
7160, 69, 70syl2anc 585 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ 𝑘 ∈ (0...𝑛)) → ( 0 (.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )
7259, 71sylan9eqr 2795 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ 𝑘 ∈ (0...𝑛)) ∧ ((coe1𝐹)‘𝑘) = 0 ) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )
7372ex 414 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))
7473ex 414 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (𝑘 ∈ (0...𝑛) → (((coe1𝐹)‘𝑘) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )))
7574com23 86 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (((coe1𝐹)‘𝑘) = 0 → (𝑘 ∈ (0...𝑛) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )))
7675a1dd 50 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (((coe1𝐹)‘𝑘) = 0 → (𝑛 ∈ ℕ → (𝑘 ∈ (0...𝑛) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
7776com14 96 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (0...𝑛) → (((coe1𝐹)‘𝑘) = 0 → (𝑛 ∈ ℕ → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
7877adantl 483 . . . . . . . . . . . . . . . . . . 19 ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘) = 0 → (𝑛 ∈ ℕ → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
7958, 78syld 47 . . . . . . . . . . . . . . . . . 18 ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → (∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 → (𝑛 ∈ ℕ → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
8079com24 95 . . . . . . . . . . . . . . . . 17 ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (𝑛 ∈ ℕ → (∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
8180ex 414 . . . . . . . . . . . . . . . 16 𝑘 = 0 → (𝑘 ∈ (0...𝑛) → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (𝑛 ∈ ℕ → (∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )))))
8281com14 96 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → (𝑘 ∈ (0...𝑛) → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (¬ 𝑘 = 0 → (∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )))))
8382imp 408 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (¬ 𝑘 = 0 → (∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
8483com14 96 . . . . . . . . . . . . 13 (∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (¬ 𝑘 = 0 → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
8584adantr 482 . . . . . . . . . . . 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 ))))
8685com13 88 . . . . . . . . . . 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 ))))
8751, 86pm2.61i 182 . . . . . . . . . 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 )))
8819, 87biimtrid 241 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 ) → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )))
8988imp 408 . . . . . . . 8 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))
9089impl 457 . . . . . . 7 (((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )
9190mpteq2dva 5206 . . . . . 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 ))
9291oveq2d 7374 . . . . 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 )))
93 ringmnd 19979 . . . . . . . . 9 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
94 ovexd 7393 . . . . . . . . 9 (𝑅 ∈ Ring → (0...𝑛) ∈ V)
9540gsumz 18651 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ (0...𝑛) ∈ V) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
9693, 94, 95syl2anc 585 . . . . . . . 8 (𝑅 ∈ Ring → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
9796adantr 482 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
9897adantr 482 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
9998adantr 482 . . . . 5 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
10018, 92, 993eqtrd 2777 . . . 4 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → ((coe1‘(𝐹 × 𝐺))‘𝑛) = 0 )
101100ralrimiva 3140 . . 3 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) → ∀𝑛 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑛) = 0 )
102 fveqeq2 6852 . . . 4 (𝑐 = 𝑛 → (((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 ↔ ((coe1‘(𝐹 × 𝐺))‘𝑛) = 0 ))
103102cbvralvw 3224 . . 3 (∀𝑐 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 ↔ ∀𝑛 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑛) = 0 )
104101, 103sylibr 233 . 2 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) → ∀𝑐 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 )
105104ex 414 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 397   = wceq 1542  wcel 2107  wne 2940  wral 3061  Vcvv 3444  cmpt 5189  cfv 6497  (class class class)co 7358  0cc0 11056  cmin 11390  cn 12158  0cn0 12418  ...cfz 13430  Basecbs 17088  .rcmulr 17139  0gc0g 17326   Σg cgsu 17327  Mndcmnd 18561  Ringcrg 19969  Poly1cpl1 21564  coe1cco1 21565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7618  df-ofr 7619  df-om 7804  df-1st 7922  df-2nd 7923  df-supp 8094  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-map 8770  df-pm 8771  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-fsupp 9309  df-sup 9383  df-oi 9451  df-card 9880  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-z 12505  df-dec 12624  df-uz 12769  df-fz 13431  df-fzo 13574  df-seq 13913  df-hash 14237  df-struct 17024  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-ress 17118  df-plusg 17151  df-mulr 17152  df-sca 17154  df-vsca 17155  df-ip 17156  df-tset 17157  df-ple 17158  df-ds 17160  df-hom 17162  df-cco 17163  df-0g 17328  df-gsum 17329  df-prds 17334  df-pws 17336  df-mre 17471  df-mrc 17472  df-acs 17474  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-mhm 18606  df-submnd 18607  df-grp 18756  df-minusg 18757  df-mulg 18878  df-ghm 19011  df-cntz 19102  df-cmn 19569  df-abl 19570  df-mgp 19902  df-ur 19919  df-ring 19971  df-psr 21327  df-mpl 21329  df-opsr 21331  df-psr1 21567  df-ply1 21569  df-coe1 21570
This theorem is referenced by:  cpmatmcllem  22083
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