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Theorem cayleyhamilton1 21504
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 21502, 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 21502 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = 0 )
98adantr 484 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿m0) ∧ 𝐹 finSupp 𝑍)) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = 0 )
10 nfv 1916 . . . . . . . 8 𝑛((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿m0) ∧ 𝐹 finSupp 𝑍))
11 nfcv 2982 . . . . . . . . . 10 𝑛𝑃
12 nfcv 2982 . . . . . . . . . 10 𝑛 Σg
13 nfmpt1 5150 . . . . . . . . . 10 𝑛(𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))
1411, 12, 13nfov 7179 . . . . . . . . 9 𝑛(𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋))))
1514nfeq2 2999 . . . . . . . 8 𝑛(𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋))))
1610, 15nfan 1901 . . . . . . 7 𝑛(((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿m0) ∧ 𝐹 finSupp 𝑍)) ∧ (𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))))
17 crngring 19309 . . . . . . . . . . . . 13 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
18173ad2ant2 1131 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 ∈ Ring)
1918adantr 484 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿m0) ∧ 𝐹 finSupp 𝑍)) → 𝑅 ∈ Ring)
20 cayleyhamilton1.p . . . . . . . . . . . . 13 𝑃 = (Poly1𝑅)
21 eqid 2824 . . . . . . . . . . . . 13 (Base‘𝑃) = (Base‘𝑃)
224, 1, 2, 20, 21chpmatply1 21444 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐶𝑀) ∈ (Base‘𝑃))
2322adantr 484 . . . . . . . . . . 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 2824 . . . . . . . . . . . 12 (0g𝑅) = (0g𝑅)
29 elmapi 8424 . . . . . . . . . . . . . 14 (𝐹 ∈ (𝐿m0) → 𝐹:ℕ0𝐿)
30 ffvelrn 6840 . . . . . . . . . . . . . . 15 ((𝐹:ℕ0𝐿𝑛 ∈ ℕ0) → (𝐹𝑛) ∈ 𝐿)
3130ralrimiva 3177 . . . . . . . . . . . . . 14 (𝐹:ℕ0𝐿 → ∀𝑛 ∈ ℕ0 (𝐹𝑛) ∈ 𝐿)
3229, 31syl 17 . . . . . . . . . . . . 13 (𝐹 ∈ (𝐿m0) → ∀𝑛 ∈ ℕ0 (𝐹𝑛) ∈ 𝐿)
3332ad2antrl 727 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿m0) ∧ 𝐹 finSupp 𝑍)) → ∀𝑛 ∈ ℕ0 (𝐹𝑛) ∈ 𝐿)
3429feqmptd 6724 . . . . . . . . . . . . . . 15 (𝐹 ∈ (𝐿m0) → 𝐹 = (𝑛 ∈ ℕ0 ↦ (𝐹𝑛)))
35 cayleyhamilton1.z . . . . . . . . . . . . . . . 16 𝑍 = (0g𝑅)
3635a1i 11 . . . . . . . . . . . . . . 15 (𝐹 ∈ (𝐿m0) → 𝑍 = (0g𝑅))
3734, 36breq12d 5065 . . . . . . . . . . . . . 14 (𝐹 ∈ (𝐿m0) → (𝐹 finSupp 𝑍 ↔ (𝑛 ∈ ℕ0 ↦ (𝐹𝑛)) finSupp (0g𝑅)))
3837biimpa 480 . . . . . . . . . . . . 13 ((𝐹 ∈ (𝐿m0) ∧ 𝐹 finSupp 𝑍) → (𝑛 ∈ ℕ0 ↦ (𝐹𝑛)) finSupp (0g𝑅))
3938adantl 485 . . . . . . . . . . . 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 20939 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿m0) ∧ 𝐹 finSupp 𝑍)) → (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))) ∈ (Base‘𝑃))
41 fveq2 6661 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑛 → (𝐹𝑖) = (𝐹𝑛))
42 oveq1 7156 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑛 → (𝑖𝐸𝑋) = (𝑛𝐸𝑋))
4341, 42oveq12d 7167 . . . . . . . . . . . . . . 15 (𝑖 = 𝑛 → ((𝐹𝑖) · (𝑖𝐸𝑋)) = ((𝐹𝑛) · (𝑛𝐸𝑋)))
4443cbvmptv 5155 . . . . . . . . . . . . . 14 (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋))) = (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))
4544oveq2i 7160 . . . . . . . . . . . . 13 (𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋))))
4645fveq2i 6664 . . . . . . . . . . . 12 (coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋))))) = (coe1‘(𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))))
4720, 21, 5, 46ply1coe1eq 20934 . . . . . . . . . . 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 1368 . . . . . . . . . 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 6661 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (𝐾𝑚) = (𝐾𝑛))
50 fveq2 6661 . . . . . . . . . . . . . . 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 3607 . . . . . . . . . . . . 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 486 . . . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿m0) ∧ 𝐹 finSupp 𝑍))) → 𝑅 ∈ Ring)
55 ffvelrn 6840 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹:ℕ0𝐿𝑖 ∈ ℕ0) → (𝐹𝑖) ∈ 𝐿)
5655ralrimiva 3177 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:ℕ0𝐿 → ∀𝑖 ∈ ℕ0 (𝐹𝑖) ∈ 𝐿)
5729, 56syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹 ∈ (𝐿m0) → ∀𝑖 ∈ ℕ0 (𝐹𝑖) ∈ 𝐿)
5857ad2antrl 727 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿m0) ∧ 𝐹 finSupp 𝑍)) → ∀𝑖 ∈ ℕ0 (𝐹𝑖) ∈ 𝐿)
5958adantl 485 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿m0) ∧ 𝐹 finSupp 𝑍))) → ∀𝑖 ∈ ℕ0 (𝐹𝑖) ∈ 𝐿)
6029feqmptd 6724 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐹 ∈ (𝐿m0) → 𝐹 = (𝑖 ∈ ℕ0 ↦ (𝐹𝑖)))
6160breq1d 5062 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹 ∈ (𝐿m0) → (𝐹 finSupp 𝑍 ↔ (𝑖 ∈ ℕ0 ↦ (𝐹𝑖)) finSupp 𝑍))
6261biimpa 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹 ∈ (𝐿m0) ∧ 𝐹 finSupp 𝑍) → (𝑖 ∈ ℕ0 ↦ (𝐹𝑖)) finSupp 𝑍)
6362adantl 485 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿m0) ∧ 𝐹 finSupp 𝑍)) → (𝑖 ∈ ℕ0 ↦ (𝐹𝑖)) finSupp 𝑍)
6463adantl 485 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿m0) ∧ 𝐹 finSupp 𝑍))) → (𝑖 ∈ ℕ0 ↦ (𝐹𝑖)) finSupp 𝑍)
65 simpl 486 . . . . . . . . . . . . . . . . . . . 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 20940 . . . . . . . . . . . . . . . . . . 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 6706 . . . . . . . . . . . . . . . . . . 19 𝑛 / 𝑖(𝐹𝑖) = (𝐹𝑛)
6866, 67syl6eq 2875 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℕ0 ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿m0) ∧ 𝐹 finSupp 𝑍))) → ((coe1‘(𝑃 Σg (𝑖 ∈ ℕ0 ↦ ((𝐹𝑖) · (𝑖𝐸𝑋)))))‘𝑛) = (𝐹𝑛))
6968adantl 485 . . . . . . . . . . . . . . . . 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 2859 . . . . . . . . . . . . . . . 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 424 . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . 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 420 . . . . . . . . . 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 263 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿m0) ∧ 𝐹 finSupp 𝑍)) → ((𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))) → (𝑛 ∈ ℕ0 → (𝐾𝑛) = (𝐹𝑛))))
7877imp31 421 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿m0) ∧ 𝐹 finSupp 𝑍)) ∧ (𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋))))) ∧ 𝑛 ∈ ℕ0) → (𝐾𝑛) = (𝐹𝑛))
7978oveq1d 7164 . . . . . . 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 5146 . . . . . 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 7165 . . . . 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 2826 . . . 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 232 . . 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 416 . 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133  csb 3866   class class class wbr 5052  cmpt 5132  wf 6339  cfv 6343  (class class class)co 7149  m cmap 8402  Fincfn 8505   finSupp cfsupp 8830  0cn0 11894  Basecbs 16483   ·𝑠 cvsca 16569  0gc0g 16713   Σg cgsu 16714  .gcmg 18224  mulGrpcmgp 19239  Ringcrg 19297  CRingccrg 19298  var1cv1 20812  Poly1cpl1 20813  coe1cco1 20814   Mat cmat 21019   CharPlyMat cchpmat 21438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-addf 10614  ax-mulf 10615
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-xor 1503  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-ot 4559  df-uni 4825  df-int 4863  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-of 7403  df-ofr 7404  df-om 7575  df-1st 7684  df-2nd 7685  df-supp 7827  df-tpos 7888  df-cur 7929  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-er 8285  df-map 8404  df-pm 8405  df-ixp 8458  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-fsupp 8831  df-sup 8903  df-oi 8971  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-nn 11635  df-2 11697  df-3 11698  df-4 11699  df-5 11700  df-6 11701  df-7 11702  df-8 11703  df-9 11704  df-n0 11895  df-xnn0 11965  df-z 11979  df-dec 12096  df-uz 12241  df-rp 12387  df-fz 12895  df-fzo 13038  df-seq 13374  df-exp 13435  df-hash 13696  df-word 13867  df-lsw 13915  df-concat 13923  df-s1 13950  df-substr 14003  df-pfx 14033  df-splice 14112  df-reverse 14121  df-s2 14210  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-mulr 16579  df-starv 16580  df-sca 16581  df-vsca 16582  df-ip 16583  df-tset 16584  df-ple 16585  df-ds 16587  df-unif 16588  df-hom 16589  df-cco 16590  df-0g 16715  df-gsum 16716  df-prds 16721  df-pws 16723  df-mre 16857  df-mrc 16858  df-acs 16860  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-mhm 17956  df-submnd 17957  df-efmnd 18034  df-grp 18106  df-minusg 18107  df-sbg 18108  df-mulg 18225  df-subg 18276  df-ghm 18356  df-gim 18399  df-cntz 18447  df-oppg 18474  df-symg 18496  df-pmtr 18570  df-psgn 18619  df-evpm 18620  df-cmn 18908  df-abl 18909  df-mgp 19240  df-ur 19252  df-srg 19256  df-ring 19299  df-cring 19300  df-oppr 19376  df-dvdsr 19394  df-unit 19395  df-invr 19425  df-dvr 19436  df-rnghom 19470  df-drng 19504  df-subrg 19533  df-lmod 19636  df-lss 19704  df-sra 19944  df-rgmod 19945  df-cnfld 20099  df-zring 20171  df-zrh 20204  df-dsmm 20428  df-frlm 20443  df-assa 20549  df-ascl 20551  df-psr 20601  df-mvr 20602  df-mpl 20603  df-opsr 20605  df-psr1 20816  df-vr1 20817  df-ply1 20818  df-coe1 20819  df-mamu 20998  df-mat 21020  df-mdet 21197  df-madu 21246  df-cpmat 21318  df-mat2pmat 21319  df-cpmat2mat 21320  df-decpmat 21375  df-pm2mp 21405  df-chpmat 21439
This theorem is referenced by: (None)
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