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Theorem cayleyhamilton1 22022
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 22020, the meaning of "inserted" is made more transparent: If the characteristic polynomial is a polynomial with coefficients (𝐹𝑛), 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 22020 . . 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 1920 . . . . . . . 8 𝑛((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿m0) ∧ 𝐹 finSupp 𝑍))
11 nfcv 2908 . . . . . . . . . 10 𝑛𝑃
12 nfcv 2908 . . . . . . . . . 10 𝑛 Σg
13 nfmpt1 5186 . . . . . . . . . 10 𝑛(𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))
1411, 12, 13nfov 7298 . . . . . . . . 9 𝑛(𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋))))
1514nfeq2 2925 . . . . . . . 8 𝑛(𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋))))
1610, 15nfan 1905 . . . . . . 7 𝑛(((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿m0) ∧ 𝐹 finSupp 𝑍)) ∧ (𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))))
17 crngring 19776 . . . . . . . . . . . . 13 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
18173ad2ant2 1132 . . . . . . . . . . . 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 2739 . . . . . . . . . . . . 13 (Base‘𝑃) = (Base‘𝑃)
224, 1, 2, 20, 21chpmatply1 21962 . . . . . . . . . . . 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 2739 . . . . . . . . . . . 12 (0g𝑅) = (0g𝑅)
29 elmapi 8611 . . . . . . . . . . . . . 14 (𝐹 ∈ (𝐿m0) → 𝐹:ℕ0𝐿)
30 ffvelrn 6953 . . . . . . . . . . . . . . 15 ((𝐹:ℕ0𝐿𝑛 ∈ ℕ0) → (𝐹𝑛) ∈ 𝐿)
3130ralrimiva 3109 . . . . . . . . . . . . . 14 (𝐹:ℕ0𝐿 → ∀𝑛 ∈ ℕ0 (𝐹𝑛) ∈ 𝐿)
3229, 31syl 17 . . . . . . . . . . . . 13 (𝐹 ∈ (𝐿m0) → ∀𝑛 ∈ ℕ0 (𝐹𝑛) ∈ 𝐿)
3332ad2antrl 724 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿m0) ∧ 𝐹 finSupp 𝑍)) → ∀𝑛 ∈ ℕ0 (𝐹𝑛) ∈ 𝐿)
3429feqmptd 6831 . . . . . . . . . . . . . . 15 (𝐹 ∈ (𝐿m0) → 𝐹 = (𝑛 ∈ ℕ0 ↦ (𝐹𝑛)))
35 cayleyhamilton1.z . . . . . . . . . . . . . . . 16 𝑍 = (0g𝑅)
3635a1i 11 . . . . . . . . . . . . . . 15 (𝐹 ∈ (𝐿m0) → 𝑍 = (0g𝑅))
3734, 36breq12d 5091 . . . . . . . . . . . . . 14 (𝐹 ∈ (𝐿m0) → (𝐹 finSupp 𝑍 ↔ (𝑛 ∈ ℕ0 ↦ (𝐹𝑛)) finSupp (0g𝑅)))
3837biimpa 476 . . . . . . . . . . . . 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 21455 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿m0) ∧ 𝐹 finSupp 𝑍)) → (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))) ∈ (Base‘𝑃))
41 fveq2 6768 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑛 → (𝐹𝑖) = (𝐹𝑛))
42 oveq1 7275 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑛 → (𝑖𝐸𝑋) = (𝑛𝐸𝑋))
4341, 42oveq12d 7286 . . . . . . . . . . . . . . 15 (𝑖 = 𝑛 → ((𝐹𝑖) · (𝑖𝐸𝑋)) = ((𝐹𝑛) · (𝑛𝐸𝑋)))
4443cbvmptv 5191 . . . . . . . . . . . . . 14 (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋))) = (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))
4544oveq2i 7279 . . . . . . . . . . . . 13 (𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋))))
4645fveq2i 6771 . . . . . . . . . . . 12 (coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋))))) = (coe1‘(𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))))
4720, 21, 5, 46ply1coe1eq 21450 . . . . . . . . . . 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 1369 . . . . . . . . . 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 6768 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (𝐾𝑚) = (𝐾𝑛))
50 fveq2 6768 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑚) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑛))
5149, 50eqeq12d 2755 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → ((𝐾𝑚) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑚) ↔ (𝐾𝑛) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑛)))
5251rspcva 3558 . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿m0) ∧ 𝐹 finSupp 𝑍))) → 𝑅 ∈ Ring)
55 ffvelrn 6953 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹:ℕ0𝐿𝑖 ∈ ℕ0) → (𝐹𝑖) ∈ 𝐿)
5655ralrimiva 3109 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:ℕ0𝐿 → ∀𝑖 ∈ ℕ0 (𝐹𝑖) ∈ 𝐿)
5729, 56syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹 ∈ (𝐿m0) → ∀𝑖 ∈ ℕ0 (𝐹𝑖) ∈ 𝐿)
5857ad2antrl 724 . . . . . . . . . . . . . . . . . . . . 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 6831 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐹 ∈ (𝐿m0) → 𝐹 = (𝑖 ∈ ℕ0 ↦ (𝐹𝑖)))
6160breq1d 5088 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹 ∈ (𝐿m0) → (𝐹 finSupp 𝑍 ↔ (𝑖 ∈ ℕ0 ↦ (𝐹𝑖)) finSupp 𝑍))
6261biimpa 476 . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . 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 21456 . . . . . . . . . . . . . . . . . . 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 6813 . . . . . . . . . . . . . . . . . . 19 𝑛 / 𝑖(𝐹𝑖) = (𝐹𝑛)
6866, 67eqtrdi 2795 . . . . . . . . . . . . . . . . . 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 2779 . . . . . . . . . . . . . . . 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 420 . . . . . . . . . . . . . . 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 416 . . . . . . . . . 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 259 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿m0) ∧ 𝐹 finSupp 𝑍)) → ((𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))) → (𝑛 ∈ ℕ0 → (𝐾𝑛) = (𝐹𝑛))))
7877imp31 417 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿m0) ∧ 𝐹 finSupp 𝑍)) ∧ (𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋))))) ∧ 𝑛 ∈ ℕ0) → (𝐾𝑛) = (𝐹𝑛))
7978oveq1d 7283 . . . . . . 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 5176 . . . . . 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 7284 . . . . 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 2741 . . . 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 228 . . 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 412 . 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 205  wa 395  w3a 1085   = wceq 1541  wcel 2109  wral 3065  csb 3836   class class class wbr 5078  cmpt 5161  wf 6426  cfv 6430  (class class class)co 7268  m cmap 8589  Fincfn 8707   finSupp cfsupp 9089  0cn0 12216  Basecbs 16893   ·𝑠 cvsca 16947  0gc0g 17131   Σg cgsu 17132  .gcmg 18681  mulGrpcmgp 19701  Ringcrg 19764  CRingccrg 19765  var1cv1 21328  Poly1cpl1 21329  coe1cco1 21330   Mat cmat 21535   CharPlyMat cchpmat 21956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579  ax-cnex 10911  ax-resscn 10912  ax-1cn 10913  ax-icn 10914  ax-addcl 10915  ax-addrcl 10916  ax-mulcl 10917  ax-mulrcl 10918  ax-mulcom 10919  ax-addass 10920  ax-mulass 10921  ax-distr 10922  ax-i2m1 10923  ax-1ne0 10924  ax-1rid 10925  ax-rnegex 10926  ax-rrecex 10927  ax-cnre 10928  ax-pre-lttri 10929  ax-pre-lttrn 10930  ax-pre-ltadd 10931  ax-pre-mulgt0 10932  ax-addf 10934  ax-mulf 10935
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-xor 1506  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-reu 3072  df-rmo 3073  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-ot 4575  df-uni 4845  df-int 4885  df-iun 4931  df-iin 4932  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-se 5544  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-pred 6199  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-isom 6439  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-of 7524  df-ofr 7525  df-om 7701  df-1st 7817  df-2nd 7818  df-supp 7962  df-tpos 8026  df-cur 8067  df-frecs 8081  df-wrecs 8112  df-recs 8186  df-rdg 8225  df-1o 8281  df-2o 8282  df-er 8472  df-map 8591  df-pm 8592  df-ixp 8660  df-en 8708  df-dom 8709  df-sdom 8710  df-fin 8711  df-fsupp 9090  df-sup 9162  df-oi 9230  df-card 9681  df-pnf 10995  df-mnf 10996  df-xr 10997  df-ltxr 10998  df-le 10999  df-sub 11190  df-neg 11191  df-div 11616  df-nn 11957  df-2 12019  df-3 12020  df-4 12021  df-5 12022  df-6 12023  df-7 12024  df-8 12025  df-9 12026  df-n0 12217  df-xnn0 12289  df-z 12303  df-dec 12420  df-uz 12565  df-rp 12713  df-fz 13222  df-fzo 13365  df-seq 13703  df-exp 13764  df-hash 14026  df-word 14199  df-lsw 14247  df-concat 14255  df-s1 14282  df-substr 14335  df-pfx 14365  df-splice 14444  df-reverse 14453  df-s2 14542  df-struct 16829  df-sets 16846  df-slot 16864  df-ndx 16876  df-base 16894  df-ress 16923  df-plusg 16956  df-mulr 16957  df-starv 16958  df-sca 16959  df-vsca 16960  df-ip 16961  df-tset 16962  df-ple 16963  df-ds 16965  df-unif 16966  df-hom 16967  df-cco 16968  df-0g 17133  df-gsum 17134  df-prds 17139  df-pws 17141  df-mre 17276  df-mrc 17277  df-acs 17279  df-mgm 18307  df-sgrp 18356  df-mnd 18367  df-mhm 18411  df-submnd 18412  df-efmnd 18489  df-grp 18561  df-minusg 18562  df-sbg 18563  df-mulg 18682  df-subg 18733  df-ghm 18813  df-gim 18856  df-cntz 18904  df-oppg 18931  df-symg 18956  df-pmtr 19031  df-psgn 19080  df-evpm 19081  df-cmn 19369  df-abl 19370  df-mgp 19702  df-ur 19719  df-srg 19723  df-ring 19766  df-cring 19767  df-oppr 19843  df-dvdsr 19864  df-unit 19865  df-invr 19895  df-dvr 19906  df-rnghom 19940  df-drng 19974  df-subrg 20003  df-lmod 20106  df-lss 20175  df-sra 20415  df-rgmod 20416  df-cnfld 20579  df-zring 20652  df-zrh 20686  df-dsmm 20920  df-frlm 20935  df-assa 21041  df-ascl 21043  df-psr 21093  df-mvr 21094  df-mpl 21095  df-opsr 21097  df-psr1 21332  df-vr1 21333  df-ply1 21334  df-coe1 21335  df-mamu 21514  df-mat 21536  df-mdet 21715  df-madu 21764  df-cpmat 21836  df-mat2pmat 21837  df-cpmat2mat 21838  df-decpmat 21893  df-pm2mp 21923  df-chpmat 21957
This theorem is referenced by: (None)
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