Theorem decpmatval0 22729

Description: The matrix consisting of the coefficients in the polynomial entries of a polynomial matrix for the same power, most general version. (Contributed by AV, 2-Dec-2019.)

Assertion

Ref	Expression
decpmatval0	⊢ ((𝑀 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) → (𝑀 decompPMat 𝐾) = (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝐾)))

Distinct variable groups:   𝑖,𝐾,𝑗   𝑖,𝑀,𝑗

Allowed substitution hints:   𝑉(𝑖,𝑗)

Proof of Theorem decpmatval0

Dummy variables 𝑘 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-decpmat 22728	. . 3 ⊢ decompPMat = (𝑚 ∈ V, 𝑘 ∈ ℕ0 ↦ (𝑖 ∈ dom dom 𝑚, 𝑗 ∈ dom dom 𝑚 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)))
2	1	a1i 11	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) → decompPMat = (𝑚 ∈ V, 𝑘 ∈ ℕ0 ↦ (𝑖 ∈ dom dom 𝑚, 𝑗 ∈ dom dom 𝑚 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘))))
3		dmeq 5858	. . . . . 6 ⊢ (𝑚 = 𝑀 → dom 𝑚 = dom 𝑀)
4	3	adantr 480	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾) → dom 𝑚 = dom 𝑀)
5	4	dmeqd 5860	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾) → dom dom 𝑚 = dom dom 𝑀)
6		oveq 7373	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
7	6	fveq2d 6844	. . . . . 6 ⊢ (𝑚 = 𝑀 → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
8	7	adantr 480	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾) → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
9		simpr 484	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾) → 𝑘 = 𝐾)
10	8, 9	fveq12d 6847	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾) → ((coe1‘(𝑖𝑚𝑗))‘𝑘) = ((coe1‘(𝑖𝑀𝑗))‘𝐾))
11	5, 5, 10	mpoeq123dv 7442	. . 3 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾) → (𝑖 ∈ dom dom 𝑚, 𝑗 ∈ dom dom 𝑚 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)) = (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝐾)))
12	11	adantl 481	. 2 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) ∧ (𝑚 = 𝑀 ∧ 𝑘 = 𝐾)) → (𝑖 ∈ dom dom 𝑚, 𝑗 ∈ dom dom 𝑚 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)) = (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝐾)))
13		elex 3450	. . 3 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V)
14	13	adantr 480	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) → 𝑀 ∈ V)
15		simpr 484	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) → 𝐾 ∈ ℕ0)
16		dmexg 7852	. . . . . 6 ⊢ (𝑀 ∈ 𝑉 → dom 𝑀 ∈ V)
17	16	dmexd 7854	. . . . 5 ⊢ (𝑀 ∈ 𝑉 → dom dom 𝑀 ∈ V)
18	17, 17	jca 511	. . . 4 ⊢ (𝑀 ∈ 𝑉 → (dom dom 𝑀 ∈ V ∧ dom dom 𝑀 ∈ V))
19	18	adantr 480	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) → (dom dom 𝑀 ∈ V ∧ dom dom 𝑀 ∈ V))
20		mpoexga 8030	. . 3 ⊢ ((dom dom 𝑀 ∈ V ∧ dom dom 𝑀 ∈ V) → (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝐾)) ∈ V)
21	19, 20	syl 17	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) → (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝐾)) ∈ V)
22	2, 12, 14, 15, 21	ovmpod 7519	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) → (𝑀 decompPMat 𝐾) = (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝐾)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3429  dom cdm 5631  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  ℕ₀cn0 12437  coe₁cco1 22141   decompPMat cdecpmat 22727

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-decpmat 22728

This theorem is referenced by:  decpmatval  22730