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Theorem decpmatval0 22113
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 ((𝑀𝑉𝐾 ∈ 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.
StepHypRef Expression
1 df-decpmat 22112 . . 3 decompPMat = (𝑚 ∈ V, 𝑘 ∈ ℕ0 ↦ (𝑖 ∈ dom dom 𝑚, 𝑗 ∈ dom dom 𝑚 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)))
21a1i 11 . 2 ((𝑀𝑉𝐾 ∈ ℕ0) → decompPMat = (𝑚 ∈ V, 𝑘 ∈ ℕ0 ↦ (𝑖 ∈ dom dom 𝑚, 𝑗 ∈ dom dom 𝑚 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘))))
3 dmeq 5859 . . . . . 6 (𝑚 = 𝑀 → dom 𝑚 = dom 𝑀)
43adantr 481 . . . . 5 ((𝑚 = 𝑀𝑘 = 𝐾) → dom 𝑚 = dom 𝑀)
54dmeqd 5861 . . . 4 ((𝑚 = 𝑀𝑘 = 𝐾) → dom dom 𝑚 = dom dom 𝑀)
6 oveq 7363 . . . . . . 7 (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
76fveq2d 6846 . . . . . 6 (𝑚 = 𝑀 → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
87adantr 481 . . . . 5 ((𝑚 = 𝑀𝑘 = 𝐾) → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
9 simpr 485 . . . . 5 ((𝑚 = 𝑀𝑘 = 𝐾) → 𝑘 = 𝐾)
108, 9fveq12d 6849 . . . 4 ((𝑚 = 𝑀𝑘 = 𝐾) → ((coe1‘(𝑖𝑚𝑗))‘𝑘) = ((coe1‘(𝑖𝑀𝑗))‘𝐾))
115, 5, 10mpoeq123dv 7432 . . 3 ((𝑚 = 𝑀𝑘 = 𝐾) → (𝑖 ∈ dom dom 𝑚, 𝑗 ∈ dom dom 𝑚 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)) = (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝐾)))
1211adantl 482 . 2 (((𝑀𝑉𝐾 ∈ ℕ0) ∧ (𝑚 = 𝑀𝑘 = 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 (𝑀𝑉𝑀 ∈ V)
1413adantr 481 . 2 ((𝑀𝑉𝐾 ∈ ℕ0) → 𝑀 ∈ V)
15 simpr 485 . 2 ((𝑀𝑉𝐾 ∈ ℕ0) → 𝐾 ∈ ℕ0)
16 dmexg 7840 . . . . . 6 (𝑀𝑉 → dom 𝑀 ∈ V)
1716dmexd 7842 . . . . 5 (𝑀𝑉 → dom dom 𝑀 ∈ V)
1817, 17jca 512 . . . 4 (𝑀𝑉 → (dom dom 𝑀 ∈ V ∧ dom dom 𝑀 ∈ V))
1918adantr 481 . . 3 ((𝑀𝑉𝐾 ∈ ℕ0) → (dom dom 𝑀 ∈ V ∧ dom dom 𝑀 ∈ V))
20 mpoexga 8010 . . 3 ((dom dom 𝑀 ∈ V ∧ dom dom 𝑀 ∈ V) → (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝐾)) ∈ V)
2119, 20syl 17 . 2 ((𝑀𝑉𝐾 ∈ ℕ0) → (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝐾)) ∈ V)
222, 12, 14, 15, 21ovmpod 7507 1 ((𝑀𝑉𝐾 ∈ ℕ0) → (𝑀 decompPMat 𝐾) = (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3445  dom cdm 5633  cfv 6496  (class class class)co 7357  cmpo 7359  0cn0 12413  coe1cco1 21549   decompPMat cdecpmat 22111
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-decpmat 22112
This theorem is referenced by:  decpmatval  22114
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