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Theorem cply1mul 20382
 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 2825 . . . . . . . . . 10 (.r𝑅) = (.r𝑅)
4 cply1mul.b . . . . . . . . . 10 𝐵 = (Base‘𝑃)
51, 2, 3, 4coe1mul 20358 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (coe1‘(𝐹 × 𝐺)) = (𝑠 ∈ ℕ0 ↦ (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑠𝑘)))))))
653expb 1114 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (coe1‘(𝐹 × 𝐺)) = (𝑠 ∈ ℕ0 ↦ (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑠𝑘)))))))
76adantr 481 . . . . . . 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 481 . . . . . 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 7159 . . . . . . . . 9 (𝑠 = 𝑛 → (0...𝑠) = (0...𝑛))
10 fvoveq1 7174 . . . . . . . . . 10 (𝑠 = 𝑛 → ((coe1𝐺)‘(𝑠𝑘)) = ((coe1𝐺)‘(𝑛𝑘)))
1110oveq2d 7167 . . . . . . . . 9 (𝑠 = 𝑛 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑠𝑘))) = (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))))
129, 11mpteq12dv 5147 . . . . . . . 8 (𝑠 = 𝑛 → (𝑘 ∈ (0...𝑠) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑠𝑘)))) = (𝑘 ∈ (0...𝑛) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘)))))
1312oveq2d 7167 . . . . . . 7 (𝑠 = 𝑛 → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑠𝑘))))) = (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))))))
1413adantl 482 . . . . . 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 11896 . . . . . . 7 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
1615adantl 482 . . . . . 6 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
17 ovexd 7186 . . . . . 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 6770 . . . . 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 3174 . . . . . . . . . 10 (∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 ) ↔ (∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 ∧ ∀𝑐 ∈ ℕ ((coe1𝐺)‘𝑐) = 0 ))
20 oveq2 7159 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (𝑛𝑘) = (𝑛 − 0))
21 nncn 11638 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
2221subid1d 10978 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ → (𝑛 − 0) = 𝑛)
2322adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (𝑛 − 0) = 𝑛)
2420, 23sylan9eqr 2882 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → (𝑛𝑘) = 𝑛)
25 simpll 763 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → 𝑛 ∈ ℕ)
2624, 25eqeltrd 2917 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → (𝑛𝑘) ∈ ℕ)
27 fveqeq2 6675 . . . . . . . . . . . . . . . . . 18 (𝑐 = (𝑛𝑘) → (((coe1𝐺)‘𝑐) = 0 ↔ ((coe1𝐺)‘(𝑛𝑘)) = 0 ))
2827rspcv 3621 . . . . . . . . . . . . . . . . 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 7159 . . . . . . . . . . . . . . . . . . . 20 (((coe1𝐺)‘(𝑛𝑘)) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = (((coe1𝐹)‘𝑘)(.r𝑅) 0 ))
31 simpll 763 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) → 𝑅 ∈ Ring)
32 simprl 767 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → 𝐹𝐵)
33 elfznn0 12993 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0)
3433adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → 𝑘 ∈ ℕ0)
3534adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → 𝑘 ∈ ℕ0)
36 eqid 2825 . . . . . . . . . . . . . . . . . . . . . . 23 (coe1𝐹) = (coe1𝐹)
37 eqid 2825 . . . . . . . . . . . . . . . . . . . . . . 23 (Base‘𝑅) = (Base‘𝑅)
3836, 4, 1, 37coe1fvalcl 20300 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹𝐵𝑘 ∈ ℕ0) → ((coe1𝐹)‘𝑘) ∈ (Base‘𝑅))
3932, 35, 38syl2an 595 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) → ((coe1𝐹)‘𝑘) ∈ (Base‘𝑅))
40 cply1mul.0 . . . . . . . . . . . . . . . . . . . . . 22 0 = (0g𝑅)
4137, 3, 40ringrz 19261 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Ring ∧ ((coe1𝐹)‘𝑘) ∈ (Base‘𝑅)) → (((coe1𝐹)‘𝑘)(.r𝑅) 0 ) = 0 )
4231, 39, 41syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) → (((coe1𝐹)‘𝑘)(.r𝑅) 0 ) = 0 )
4330, 42sylan9eqr 2882 . . . . . . . . . . . . . . . . . . 19 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) ∧ ((coe1𝐺)‘(𝑛𝑘)) = 0 ) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )
4443ex 413 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) → (((coe1𝐺)‘(𝑛𝑘)) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))
4544expcom 414 . . . . . . . . . . . . . . . . 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 416 . . . . . . . . . . . . . 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 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 ))))
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 3028 . . . . . . . . . . . . . . . . . . . . . 22 𝑘 = 0 → 𝑘 ≠ 0)
5352, 33anim12ci 613 . . . . . . . . . . . . . . . . . . . . 21 ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → (𝑘 ∈ ℕ0𝑘 ≠ 0))
54 elnnne0 11903 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ ℕ ↔ (𝑘 ∈ ℕ0𝑘 ≠ 0))
5553, 54sylibr 235 . . . . . . . . . . . . . . . . . . . 20 ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → 𝑘 ∈ ℕ)
56 fveqeq2 6675 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = 𝑘 → (((coe1𝐹)‘𝑐) = 0 ↔ ((coe1𝐹)‘𝑘) = 0 ))
5756rspcv 3621 . . . . . . . . . . . . . . . . . . . 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 7158 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((coe1𝐹)‘𝑘) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = ( 0 (.r𝑅)((coe1𝐺)‘(𝑛𝑘))))
60 simpll 763 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ 𝑘 ∈ (0...𝑛)) → 𝑅 ∈ Ring)
614eleq2i 2908 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐺𝐵𝐺 ∈ (Base‘𝑃))
6261biimpi 217 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐺𝐵𝐺 ∈ (Base‘𝑃))
6362adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐹𝐵𝐺𝐵) → 𝐺 ∈ (Base‘𝑃))
6463adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → 𝐺 ∈ (Base‘𝑃))
65 fznn0sub 12932 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 ∈ (0...𝑛) → (𝑛𝑘) ∈ ℕ0)
66 eqid 2825 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (coe1𝐺) = (coe1𝐺)
67 eqid 2825 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (Base‘𝑃) = (Base‘𝑃)
6866, 67, 1, 37coe1fvalcl 20300 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐺 ∈ (Base‘𝑃) ∧ (𝑛𝑘) ∈ ℕ0) → ((coe1𝐺)‘(𝑛𝑘)) ∈ (Base‘𝑅))
6964, 65, 68syl2an 595 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ 𝑘 ∈ (0...𝑛)) → ((coe1𝐺)‘(𝑛𝑘)) ∈ (Base‘𝑅))
7037, 3, 40ringlz 19260 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑅 ∈ Ring ∧ ((coe1𝐺)‘(𝑛𝑘)) ∈ (Base‘𝑅)) → ( 0 (.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )
7160, 69, 70syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ 𝑘 ∈ (0...𝑛)) → ( 0 (.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )
7259, 71sylan9eqr 2882 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ 𝑘 ∈ (0...𝑛)) ∧ ((coe1𝐹)‘𝑘) = 0 ) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )
7372ex 413 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))
7473ex 413 . . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . 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 183 . . . . . . . . . 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, 87syl5bi 243 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 ) → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )))
8988imp 407 . . . . . . . 8 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))
9089impl 456 . . . . . . 7 (((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )
9190mpteq2dva 5157 . . . . . 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 7167 . . . . 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 19229 . . . . . . . . 9 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
94 ovexd 7186 . . . . . . . . 9 (𝑅 ∈ Ring → (0...𝑛) ∈ V)
9540gsumz 17986 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ (0...𝑛) ∈ V) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
9693, 94, 95syl2anc 584 . . . . . . . 8 (𝑅 ∈ Ring → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
9796adantr 481 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
9897adantr 481 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
9998adantr 481 . . . . 5 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
10018, 92, 993eqtrd 2864 . . . 4 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → ((coe1‘(𝐹 × 𝐺))‘𝑛) = 0 )
101100ralrimiva 3186 . . 3 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) → ∀𝑛 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑛) = 0 )
102 fveqeq2 6675 . . . 4 (𝑐 = 𝑛 → (((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 ↔ ((coe1‘(𝐹 × 𝐺))‘𝑛) = 0 ))
103102cbvralvw 3454 . . 3 (∀𝑐 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 ↔ ∀𝑛 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑛) = 0 )
104101, 103sylibr 235 . 2 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) → ∀𝑐 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 )
105104ex 413 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 396   = wceq 1530   ∈ wcel 2107   ≠ wne 3020  ∀wral 3142  Vcvv 3499   ↦ cmpt 5142  ‘cfv 6351  (class class class)co 7151  0cc0 10529   − cmin 10862  ℕcn 11630  ℕ0cn0 11889  ...cfz 12885  Basecbs 16476  .rcmulr 16559  0gc0g 16706   Σg cgsu 16707  Mndcmnd 17903  Ringcrg 19220  Poly1cpl1 20265  coe1cco1 20266 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-iin 4919  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-of 7402  df-ofr 7403  df-om 7572  df-1st 7683  df-2nd 7684  df-supp 7825  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8282  df-map 8401  df-pm 8402  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fsupp 8826  df-oi 8966  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12886  df-fzo 13027  df-seq 13363  df-hash 13684  df-struct 16478  df-ndx 16479  df-slot 16480  df-base 16482  df-sets 16483  df-ress 16484  df-plusg 16571  df-mulr 16572  df-sca 16574  df-vsca 16575  df-tset 16577  df-ple 16578  df-0g 16708  df-gsum 16709  df-mre 16850  df-mrc 16851  df-acs 16853  df-mgm 17845  df-sgrp 17893  df-mnd 17904  df-mhm 17947  df-submnd 17948  df-grp 18039  df-minusg 18040  df-mulg 18158  df-ghm 18289  df-cntz 18380  df-cmn 18831  df-abl 18832  df-mgp 19163  df-ur 19175  df-ring 19222  df-psr 20058  df-mpl 20060  df-opsr 20062  df-psr1 20268  df-ply1 20270  df-coe1 20271 This theorem is referenced by:  cpmatmcllem  21245
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