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Theorem cply1mul 19712
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 2651 . . . . . . . . . 10 (.r𝑅) = (.r𝑅)
4 cply1mul.b . . . . . . . . . 10 𝐵 = (Base‘𝑃)
51, 2, 3, 4coe1mul 19688 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (coe1‘(𝐹 × 𝐺)) = (𝑠 ∈ ℕ0 ↦ (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑠𝑘)))))))
653expb 1285 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (coe1‘(𝐹 × 𝐺)) = (𝑠 ∈ ℕ0 ↦ (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑠𝑘)))))))
76adantr 480 . . . . . . 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 480 . . . . . 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 6698 . . . . . . . . 9 (𝑠 = 𝑛 → (0...𝑠) = (0...𝑛))
10 oveq1 6697 . . . . . . . . . . 11 (𝑠 = 𝑛 → (𝑠𝑘) = (𝑛𝑘))
1110fveq2d 6233 . . . . . . . . . 10 (𝑠 = 𝑛 → ((coe1𝐺)‘(𝑠𝑘)) = ((coe1𝐺)‘(𝑛𝑘)))
1211oveq2d 6706 . . . . . . . . 9 (𝑠 = 𝑛 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑠𝑘))) = (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))))
139, 12mpteq12dv 4766 . . . . . . . 8 (𝑠 = 𝑛 → (𝑘 ∈ (0...𝑠) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑠𝑘)))) = (𝑘 ∈ (0...𝑛) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘)))))
1413oveq2d 6706 . . . . . . 7 (𝑠 = 𝑛 → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑠𝑘))))) = (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))))))
1514adantl 481 . . . . . 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 11337 . . . . . . 7 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
1716adantl 481 . . . . . 6 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
18 ovexd 6720 . . . . . 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)
198, 15, 17, 18fvmptd 6327 . . . . 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))))))
20 r19.26 3093 . . . . . . . . . 10 (∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 ) ↔ (∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 ∧ ∀𝑐 ∈ ℕ ((coe1𝐺)‘𝑐) = 0 ))
21 oveq2 6698 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (𝑛𝑘) = (𝑛 − 0))
22 nncn 11066 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
2322subid1d 10419 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ → (𝑛 − 0) = 𝑛)
2423adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (𝑛 − 0) = 𝑛)
2521, 24sylan9eqr 2707 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → (𝑛𝑘) = 𝑛)
26 simpll 805 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → 𝑛 ∈ ℕ)
2725, 26eqeltrd 2730 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → (𝑛𝑘) ∈ ℕ)
28 fveq2 6229 . . . . . . . . . . . . . . . . . . 19 (𝑐 = (𝑛𝑘) → ((coe1𝐺)‘𝑐) = ((coe1𝐺)‘(𝑛𝑘)))
2928eqeq1d 2653 . . . . . . . . . . . . . . . . . 18 (𝑐 = (𝑛𝑘) → (((coe1𝐺)‘𝑐) = 0 ↔ ((coe1𝐺)‘(𝑛𝑘)) = 0 ))
3029rspcv 3336 . . . . . . . . . . . . . . . . 17 ((𝑛𝑘) ∈ ℕ → (∀𝑐 ∈ ℕ ((coe1𝐺)‘𝑐) = 0 → ((coe1𝐺)‘(𝑛𝑘)) = 0 ))
3127, 30syl 17 . . . . . . . . . . . . . . . 16 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → (∀𝑐 ∈ ℕ ((coe1𝐺)‘𝑐) = 0 → ((coe1𝐺)‘(𝑛𝑘)) = 0 ))
32 oveq2 6698 . . . . . . . . . . . . . . . . . . . 20 (((coe1𝐺)‘(𝑛𝑘)) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = (((coe1𝐹)‘𝑘)(.r𝑅) 0 ))
33 simpll 805 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) → 𝑅 ∈ Ring)
34 simpl 472 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹𝐵𝐺𝐵) → 𝐹𝐵)
3534adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → 𝐹𝐵)
36 elfznn0 12471 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0)
3736adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → 𝑘 ∈ ℕ0)
3837adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → 𝑘 ∈ ℕ0)
39 eqid 2651 . . . . . . . . . . . . . . . . . . . . . . 23 (coe1𝐹) = (coe1𝐹)
40 eqid 2651 . . . . . . . . . . . . . . . . . . . . . . 23 (Base‘𝑅) = (Base‘𝑅)
4139, 4, 1, 40coe1fvalcl 19630 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹𝐵𝑘 ∈ ℕ0) → ((coe1𝐹)‘𝑘) ∈ (Base‘𝑅))
4235, 38, 41syl2an 493 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) → ((coe1𝐹)‘𝑘) ∈ (Base‘𝑅))
43 cply1mul.0 . . . . . . . . . . . . . . . . . . . . . 22 0 = (0g𝑅)
4440, 3, 43ringrz 18634 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Ring ∧ ((coe1𝐹)‘𝑘) ∈ (Base‘𝑅)) → (((coe1𝐹)‘𝑘)(.r𝑅) 0 ) = 0 )
4533, 42, 44syl2anc 694 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) → (((coe1𝐹)‘𝑘)(.r𝑅) 0 ) = 0 )
4632, 45sylan9eqr 2707 . . . . . . . . . . . . . . . . . . 19 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) ∧ ((coe1𝐺)‘(𝑛𝑘)) = 0 ) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )
4746ex 449 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) → (((coe1𝐺)‘(𝑛𝑘)) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))
4847expcom 450 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (((coe1𝐺)‘(𝑛𝑘)) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )))
4948com23 86 . . . . . . . . . . . . . . . 16 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → (((coe1𝐺)‘(𝑛𝑘)) = 0 → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )))
5031, 49syldc 48 . . . . . . . . . . . . . . 15 (∀𝑐 ∈ ℕ ((coe1𝐺)‘𝑐) = 0 → (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )))
5150expd 451 . . . . . . . . . . . . . 14 (∀𝑐 ∈ ℕ ((coe1𝐺)‘𝑐) = 0 → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (𝑘 = 0 → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
5251com24 95 . . . . . . . . . . . . 13 (∀𝑐 ∈ ℕ ((coe1𝐺)‘𝑐) = 0 → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (𝑘 = 0 → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
5352adantl 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 ))))
5453com13 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 ))))
55 df-ne 2824 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ≠ 0 ↔ ¬ 𝑘 = 0)
5655biimpri 218 . . . . . . . . . . . . . . . . . . . . . 22 𝑘 = 0 → 𝑘 ≠ 0)
5756, 36anim12ci 590 . . . . . . . . . . . . . . . . . . . . 21 ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → (𝑘 ∈ ℕ0𝑘 ≠ 0))
58 elnnne0 11344 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ ℕ ↔ (𝑘 ∈ ℕ0𝑘 ≠ 0))
5957, 58sylibr 224 . . . . . . . . . . . . . . . . . . . 20 ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → 𝑘 ∈ ℕ)
60 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . 22 (𝑐 = 𝑘 → ((coe1𝐹)‘𝑐) = ((coe1𝐹)‘𝑘))
6160eqeq1d 2653 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = 𝑘 → (((coe1𝐹)‘𝑐) = 0 ↔ ((coe1𝐹)‘𝑘) = 0 ))
6261rspcv 3336 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ ℕ → (∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 → ((coe1𝐹)‘𝑘) = 0 ))
6359, 62syl 17 . . . . . . . . . . . . . . . . . . 19 ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → (∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 → ((coe1𝐹)‘𝑘) = 0 ))
64 oveq1 6697 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((coe1𝐹)‘𝑘) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = ( 0 (.r𝑅)((coe1𝐺)‘(𝑛𝑘))))
65 simpll 805 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ 𝑘 ∈ (0...𝑛)) → 𝑅 ∈ Ring)
664eleq2i 2722 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐺𝐵𝐺 ∈ (Base‘𝑃))
6766biimpi 206 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐺𝐵𝐺 ∈ (Base‘𝑃))
6867adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐹𝐵𝐺𝐵) → 𝐺 ∈ (Base‘𝑃))
6968adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → 𝐺 ∈ (Base‘𝑃))
70 fznn0sub 12411 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 ∈ (0...𝑛) → (𝑛𝑘) ∈ ℕ0)
71 eqid 2651 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (coe1𝐺) = (coe1𝐺)
72 eqid 2651 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (Base‘𝑃) = (Base‘𝑃)
7371, 72, 1, 40coe1fvalcl 19630 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐺 ∈ (Base‘𝑃) ∧ (𝑛𝑘) ∈ ℕ0) → ((coe1𝐺)‘(𝑛𝑘)) ∈ (Base‘𝑅))
7469, 70, 73syl2an 493 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ 𝑘 ∈ (0...𝑛)) → ((coe1𝐺)‘(𝑛𝑘)) ∈ (Base‘𝑅))
7540, 3, 43ringlz 18633 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑅 ∈ Ring ∧ ((coe1𝐺)‘(𝑛𝑘)) ∈ (Base‘𝑅)) → ( 0 (.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )
7665, 74, 75syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ 𝑘 ∈ (0...𝑛)) → ( 0 (.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )
7764, 76sylan9eqr 2707 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ 𝑘 ∈ (0...𝑛)) ∧ ((coe1𝐹)‘𝑘) = 0 ) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )
7877ex 449 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))
7978ex 449 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (𝑘 ∈ (0...𝑛) → (((coe1𝐹)‘𝑘) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )))
8079com23 86 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (((coe1𝐹)‘𝑘) = 0 → (𝑘 ∈ (0...𝑛) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )))
8180a1dd 50 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (((coe1𝐹)‘𝑘) = 0 → (𝑛 ∈ ℕ → (𝑘 ∈ (0...𝑛) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
8281com14 96 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (0...𝑛) → (((coe1𝐹)‘𝑘) = 0 → (𝑛 ∈ ℕ → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
8382adantl 481 . . . . . . . . . . . . . . . . . . 19 ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘) = 0 → (𝑛 ∈ ℕ → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
8463, 83syld 47 . . . . . . . . . . . . . . . . . 18 ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → (∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 → (𝑛 ∈ ℕ → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
8584com24 95 . . . . . . . . . . . . . . . . 17 ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (𝑛 ∈ ℕ → (∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
8685ex 449 . . . . . . . . . . . . . . . 16 𝑘 = 0 → (𝑘 ∈ (0...𝑛) → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (𝑛 ∈ ℕ → (∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )))))
8786com14 96 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → (𝑘 ∈ (0...𝑛) → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (¬ 𝑘 = 0 → (∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )))))
8887imp 444 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (¬ 𝑘 = 0 → (∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
8988com14 96 . . . . . . . . . . . . 13 (∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 → ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (¬ 𝑘 = 0 → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))))
9089adantr 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 ))))
9190com13 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 ))))
9254, 91pm2.61i 176 . . . . . . . . . 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 )))
9320, 92syl5bi 232 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 ) → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )))
9493imp 444 . . . . . . . 8 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))
9594impl 649 . . . . . . 7 (((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )
9695mpteq2dva 4777 . . . . . 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 ))
9796oveq2d 6706 . . . . 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 )))
98 ringmnd 18602 . . . . . . . . 9 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
99 ovexd 6720 . . . . . . . . 9 (𝑅 ∈ Ring → (0...𝑛) ∈ V)
10043gsumz 17421 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ (0...𝑛) ∈ V) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
10198, 99, 100syl2anc 694 . . . . . . . 8 (𝑅 ∈ Ring → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
102101adantr 480 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
103102adantr 480 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
104103adantr 480 . . . . 5 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
10519, 97, 1043eqtrd 2689 . . . 4 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → ((coe1‘(𝐹 × 𝐺))‘𝑛) = 0 )
106105ralrimiva 2995 . . 3 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) → ∀𝑛 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑛) = 0 )
107 fveq2 6229 . . . . 5 (𝑐 = 𝑛 → ((coe1‘(𝐹 × 𝐺))‘𝑐) = ((coe1‘(𝐹 × 𝐺))‘𝑛))
108107eqeq1d 2653 . . . 4 (𝑐 = 𝑛 → (((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 ↔ ((coe1‘(𝐹 × 𝐺))‘𝑛) = 0 ))
109108cbvralv 3201 . . 3 (∀𝑐 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 ↔ ∀𝑛 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑛) = 0 )
110106, 109sylibr 224 . 2 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) → ∀𝑐 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 )
111110ex 449 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 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  Vcvv 3231  cmpt 4762  cfv 5926  (class class class)co 6690  0cc0 9974  cmin 10304  cn 11058  0cn0 11330  ...cfz 12364  Basecbs 15904  .rcmulr 15989  0gc0g 16147   Σg cgsu 16148  Mndcmnd 17341  Ringcrg 18593  Poly1cpl1 19595  coe1cco1 19596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-ofr 6940  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-tset 16007  df-ple 16008  df-0g 16149  df-gsum 16150  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-submnd 17383  df-grp 17472  df-minusg 17473  df-mulg 17588  df-ghm 17705  df-cntz 17796  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-psr 19404  df-mpl 19406  df-opsr 19408  df-psr1 19598  df-ply1 19600  df-coe1 19601
This theorem is referenced by:  cpmatmcllem  20571
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