Theorem decpmatval0 21371

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 21370	. . 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 5771	. . . . . 6 ⊢ (𝑚 = 𝑀 → dom 𝑚 = dom 𝑀)
4	3	adantr 483	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾) → dom 𝑚 = dom 𝑀)
5	4	dmeqd 5773	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾) → dom dom 𝑚 = dom dom 𝑀)
6		oveq 7161	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
7	6	fveq2d 6673	. . . . . 6 ⊢ (𝑚 = 𝑀 → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
8	7	adantr 483	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾) → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
9		simpr 487	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾) → 𝑘 = 𝐾)
10	8, 9	fveq12d 6676	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾) → ((coe1‘(𝑖𝑚𝑗))‘𝑘) = ((coe1‘(𝑖𝑀𝑗))‘𝐾))
11	5, 5, 10	mpoeq123dv 7228	. . 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 484	. 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 3512	. . 3 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V)
14	13	adantr 483	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) → 𝑀 ∈ V)
15		simpr 487	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) → 𝐾 ∈ ℕ0)
16		dmexg 7612	. . . . . 6 ⊢ (𝑀 ∈ 𝑉 → dom 𝑀 ∈ V)
17	16	dmexd 7614	. . . . 5 ⊢ (𝑀 ∈ 𝑉 → dom dom 𝑀 ∈ V)
18	17, 17	jca 514	. . . 4 ⊢ (𝑀 ∈ 𝑉 → (dom dom 𝑀 ∈ V ∧ dom dom 𝑀 ∈ V))
19	18	adantr 483	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) → (dom dom 𝑀 ∈ V ∧ dom dom 𝑀 ∈ V))
20		mpoexga 7774	. . 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 7301	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) → (𝑀 decompPMat 𝐾) = (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝐾)))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  Vcvv 3494  dom cdm 5554  ‘cfv 6354  (class class class)co 7155   ∈ cmpo 7157  ℕ₀cn0 11896  coe₁cco1 20345   decompPMat cdecpmat 21369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7688  df-2nd 7689  df-decpmat 21370

This theorem is referenced by:  decpmatval  21372