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Theorem cayleyhamilton1 21067
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 21065, 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 21065 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = 0 )
98adantr 474 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = 0 )
10 nfv 2013 . . . . . . . 8 𝑛((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍))
11 nfcv 2969 . . . . . . . . . 10 𝑛𝑃
12 nfcv 2969 . . . . . . . . . 10 𝑛 Σg
13 nfmpt1 4970 . . . . . . . . . 10 𝑛(𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))
1411, 12, 13nfov 6935 . . . . . . . . 9 𝑛(𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋))))
1514nfeq2 2985 . . . . . . . 8 𝑛(𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋))))
1610, 15nfan 2002 . . . . . . 7 𝑛(((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) ∧ (𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))))
17 crngring 18912 . . . . . . . . . . . . 13 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
18173ad2ant2 1168 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 ∈ Ring)
1918adantr 474 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → 𝑅 ∈ Ring)
20 cayleyhamilton1.p . . . . . . . . . . . . 13 𝑃 = (Poly1𝑅)
21 eqid 2825 . . . . . . . . . . . . 13 (Base‘𝑃) = (Base‘𝑃)
224, 1, 2, 20, 21chpmatply1 21007 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐶𝑀) ∈ (Base‘𝑃))
2322adantr 474 . . . . . . . . . . 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 2825 . . . . . . . . . . . 12 (0g𝑅) = (0g𝑅)
29 elmapi 8144 . . . . . . . . . . . . . 14 (𝐹 ∈ (𝐿𝑚0) → 𝐹:ℕ0𝐿)
30 ffvelrn 6606 . . . . . . . . . . . . . . 15 ((𝐹:ℕ0𝐿𝑛 ∈ ℕ0) → (𝐹𝑛) ∈ 𝐿)
3130ralrimiva 3175 . . . . . . . . . . . . . 14 (𝐹:ℕ0𝐿 → ∀𝑛 ∈ ℕ0 (𝐹𝑛) ∈ 𝐿)
3229, 31syl 17 . . . . . . . . . . . . 13 (𝐹 ∈ (𝐿𝑚0) → ∀𝑛 ∈ ℕ0 (𝐹𝑛) ∈ 𝐿)
3332ad2antrl 719 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → ∀𝑛 ∈ ℕ0 (𝐹𝑛) ∈ 𝐿)
3429feqmptd 6496 . . . . . . . . . . . . . . 15 (𝐹 ∈ (𝐿𝑚0) → 𝐹 = (𝑛 ∈ ℕ0 ↦ (𝐹𝑛)))
35 cayleyhamilton1.z . . . . . . . . . . . . . . . 16 𝑍 = (0g𝑅)
3635a1i 11 . . . . . . . . . . . . . . 15 (𝐹 ∈ (𝐿𝑚0) → 𝑍 = (0g𝑅))
3734, 36breq12d 4886 . . . . . . . . . . . . . 14 (𝐹 ∈ (𝐿𝑚0) → (𝐹 finSupp 𝑍 ↔ (𝑛 ∈ ℕ0 ↦ (𝐹𝑛)) finSupp (0g𝑅)))
3837biimpa 470 . . . . . . . . . . . . 13 ((𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍) → (𝑛 ∈ ℕ0 ↦ (𝐹𝑛)) finSupp (0g𝑅))
3938adantl 475 . . . . . . . . . . . 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 20033 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))) ∈ (Base‘𝑃))
41 fveq2 6433 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑛 → (𝐹𝑖) = (𝐹𝑛))
42 oveq1 6912 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑛 → (𝑖𝐸𝑋) = (𝑛𝐸𝑋))
4341, 42oveq12d 6923 . . . . . . . . . . . . . . 15 (𝑖 = 𝑛 → ((𝐹𝑖) · (𝑖𝐸𝑋)) = ((𝐹𝑛) · (𝑛𝐸𝑋)))
4443cbvmptv 4973 . . . . . . . . . . . . . 14 (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋))) = (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))
4544oveq2i 6916 . . . . . . . . . . . . 13 (𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋))))
4645fveq2i 6436 . . . . . . . . . . . 12 (coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋))))) = (coe1‘(𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))))
4720, 21, 5, 46ply1coe1eq 20028 . . . . . . . . . . 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 1494 . . . . . . . . . 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 6433 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (𝐾𝑚) = (𝐾𝑛))
50 fveq2 6433 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑚) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑛))
5149, 50eqeq12d 2840 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → ((𝐾𝑚) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑚) ↔ (𝐾𝑛) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑛)))
5251rspcva 3524 . . . . . . . . . . . . 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 476 . . . . . . . . . . . . . . . . 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 719 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍))) → 𝑅 ∈ Ring)
55 ffvelrn 6606 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹:ℕ0𝐿𝑖 ∈ ℕ0) → (𝐹𝑖) ∈ 𝐿)
5655ralrimiva 3175 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:ℕ0𝐿 → ∀𝑖 ∈ ℕ0 (𝐹𝑖) ∈ 𝐿)
5729, 56syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹 ∈ (𝐿𝑚0) → ∀𝑖 ∈ ℕ0 (𝐹𝑖) ∈ 𝐿)
5857ad2antrl 719 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → ∀𝑖 ∈ ℕ0 (𝐹𝑖) ∈ 𝐿)
5958adantl 475 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍))) → ∀𝑖 ∈ ℕ0 (𝐹𝑖) ∈ 𝐿)
6029feqmptd 6496 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐹 ∈ (𝐿𝑚0) → 𝐹 = (𝑖 ∈ ℕ0 ↦ (𝐹𝑖)))
6160breq1d 4883 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹 ∈ (𝐿𝑚0) → (𝐹 finSupp 𝑍 ↔ (𝑖 ∈ ℕ0 ↦ (𝐹𝑖)) finSupp 𝑍))
6261biimpa 470 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍) → (𝑖 ∈ ℕ0 ↦ (𝐹𝑖)) finSupp 𝑍)
6362adantl 475 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → (𝑖 ∈ ℕ0 ↦ (𝐹𝑖)) finSupp 𝑍)
6463adantl 475 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍))) → (𝑖 ∈ ℕ0 ↦ (𝐹𝑖)) finSupp 𝑍)
65 simpl 476 . . . . . . . . . . . . . . . . . . . 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 20034 . . . . . . . . . . . . . . . . . . 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 6479 . . . . . . . . . . . . . . . . . . 19 𝑛 / 𝑖(𝐹𝑖) = (𝐹𝑛)
6866, 67syl6eq 2877 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍))) → ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑛) = (𝐹𝑛))
6968adantl 475 . . . . . . . . . . . . . . . . 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 2861 . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . 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 408 . . . . . . . . . 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 252 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → ((𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))) → (𝑛 ∈ ℕ0 → (𝐾𝑛) = (𝐹𝑛))))
7877imp31 410 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) ∧ (𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋))))) ∧ 𝑛 ∈ ℕ0) → (𝐾𝑛) = (𝐹𝑛))
7978oveq1d 6920 . . . . . . 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 4966 . . . . . 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 6921 . . . . 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 2827 . . . 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 221 . . 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 403 . 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 198  wa 386  w3a 1111   = wceq 1656  wcel 2164  wral 3117  csb 3757   class class class wbr 4873  cmpt 4952  wf 6119  cfv 6123  (class class class)co 6905  𝑚 cmap 8122  Fincfn 8222   finSupp cfsupp 8544  0cn0 11618  Basecbs 16222   ·𝑠 cvsca 16309  0gc0g 16453   Σg cgsu 16454  .gcmg 17894  mulGrpcmgp 18843  Ringcrg 18901  CRingccrg 18902  var1cv1 19906  Poly1cpl1 19907  coe1cco1 19908   Mat cmat 20580   CharPlyMat cchpmat 21001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-inf2 8815  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329  ax-addf 10331  ax-mulf 10332
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-xor 1638  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-ot 4406  df-uni 4659  df-int 4698  df-iun 4742  df-iin 4743  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-se 5302  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-isom 6132  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-of 7157  df-ofr 7158  df-om 7327  df-1st 7428  df-2nd 7429  df-supp 7560  df-tpos 7617  df-cur 7658  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-2o 7827  df-oadd 7830  df-er 8009  df-map 8124  df-pm 8125  df-ixp 8176  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-fsupp 8545  df-sup 8617  df-oi 8684  df-card 9078  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-div 11010  df-nn 11351  df-2 11414  df-3 11415  df-4 11416  df-5 11417  df-6 11418  df-7 11419  df-8 11420  df-9 11421  df-n0 11619  df-xnn0 11691  df-z 11705  df-dec 11822  df-uz 11969  df-rp 12113  df-fz 12620  df-fzo 12761  df-seq 13096  df-exp 13155  df-hash 13411  df-word 13575  df-lsw 13623  df-concat 13631  df-s1 13656  df-substr 13701  df-pfx 13750  df-splice 13857  df-reverse 13875  df-s2 13969  df-struct 16224  df-ndx 16225  df-slot 16226  df-base 16228  df-sets 16229  df-ress 16230  df-plusg 16318  df-mulr 16319  df-starv 16320  df-sca 16321  df-vsca 16322  df-ip 16323  df-tset 16324  df-ple 16325  df-ds 16327  df-unif 16328  df-hom 16329  df-cco 16330  df-0g 16455  df-gsum 16456  df-prds 16461  df-pws 16463  df-mre 16599  df-mrc 16600  df-acs 16602  df-mgm 17595  df-sgrp 17637  df-mnd 17648  df-mhm 17688  df-submnd 17689  df-grp 17779  df-minusg 17780  df-sbg 17781  df-mulg 17895  df-subg 17942  df-ghm 18009  df-gim 18052  df-cntz 18100  df-oppg 18126  df-symg 18148  df-pmtr 18212  df-psgn 18261  df-evpm 18262  df-cmn 18548  df-abl 18549  df-mgp 18844  df-ur 18856  df-srg 18860  df-ring 18903  df-cring 18904  df-oppr 18977  df-dvdsr 18995  df-unit 18996  df-invr 19026  df-dvr 19037  df-rnghom 19071  df-drng 19105  df-subrg 19134  df-lmod 19221  df-lss 19289  df-sra 19533  df-rgmod 19534  df-assa 19673  df-ascl 19675  df-psr 19717  df-mvr 19718  df-mpl 19719  df-opsr 19721  df-psr1 19910  df-vr1 19911  df-ply1 19912  df-coe1 19913  df-cnfld 20107  df-zring 20179  df-zrh 20212  df-dsmm 20439  df-frlm 20454  df-mamu 20557  df-mat 20581  df-mdet 20759  df-madu 20808  df-cpmat 20881  df-mat2pmat 20882  df-cpmat2mat 20883  df-decpmat 20938  df-pm2mp 20968  df-chpmat 21002
This theorem is referenced by: (None)
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