Theorem decpmatval0 22824

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 22823	. . 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 5879	. . . . . 6 ⊢ (𝑚 = 𝑀 → dom 𝑚 = dom 𝑀)
4	3	adantr 484	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾) → dom 𝑚 = dom 𝑀)
5	4	dmeqd 5881	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾) → dom dom 𝑚 = dom dom 𝑀)
6		oveq 7402	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
7	6	fveq2d 6871	. . . . . 6 ⊢ (𝑚 = 𝑀 → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
8	7	adantr 484	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾) → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
9		simpr 488	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾) → 𝑘 = 𝐾)
10	8, 9	fveq12d 6874	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾) → ((coe1‘(𝑖𝑚𝑗))‘𝑘) = ((coe1‘(𝑖𝑀𝑗))‘𝐾))
11	5, 5, 10	mpoeq123dv 7471	. . 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 485	. 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 3475	. . 3 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V)
14	13	adantr 484	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) → 𝑀 ∈ V)
15		simpr 488	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) → 𝐾 ∈ ℕ0)
16		dmexg 7882	. . . . . 6 ⊢ (𝑀 ∈ 𝑉 → dom 𝑀 ∈ V)
17	16	dmexd 7884	. . . . 5 ⊢ (𝑀 ∈ 𝑉 → dom dom 𝑀 ∈ V)
18	17, 17	jca 519	. . . 4 ⊢ (𝑀 ∈ 𝑉 → (dom dom 𝑀 ∈ V ∧ dom dom 𝑀 ∈ V))
19	18	adantr 484	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) → (dom dom 𝑀 ∈ V ∧ dom dom 𝑀 ∈ V))
20		mpoexga 8058	. . 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 7548	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) → (𝑀 decompPMat 𝐾) = (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝐾)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  Vcvv 3454  dom cdm 5647  ‘cfv 6521  (class class class)co 7396   ∈ cmpo 7398  ℕ₀cn0 12481  coe₁cco1 22240   decompPMat cdecpmat 22822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-decpmat 22823

This theorem is referenced by:  decpmatval  22825