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Theorem cayleyhamilton1 20629
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 20627, 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 20627 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = 0 )
98adantr 481 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = 0 )
10 nfv 1840 . . . . . . . 8 𝑛((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍))
11 nfcv 2761 . . . . . . . . . 10 𝑛𝑃
12 nfcv 2761 . . . . . . . . . 10 𝑛 Σg
13 nfmpt1 4712 . . . . . . . . . 10 𝑛(𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))
1411, 12, 13nfov 6636 . . . . . . . . 9 𝑛(𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋))))
1514nfeq2 2776 . . . . . . . 8 𝑛(𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋))))
1610, 15nfan 1825 . . . . . . 7 𝑛(((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) ∧ (𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))))
17 crngring 18490 . . . . . . . . . . . . 13 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
18173ad2ant2 1081 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 ∈ Ring)
1918adantr 481 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → 𝑅 ∈ Ring)
20 cayleyhamilton1.p . . . . . . . . . . . . 13 𝑃 = (Poly1𝑅)
21 eqid 2621 . . . . . . . . . . . . 13 (Base‘𝑃) = (Base‘𝑃)
224, 1, 2, 20, 21chpmatply1 20569 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐶𝑀) ∈ (Base‘𝑃))
2322adantr 481 . . . . . . . . . . 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 2621 . . . . . . . . . . . 12 (0g𝑅) = (0g𝑅)
29 elmapi 7831 . . . . . . . . . . . . . 14 (𝐹 ∈ (𝐿𝑚0) → 𝐹:ℕ0𝐿)
30 ffvelrn 6318 . . . . . . . . . . . . . . 15 ((𝐹:ℕ0𝐿𝑛 ∈ ℕ0) → (𝐹𝑛) ∈ 𝐿)
3130ralrimiva 2961 . . . . . . . . . . . . . 14 (𝐹:ℕ0𝐿 → ∀𝑛 ∈ ℕ0 (𝐹𝑛) ∈ 𝐿)
3229, 31syl 17 . . . . . . . . . . . . 13 (𝐹 ∈ (𝐿𝑚0) → ∀𝑛 ∈ ℕ0 (𝐹𝑛) ∈ 𝐿)
3332ad2antrl 763 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → ∀𝑛 ∈ ℕ0 (𝐹𝑛) ∈ 𝐿)
3429feqmptd 6211 . . . . . . . . . . . . . . 15 (𝐹 ∈ (𝐿𝑚0) → 𝐹 = (𝑛 ∈ ℕ0 ↦ (𝐹𝑛)))
35 cayleyhamilton1.z . . . . . . . . . . . . . . . 16 𝑍 = (0g𝑅)
3635a1i 11 . . . . . . . . . . . . . . 15 (𝐹 ∈ (𝐿𝑚0) → 𝑍 = (0g𝑅))
3734, 36breq12d 4631 . . . . . . . . . . . . . 14 (𝐹 ∈ (𝐿𝑚0) → (𝐹 finSupp 𝑍 ↔ (𝑛 ∈ ℕ0 ↦ (𝐹𝑛)) finSupp (0g𝑅)))
3837biimpa 501 . . . . . . . . . . . . 13 ((𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍) → (𝑛 ∈ ℕ0 ↦ (𝐹𝑛)) finSupp (0g𝑅))
3938adantl 482 . . . . . . . . . . . 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 19605 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))) ∈ (Base‘𝑃))
41 fveq2 6153 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑛 → (𝐹𝑖) = (𝐹𝑛))
42 oveq1 6617 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑛 → (𝑖𝐸𝑋) = (𝑛𝐸𝑋))
4341, 42oveq12d 6628 . . . . . . . . . . . . . . 15 (𝑖 = 𝑛 → ((𝐹𝑖) · (𝑖𝐸𝑋)) = ((𝐹𝑛) · (𝑛𝐸𝑋)))
4443cbvmptv 4715 . . . . . . . . . . . . . 14 (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋))) = (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))
4544oveq2i 6621 . . . . . . . . . . . . 13 (𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋))))
4645fveq2i 6156 . . . . . . . . . . . 12 (coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋))))) = (coe1‘(𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))))
4720, 21, 5, 46ply1coe1eq 19600 . . . . . . . . . . 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 1323 . . . . . . . . . 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 6153 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (𝐾𝑚) = (𝐾𝑛))
50 fveq2 6153 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑚) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑛))
5149, 50eqeq12d 2636 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → ((𝐾𝑚) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑚) ↔ (𝐾𝑛) = ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑛)))
5251rspcva 3296 . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . 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 763 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍))) → 𝑅 ∈ Ring)
55 ffvelrn 6318 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹:ℕ0𝐿𝑖 ∈ ℕ0) → (𝐹𝑖) ∈ 𝐿)
5655ralrimiva 2961 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:ℕ0𝐿 → ∀𝑖 ∈ ℕ0 (𝐹𝑖) ∈ 𝐿)
5729, 56syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹 ∈ (𝐿𝑚0) → ∀𝑖 ∈ ℕ0 (𝐹𝑖) ∈ 𝐿)
5857ad2antrl 763 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → ∀𝑖 ∈ ℕ0 (𝐹𝑖) ∈ 𝐿)
5958adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍))) → ∀𝑖 ∈ ℕ0 (𝐹𝑖) ∈ 𝐿)
6029feqmptd 6211 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐹 ∈ (𝐿𝑚0) → 𝐹 = (𝑖 ∈ ℕ0 ↦ (𝐹𝑖)))
6160breq1d 4628 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹 ∈ (𝐿𝑚0) → (𝐹 finSupp 𝑍 ↔ (𝑖 ∈ ℕ0 ↦ (𝐹𝑖)) finSupp 𝑍))
6261biimpa 501 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍) → (𝑖 ∈ ℕ0 ↦ (𝐹𝑖)) finSupp 𝑍)
6362adantl 482 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → (𝑖 ∈ ℕ0 ↦ (𝐹𝑖)) finSupp 𝑍)
6463adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍))) → (𝑖 ∈ ℕ0 ↦ (𝐹𝑖)) finSupp 𝑍)
65 simpl 473 . . . . . . . . . . . . . . . . . . . 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 19606 . . . . . . . . . . . . . . . . . . 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 6195 . . . . . . . . . . . . . . . . . . 19 𝑛 / 𝑖(𝐹𝑖) = (𝐹𝑛)
6866, 67syl6eq 2671 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍))) → ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑛) = (𝐹𝑛))
6968adantl 482 . . . . . . . . . . . . . . . . 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 2655 . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . 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 454 . . . . . . . . . 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 448 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) ∧ (𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋))))) ∧ 𝑛 ∈ ℕ0) → (𝐾𝑛) = (𝐹𝑛))
7978oveq1d 6625 . . . . . . 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 4708 . . . . . 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 6626 . . . . 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 2623 . . . 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 450 . 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 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  csb 3518   class class class wbr 4618  cmpt 4678  wf 5848  cfv 5852  (class class class)co 6610  𝑚 cmap 7809  Fincfn 7907   finSupp cfsupp 8227  0cn0 11244  Basecbs 15792   ·𝑠 cvsca 15877  0gc0g 16032   Σg cgsu 16033  .gcmg 17472  mulGrpcmgp 18421  Ringcrg 18479  CRingccrg 18480  var1cv1 19478  Poly1cpl1 19479  coe1cco1 19480   Mat cmat 20145   CharPlyMat cchpmat 20563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8490  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965  ax-addf 9967  ax-mulf 9968
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-xor 1462  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-ot 4162  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-ofr 6858  df-om 7020  df-1st 7120  df-2nd 7121  df-supp 7248  df-tpos 7304  df-cur 7345  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-ixp 7861  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-fsupp 8228  df-sup 8300  df-oi 8367  df-card 8717  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-div 10637  df-nn 10973  df-2 11031  df-3 11032  df-4 11033  df-5 11034  df-6 11035  df-7 11036  df-8 11037  df-9 11038  df-n0 11245  df-xnn0 11316  df-z 11330  df-dec 11446  df-uz 11640  df-rp 11785  df-fz 12277  df-fzo 12415  df-seq 12750  df-exp 12809  df-hash 13066  df-word 13246  df-lsw 13247  df-concat 13248  df-s1 13249  df-substr 13250  df-splice 13251  df-reverse 13252  df-s2 13538  df-struct 15794  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-plusg 15886  df-mulr 15887  df-starv 15888  df-sca 15889  df-vsca 15890  df-ip 15891  df-tset 15892  df-ple 15893  df-ds 15896  df-unif 15897  df-hom 15898  df-cco 15899  df-0g 16034  df-gsum 16035  df-prds 16040  df-pws 16042  df-mre 16178  df-mrc 16179  df-acs 16181  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-mhm 17267  df-submnd 17268  df-grp 17357  df-minusg 17358  df-sbg 17359  df-mulg 17473  df-subg 17523  df-ghm 17590  df-gim 17633  df-cntz 17682  df-oppg 17708  df-symg 17730  df-pmtr 17794  df-psgn 17843  df-evpm 17844  df-cmn 18127  df-abl 18128  df-mgp 18422  df-ur 18434  df-srg 18438  df-ring 18481  df-cring 18482  df-oppr 18555  df-dvdsr 18573  df-unit 18574  df-invr 18604  df-dvr 18615  df-rnghom 18647  df-drng 18681  df-subrg 18710  df-lmod 18797  df-lss 18865  df-sra 19104  df-rgmod 19105  df-assa 19244  df-ascl 19246  df-psr 19288  df-mvr 19289  df-mpl 19290  df-opsr 19292  df-psr1 19482  df-vr1 19483  df-ply1 19484  df-coe1 19485  df-cnfld 19679  df-zring 19751  df-zrh 19784  df-dsmm 20008  df-frlm 20023  df-mamu 20122  df-mat 20146  df-mdet 20323  df-madu 20372  df-cpmat 20443  df-mat2pmat 20444  df-cpmat2mat 20445  df-decpmat 20500  df-pm2mp 20530  df-chpmat 20564
This theorem is referenced by: (None)
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