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Theorem decpmatval0 20617
 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 20616 . . 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 5356 . . . . . 6 (𝑚 = 𝑀 → dom 𝑚 = dom 𝑀)
43adantr 480 . . . . 5 ((𝑚 = 𝑀𝑘 = 𝐾) → dom 𝑚 = dom 𝑀)
54dmeqd 5358 . . . 4 ((𝑚 = 𝑀𝑘 = 𝐾) → dom dom 𝑚 = dom dom 𝑀)
6 oveq 6696 . . . . . . 7 (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
76fveq2d 6233 . . . . . 6 (𝑚 = 𝑀 → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
87adantr 480 . . . . 5 ((𝑚 = 𝑀𝑘 = 𝐾) → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
9 simpr 476 . . . . 5 ((𝑚 = 𝑀𝑘 = 𝐾) → 𝑘 = 𝐾)
108, 9fveq12d 6235 . . . 4 ((𝑚 = 𝑀𝑘 = 𝐾) → ((coe1‘(𝑖𝑚𝑗))‘𝑘) = ((coe1‘(𝑖𝑀𝑗))‘𝐾))
115, 5, 10mpt2eq123dv 6759 . . 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 481 . 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 3243 . . 3 (𝑀𝑉𝑀 ∈ V)
1413adantr 480 . 2 ((𝑀𝑉𝐾 ∈ ℕ0) → 𝑀 ∈ V)
15 simpr 476 . 2 ((𝑀𝑉𝐾 ∈ ℕ0) → 𝐾 ∈ ℕ0)
16 dmexg 7139 . . . . . 6 (𝑀𝑉 → dom 𝑀 ∈ V)
17 dmexg 7139 . . . . . 6 (dom 𝑀 ∈ V → dom dom 𝑀 ∈ V)
1816, 17syl 17 . . . . 5 (𝑀𝑉 → dom dom 𝑀 ∈ V)
1918, 18jca 553 . . . 4 (𝑀𝑉 → (dom dom 𝑀 ∈ V ∧ dom dom 𝑀 ∈ V))
2019adantr 480 . . 3 ((𝑀𝑉𝐾 ∈ ℕ0) → (dom dom 𝑀 ∈ V ∧ dom dom 𝑀 ∈ V))
21 mpt2exga 7291 . . 3 ((dom dom 𝑀 ∈ V ∧ dom dom 𝑀 ∈ V) → (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝐾)) ∈ V)
2220, 21syl 17 . 2 ((𝑀𝑉𝐾 ∈ ℕ0) → (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝐾)) ∈ V)
232, 12, 14, 15, 22ovmpt2d 6830 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 383   = wceq 1523   ∈ wcel 2030  Vcvv 3231  dom cdm 5143  ‘cfv 5926  (class class class)co 6690   ↦ cmpt2 6692  ℕ0cn0 11330  coe1cco1 19596   decompPMat cdecpmat 20615 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-decpmat 20616 This theorem is referenced by:  decpmatval  20618
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