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Theorem cply1mul 22315
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 2734 . . . . . . . . . 10 (.r𝑅) = (.r𝑅)
4 cply1mul.b . . . . . . . . . 10 𝐵 = (Base‘𝑃)
51, 2, 3, 4coe1mul 22288 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐵) → (coe1‘(𝐹 × 𝐺)) = (𝑠 ∈ ℕ0 ↦ (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑠𝑘)))))))
653expb 1119 . . . . . . . 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 7438 . . . . . . . . 9 (𝑠 = 𝑛 → (0...𝑠) = (0...𝑛))
10 fvoveq1 7453 . . . . . . . . . 10 (𝑠 = 𝑛 → ((coe1𝐺)‘(𝑠𝑘)) = ((coe1𝐺)‘(𝑛𝑘)))
1110oveq2d 7446 . . . . . . . . 9 (𝑠 = 𝑛 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑠𝑘))) = (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))))
129, 11mpteq12dv 5238 . . . . . . . 8 (𝑠 = 𝑛 → (𝑘 ∈ (0...𝑠) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑠𝑘)))) = (𝑘 ∈ (0...𝑛) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘)))))
1312oveq2d 7446 . . . . . . 7 (𝑠 = 𝑛 → (𝑅 Σg (𝑘 ∈ (0...𝑠) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑠𝑘))))) = (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))))))
1413adantl 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))))))
15 nnnn0 12530 . . . . . . 7 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
1615adantl 481 . . . . . 6 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
17 ovexd 7465 . . . . . 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 7022 . . . . 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 3108 . . . . . . . . . 10 (∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 ) ↔ (∀𝑐 ∈ ℕ ((coe1𝐹)‘𝑐) = 0 ∧ ∀𝑐 ∈ ℕ ((coe1𝐺)‘𝑐) = 0 ))
20 oveq2 7438 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (𝑛𝑘) = (𝑛 − 0))
21 nncn 12271 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
2221subid1d 11606 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ → (𝑛 − 0) = 𝑛)
2322adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (𝑛 − 0) = 𝑛)
2420, 23sylan9eqr 2796 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → (𝑛𝑘) = 𝑛)
25 simpll 767 . . . . . . . . . . . . . . . . . 18 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → 𝑛 ∈ ℕ)
2624, 25eqeltrd 2838 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → (𝑛𝑘) ∈ ℕ)
27 fveqeq2 6915 . . . . . . . . . . . . . . . . . 18 (𝑐 = (𝑛𝑘) → (((coe1𝐺)‘𝑐) = 0 ↔ ((coe1𝐺)‘(𝑛𝑘)) = 0 ))
2827rspcv 3617 . . . . . . . . . . . . . . . . 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 7438 . . . . . . . . . . . . . . . . . . . 20 (((coe1𝐺)‘(𝑛𝑘)) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = (((coe1𝐹)‘𝑘)(.r𝑅) 0 ))
31 simpll 767 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) → 𝑅 ∈ Ring)
32 simprl 771 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → 𝐹𝐵)
33 elfznn0 13656 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0)
3433adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → 𝑘 ∈ ℕ0)
3534adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0) → 𝑘 ∈ ℕ0)
36 eqid 2734 . . . . . . . . . . . . . . . . . . . . . . 23 (coe1𝐹) = (coe1𝐹)
37 eqid 2734 . . . . . . . . . . . . . . . . . . . . . . 23 (Base‘𝑅) = (Base‘𝑅)
3836, 4, 1, 37coe1fvalcl 22229 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹𝐵𝑘 ∈ ℕ0) → ((coe1𝐹)‘𝑘) ∈ (Base‘𝑅))
3932, 35, 38syl2an 596 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) → ((coe1𝐹)‘𝑘) ∈ (Base‘𝑅))
40 cply1mul.0 . . . . . . . . . . . . . . . . . . . . . 22 0 = (0g𝑅)
4137, 3, 40ringrz 20307 . . . . . . . . . . . . . . . . . . . . 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 2796 . . . . . . . . . . . . . . . . . . 19 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) ∧ ((coe1𝐺)‘(𝑛𝑘)) = 0 ) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )
4443ex 412 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) ∧ 𝑘 = 0)) → (((coe1𝐺)‘(𝑛𝑘)) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))
4544expcom 413 . . . . . . . . . . . . . . . . 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 415 . . . . . . . . . . . . . 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 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 ))))
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 2945 . . . . . . . . . . . . . . . . . . . . . 22 𝑘 = 0 → 𝑘 ≠ 0)
5352, 33anim12ci 614 . . . . . . . . . . . . . . . . . . . . 21 ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → (𝑘 ∈ ℕ0𝑘 ≠ 0))
54 elnnne0 12537 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ ℕ ↔ (𝑘 ∈ ℕ0𝑘 ≠ 0))
5553, 54sylibr 234 . . . . . . . . . . . . . . . . . . . 20 ((¬ 𝑘 = 0 ∧ 𝑘 ∈ (0...𝑛)) → 𝑘 ∈ ℕ)
56 fveqeq2 6915 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = 𝑘 → (((coe1𝐹)‘𝑐) = 0 ↔ ((coe1𝐹)‘𝑘) = 0 ))
5756rspcv 3617 . . . . . . . . . . . . . . . . . . . 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 7437 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((coe1𝐹)‘𝑘) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = ( 0 (.r𝑅)((coe1𝐺)‘(𝑛𝑘))))
60 simpll 767 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ 𝑘 ∈ (0...𝑛)) → 𝑅 ∈ Ring)
614eleq2i 2830 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐺𝐵𝐺 ∈ (Base‘𝑃))
6261biimpi 216 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐺𝐵𝐺 ∈ (Base‘𝑃))
6362adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐹𝐵𝐺𝐵) → 𝐺 ∈ (Base‘𝑃))
6463adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → 𝐺 ∈ (Base‘𝑃))
65 fznn0sub 13592 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 ∈ (0...𝑛) → (𝑛𝑘) ∈ ℕ0)
66 eqid 2734 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (coe1𝐺) = (coe1𝐺)
67 eqid 2734 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (Base‘𝑃) = (Base‘𝑃)
6866, 67, 1, 37coe1fvalcl 22229 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐺 ∈ (Base‘𝑃) ∧ (𝑛𝑘) ∈ ℕ0) → ((coe1𝐺)‘(𝑛𝑘)) ∈ (Base‘𝑅))
6964, 65, 68syl2an 596 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ 𝑘 ∈ (0...𝑛)) → ((coe1𝐺)‘(𝑛𝑘)) ∈ (Base‘𝑅))
7037, 3, 40ringlz 20306 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2796 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ 𝑘 ∈ (0...𝑛)) ∧ ((coe1𝐹)‘𝑘) = 0 ) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )
7372ex 412 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘) = 0 → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))
7473ex 412 . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . 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 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 ))))
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 242 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 ) → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )))
8988imp 406 . . . . . . . 8 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) → ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 ))
9089impl 455 . . . . . . 7 (((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑛)) → (((coe1𝐹)‘𝑘)(.r𝑅)((coe1𝐺)‘(𝑛𝑘))) = 0 )
9190mpteq2dva 5247 . . . . . 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 7446 . . . . 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 20260 . . . . . . . . 9 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
94 ovexd 7465 . . . . . . . . 9 (𝑅 ∈ Ring → (0...𝑛) ∈ V)
9540gsumz 18861 . . . . . . . . 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 480 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
9897adantr 480 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
9998adantr 480 . . . . 5 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → (𝑅 Σg (𝑘 ∈ (0...𝑛) ↦ 0 )) = 0 )
10018, 92, 993eqtrd 2778 . . . 4 ((((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) ∧ 𝑛 ∈ ℕ) → ((coe1‘(𝐹 × 𝐺))‘𝑛) = 0 )
101100ralrimiva 3143 . . 3 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) → ∀𝑛 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑛) = 0 )
102 fveqeq2 6915 . . . 4 (𝑐 = 𝑛 → (((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 ↔ ((coe1‘(𝐹 × 𝐺))‘𝑛) = 0 ))
103102cbvralvw 3234 . . 3 (∀𝑐 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 ↔ ∀𝑛 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑛) = 0 )
104101, 103sylibr 234 . 2 (((𝑅 ∈ Ring ∧ (𝐹𝐵𝐺𝐵)) ∧ ∀𝑐 ∈ ℕ (((coe1𝐹)‘𝑐) = 0 ∧ ((coe1𝐺)‘𝑐) = 0 )) → ∀𝑐 ∈ ℕ ((coe1‘(𝐹 × 𝐺))‘𝑐) = 0 )
105104ex 412 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 395   = wceq 1536  wcel 2105  wne 2937  wral 3058  Vcvv 3477  cmpt 5230  cfv 6562  (class class class)co 7430  0cc0 11152  cmin 11489  cn 12263  0cn0 12523  ...cfz 13543  Basecbs 17244  .rcmulr 17298  0gc0g 17485   Σg cgsu 17486  Mndcmnd 18759  Ringcrg 20250  Poly1cpl1 22193  coe1cco1 22194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-ofr 7697  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-sup 9479  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-fzo 13691  df-seq 14039  df-hash 14366  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-submnd 18809  df-grp 18966  df-minusg 18967  df-mulg 19098  df-ghm 19243  df-cntz 19347  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-ring 20252  df-psr 21946  df-mpl 21948  df-opsr 21950  df-psr1 22196  df-ply1 22198  df-coe1 22199
This theorem is referenced by:  cpmatmcllem  22739
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