Theorem decpmatval0 22679

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 22678	. . 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 5901	. . . . . 6 ⊢ (𝑚 = 𝑀 → dom 𝑚 = dom 𝑀)
4	3	adantr 479	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾) → dom 𝑚 = dom 𝑀)
5	4	dmeqd 5903	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾) → dom dom 𝑚 = dom dom 𝑀)
6		oveq 7419	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
7	6	fveq2d 6894	. . . . . 6 ⊢ (𝑚 = 𝑀 → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
8	7	adantr 479	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾) → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
9		simpr 483	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾) → 𝑘 = 𝐾)
10	8, 9	fveq12d 6897	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑘 = 𝐾) → ((coe1‘(𝑖𝑚𝑗))‘𝑘) = ((coe1‘(𝑖𝑀𝑗))‘𝐾))
11	5, 5, 10	mpoeq123dv 7489	. . 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 480	. 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 3482	. . 3 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V)
14	13	adantr 479	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) → 𝑀 ∈ V)
15		simpr 483	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) → 𝐾 ∈ ℕ0)
16		dmexg 7903	. . . . . 6 ⊢ (𝑀 ∈ 𝑉 → dom 𝑀 ∈ V)
17	16	dmexd 7905	. . . . 5 ⊢ (𝑀 ∈ 𝑉 → dom dom 𝑀 ∈ V)
18	17, 17	jca 510	. . . 4 ⊢ (𝑀 ∈ 𝑉 → (dom dom 𝑀 ∈ V ∧ dom dom 𝑀 ∈ V))
19	18	adantr 479	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) → (dom dom 𝑀 ∈ V ∧ dom dom 𝑀 ∈ V))
20		mpoexga 8075	. . 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 7567	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) → (𝑀 decompPMat 𝐾) = (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝐾)))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3463  dom cdm 5673  ‘cfv 6543  (class class class)co 7413   ∈ cmpo 7415  ℕ₀cn0 12497  coe₁cco1 22100   decompPMat cdecpmat 22677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7987  df-2nd 7988  df-decpmat 22678

This theorem is referenced by:  decpmatval  22680