Theorem decpmatval0 22720

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 22719	. . 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 5860	. . . . . 6 ⊢ (𝑚 = 𝑀 → dom 𝑚 = dom 𝑀)
4	3	adantr 480	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾) → dom 𝑚 = dom 𝑀)
5	4	dmeqd 5862	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾) → dom dom 𝑚 = dom dom 𝑀)
6		oveq 7374	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
7	6	fveq2d 6846	. . . . . 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 6849	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾) → ((coe1‘(𝑖𝑚𝑗))‘𝑘) = ((coe1‘(𝑖𝑀𝑗))‘𝐾))
11	5, 5, 10	mpoeq123dv 7443	. . 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 3463	. . 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 7853	. . . . . 6 ⊢ (𝑀 ∈ 𝑉 → dom 𝑀 ∈ V)
17	16	dmexd 7855	. . . . 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 8031	. . 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 7520	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 3442  dom cdm 5632  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  ℕ₀cn0 12413  coe₁cco1 22130   decompPMat cdecpmat 22718

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 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-decpmat 22719

This theorem is referenced by:  decpmatval  22721