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Theorem cayleyhamilton1 20745
 Description: The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation", or, in other words, a matrix over a commutative ring "inserted" into its characteristic polynomial results in zero. In this variant of cayleyhamilton 20743, the meaning of "inserted" is made more transparent: If the characteristic polynomial is a polynomial with coefficients (𝐹‘𝑛), then a matrix over a commutative ring "inserted" into its characteristic polynomial is the sum of these coefficients multiplied with the corresponding power of the matrix. (Contributed by AV, 25-Nov-2019.)
Hypotheses
Ref Expression
cayleyhamilton.a 𝐴 = (𝑁 Mat 𝑅)
cayleyhamilton.b 𝐵 = (Base‘𝐴)
cayleyhamilton.0 0 = (0g𝐴)
cayleyhamilton.c 𝐶 = (𝑁 CharPlyMat 𝑅)
cayleyhamilton.k 𝐾 = (coe1‘(𝐶𝑀))
cayleyhamilton.m = ( ·𝑠𝐴)
cayleyhamilton.e = (.g‘(mulGrp‘𝐴))
cayleyhamilton1.l 𝐿 = (Base‘𝑅)
cayleyhamilton1.x 𝑋 = (var1𝑅)
cayleyhamilton1.p 𝑃 = (Poly1𝑅)
cayleyhamilton1.m · = ( ·𝑠𝑃)
cayleyhamilton1.e 𝐸 = (.g‘(mulGrp‘𝑃))
cayleyhamilton1.z 𝑍 = (0g𝑅)
Assertion
Ref Expression
cayleyhamilton1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → ((𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) (𝑛 𝑀)))) = 0 ))
Distinct variable groups:   𝐴,𝑛   𝐵,𝑛   𝐶,𝑛   𝑛,𝑀   𝑛,𝑁   𝑅,𝑛   ,𝑛   ,𝑛   𝑛,𝐸   𝑛,𝐹   𝑛,𝐿   𝑃,𝑛   𝑛,𝑋   𝑛,𝑍   · ,𝑛
Allowed substitution hints:   𝐾(𝑛)   0 (𝑛)

Proof of Theorem cayleyhamilton1
Dummy variables 𝑚 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cayleyhamilton.a . . . 4 𝐴 = (𝑁 Mat 𝑅)
2 cayleyhamilton.b . . . 4 𝐵 = (Base‘𝐴)
3 cayleyhamilton.0 . . . 4 0 = (0g𝐴)
4 cayleyhamilton.c . . . 4 𝐶 = (𝑁 CharPlyMat 𝑅)
5 cayleyhamilton.k . . . 4 𝐾 = (coe1‘(𝐶𝑀))
6 cayleyhamilton.m . . . 4 = ( ·𝑠𝐴)
7 cayleyhamilton.e . . . 4 = (.g‘(mulGrp‘𝐴))
81, 2, 3, 4, 5, 6, 7cayleyhamilton 20743 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = 0 )
98adantr 480 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = 0 )
10 nfv 1883 . . . . . . . 8 𝑛((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍))
11 nfcv 2793 . . . . . . . . . 10 𝑛𝑃
12 nfcv 2793 . . . . . . . . . 10 𝑛 Σg
13 nfmpt1 4780 . . . . . . . . . 10 𝑛(𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))
1411, 12, 13nfov 6716 . . . . . . . . 9 𝑛(𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋))))
1514nfeq2 2809 . . . . . . . 8 𝑛(𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋))))
1610, 15nfan 1868 . . . . . . 7 𝑛(((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) ∧ (𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))))
17 crngring 18604 . . . . . . . . . . . . 13 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
18173ad2ant2 1103 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 ∈ Ring)
1918adantr 480 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → 𝑅 ∈ Ring)
20 cayleyhamilton1.p . . . . . . . . . . . . 13 𝑃 = (Poly1𝑅)
21 eqid 2651 . . . . . . . . . . . . 13 (Base‘𝑃) = (Base‘𝑃)
224, 1, 2, 20, 21chpmatply1 20685 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐶𝑀) ∈ (Base‘𝑃))
2322adantr 480 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → (𝐶𝑀) ∈ (Base‘𝑃))
24 cayleyhamilton1.x . . . . . . . . . . . 12 𝑋 = (var1𝑅)
25 cayleyhamilton1.e . . . . . . . . . . . 12 𝐸 = (.g‘(mulGrp‘𝑃))
26 cayleyhamilton1.l . . . . . . . . . . . 12 𝐿 = (Base‘𝑅)
27 cayleyhamilton1.m . . . . . . . . . . . 12 · = ( ·𝑠𝑃)
28 eqid 2651 . . . . . . . . . . . 12 (0g𝑅) = (0g𝑅)
29 elmapi 7921 . . . . . . . . . . . . . 14 (𝐹 ∈ (𝐿𝑚0) → 𝐹:ℕ0𝐿)
30 ffvelrn 6397 . . . . . . . . . . . . . . 15 ((𝐹:ℕ0𝐿𝑛 ∈ ℕ0) → (𝐹𝑛) ∈ 𝐿)
3130ralrimiva 2995 . . . . . . . . . . . . . 14 (𝐹:ℕ0𝐿 → ∀𝑛 ∈ ℕ0 (𝐹𝑛) ∈ 𝐿)
3229, 31syl 17 . . . . . . . . . . . . 13 (𝐹 ∈ (𝐿𝑚0) → ∀𝑛 ∈ ℕ0 (𝐹𝑛) ∈ 𝐿)
3332ad2antrl 764 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → ∀𝑛 ∈ ℕ0 (𝐹𝑛) ∈ 𝐿)
3429feqmptd 6288 . . . . . . . . . . . . . . 15 (𝐹 ∈ (𝐿𝑚0) → 𝐹 = (𝑛 ∈ ℕ0 ↦ (𝐹𝑛)))
35 cayleyhamilton1.z . . . . . . . . . . . . . . . 16 𝑍 = (0g𝑅)
3635a1i 11 . . . . . . . . . . . . . . 15 (𝐹 ∈ (𝐿𝑚0) → 𝑍 = (0g𝑅))
3734, 36breq12d 4698 . . . . . . . . . . . . . 14 (𝐹 ∈ (𝐿𝑚0) → (𝐹 finSupp 𝑍 ↔ (𝑛 ∈ ℕ0 ↦ (𝐹𝑛)) finSupp (0g𝑅)))
3837biimpa 500 . . . . . . . . . . . . 13 ((𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍) → (𝑛 ∈ ℕ0 ↦ (𝐹𝑛)) finSupp (0g𝑅))
3938adantl 481 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → (𝑛 ∈ ℕ0 ↦ (𝐹𝑛)) finSupp (0g𝑅))
4020, 21, 24, 25, 19, 26, 27, 28, 33, 39gsumsmonply1 19721 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))) ∈ (Base‘𝑃))
41 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑛 → (𝐹𝑖) = (𝐹𝑛))
42 oveq1 6697 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑛 → (𝑖𝐸𝑋) = (𝑛𝐸𝑋))
4341, 42oveq12d 6708 . . . . . . . . . . . . . . 15 (𝑖 = 𝑛 → ((𝐹𝑖) · (𝑖𝐸𝑋)) = ((𝐹𝑛) · (𝑛𝐸𝑋)))
4443cbvmptv 4783 . . . . . . . . . . . . . 14 (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋))) = (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))
4544oveq2i 6701 . . . . . . . . . . . . 13 (𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋))))
4645fveq2i 6232 . . . . . . . . . . . 12 (coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋))))) = (coe1‘(𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))))
4720, 21, 5, 46ply1coe1eq 19716 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ (𝐶𝑀) ∈ (Base‘𝑃) ∧ (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))) ∈ (Base‘𝑃)) → (∀𝑚 ∈ ℕ0 (𝐾𝑚) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑚) ↔ (𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋))))))
4819, 23, 40, 47syl3anc 1366 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → (∀𝑚 ∈ ℕ0 (𝐾𝑚) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑚) ↔ (𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋))))))
49 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (𝐾𝑚) = (𝐾𝑛))
50 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑚) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑛))
5149, 50eqeq12d 2666 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → ((𝐾𝑚) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑚) ↔ (𝐾𝑛) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑛)))
5251rspcva 3338 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ0 ∧ ∀𝑚 ∈ ℕ0 (𝐾𝑚) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑚)) → (𝐾𝑛) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑛))
53 simpl 472 . . . . . . . . . . . . . . . . 17 (((𝐾𝑛) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑛) ∧ (𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)))) → (𝐾𝑛) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑛))
5418ad2antrl 764 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍))) → 𝑅 ∈ Ring)
55 ffvelrn 6397 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹:ℕ0𝐿𝑖 ∈ ℕ0) → (𝐹𝑖) ∈ 𝐿)
5655ralrimiva 2995 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:ℕ0𝐿 → ∀𝑖 ∈ ℕ0 (𝐹𝑖) ∈ 𝐿)
5729, 56syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹 ∈ (𝐿𝑚0) → ∀𝑖 ∈ ℕ0 (𝐹𝑖) ∈ 𝐿)
5857ad2antrl 764 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → ∀𝑖 ∈ ℕ0 (𝐹𝑖) ∈ 𝐿)
5958adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍))) → ∀𝑖 ∈ ℕ0 (𝐹𝑖) ∈ 𝐿)
6029feqmptd 6288 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐹 ∈ (𝐿𝑚0) → 𝐹 = (𝑖 ∈ ℕ0 ↦ (𝐹𝑖)))
6160breq1d 4695 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹 ∈ (𝐿𝑚0) → (𝐹 finSupp 𝑍 ↔ (𝑖 ∈ ℕ0 ↦ (𝐹𝑖)) finSupp 𝑍))
6261biimpa 500 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍) → (𝑖 ∈ ℕ0 ↦ (𝐹𝑖)) finSupp 𝑍)
6362adantl 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → (𝑖 ∈ ℕ0 ↦ (𝐹𝑖)) finSupp 𝑍)
6463adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍))) → (𝑖 ∈ ℕ0 ↦ (𝐹𝑖)) finSupp 𝑍)
65 simpl 472 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍))) → 𝑛 ∈ ℕ0)
6620, 21, 24, 25, 54, 26, 27, 35, 59, 64, 65gsummoncoe1 19722 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍))) → ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑛) = 𝑛 / 𝑖(𝐹𝑖))
67 csbfv 6271 . . . . . . . . . . . . . . . . . . 19 𝑛 / 𝑖(𝐹𝑖) = (𝐹𝑛)
6866, 67syl6eq 2701 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍))) → ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑛) = (𝐹𝑛))
6968adantl 481 . . . . . . . . . . . . . . . . 17 (((𝐾𝑛) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑛) ∧ (𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)))) → ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑛) = (𝐹𝑛))
7053, 69eqtrd 2685 . . . . . . . . . . . . . . . 16 (((𝐾𝑛) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑛) ∧ (𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)))) → (𝐾𝑛) = (𝐹𝑛))
7170exp32 630 . . . . . . . . . . . . . . 15 ((𝐾𝑛) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑛) → (𝑛 ∈ ℕ0 → (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → (𝐾𝑛) = (𝐹𝑛))))
7271com12 32 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ0 → ((𝐾𝑛) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑛) → (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → (𝐾𝑛) = (𝐹𝑛))))
7372adantr 480 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ0 ∧ ∀𝑚 ∈ ℕ0 (𝐾𝑚) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑚)) → ((𝐾𝑛) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑛) → (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → (𝐾𝑛) = (𝐹𝑛))))
7452, 73mpd 15 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ0 ∧ ∀𝑚 ∈ ℕ0 (𝐾𝑚) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑚)) → (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → (𝐾𝑛) = (𝐹𝑛)))
7574com12 32 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → ((𝑛 ∈ ℕ0 ∧ ∀𝑚 ∈ ℕ0 (𝐾𝑚) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑚)) → (𝐾𝑛) = (𝐹𝑛)))
7675expcomd 453 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → (∀𝑚 ∈ ℕ0 (𝐾𝑚) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑚) → (𝑛 ∈ ℕ0 → (𝐾𝑛) = (𝐹𝑛))))
7748, 76sylbird 250 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → ((𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))) → (𝑛 ∈ ℕ0 → (𝐾𝑛) = (𝐹𝑛))))
7877imp31 447 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) ∧ (𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋))))) ∧ 𝑛 ∈ ℕ0) → (𝐾𝑛) = (𝐹𝑛))
7978oveq1d 6705 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) ∧ (𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋))))) ∧ 𝑛 ∈ ℕ0) → ((𝐾𝑛) (𝑛 𝑀)) = ((𝐹𝑛) (𝑛 𝑀)))
8016, 79mpteq2da 4776 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) ∧ (𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋))))) → (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀))) = (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) (𝑛 𝑀))))
8180oveq2d 6706 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) ∧ (𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋))))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) (𝑛 𝑀)))))
8281eqeq1d 2653 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) ∧ (𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋))))) → ((𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = 0 ↔ (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) (𝑛 𝑀)))) = 0 ))
8382biimpd 219 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) ∧ (𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋))))) → ((𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = 0 → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) (𝑛 𝑀)))) = 0 ))
8483ex 449 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → ((𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))) → ((𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = 0 → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) (𝑛 𝑀)))) = 0 )))
859, 84mpid 44 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → ((𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) (𝑛 𝑀)))) = 0 ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ⦋csb 3566   class class class wbr 4685   ↦ cmpt 4762  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690   ↑𝑚 cmap 7899  Fincfn 7997   finSupp cfsupp 8316  ℕ0cn0 11330  Basecbs 15904   ·𝑠 cvsca 15992  0gc0g 16147   Σg cgsu 16148  .gcmg 17587  mulGrpcmgp 18535  Ringcrg 18593  CRingccrg 18594  var1cv1 19594  Poly1cpl1 19595  coe1cco1 19596   Mat cmat 20261   CharPlyMat cchpmat 20679 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  ax-addf 10053  ax-mulf 10054 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-xor 1505  df-tru 1526  df-fal 1529  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-ot 4219  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-tpos 7397  df-cur 7438  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-sup 8389  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-div 10723  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-xnn0 11402  df-z 11416  df-dec 11532  df-uz 11726  df-rp 11871  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-word 13331  df-lsw 13332  df-concat 13333  df-s1 13334  df-substr 13335  df-splice 13336  df-reverse 13337  df-s2 13639  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-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-0g 16149  df-gsum 16150  df-prds 16155  df-pws 16157  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-sbg 17474  df-mulg 17588  df-subg 17638  df-ghm 17705  df-gim 17748  df-cntz 17796  df-oppg 17822  df-symg 17844  df-pmtr 17908  df-psgn 17957  df-evpm 17958  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-srg 18552  df-ring 18595  df-cring 18596  df-oppr 18669  df-dvdsr 18687  df-unit 18688  df-invr 18718  df-dvr 18729  df-rnghom 18763  df-drng 18797  df-subrg 18826  df-lmod 18913  df-lss 18981  df-sra 19220  df-rgmod 19221  df-assa 19360  df-ascl 19362  df-psr 19404  df-mvr 19405  df-mpl 19406  df-opsr 19408  df-psr1 19598  df-vr1 19599  df-ply1 19600  df-coe1 19601  df-cnfld 19795  df-zring 19867  df-zrh 19900  df-dsmm 20124  df-frlm 20139  df-mamu 20238  df-mat 20262  df-mdet 20439  df-madu 20488  df-cpmat 20559  df-mat2pmat 20560  df-cpmat2mat 20561  df-decpmat 20616  df-pm2mp 20646  df-chpmat 20680 This theorem is referenced by: (None)
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