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Theorem decpmatval0 20483
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 20482 . . 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 5289 . . . . . 6 (𝑚 = 𝑀 → dom 𝑚 = dom 𝑀)
43adantr 481 . . . . 5 ((𝑚 = 𝑀𝑘 = 𝐾) → dom 𝑚 = dom 𝑀)
54dmeqd 5291 . . . 4 ((𝑚 = 𝑀𝑘 = 𝐾) → dom dom 𝑚 = dom dom 𝑀)
6 oveq 6611 . . . . . . 7 (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
76fveq2d 6154 . . . . . 6 (𝑚 = 𝑀 → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
87adantr 481 . . . . 5 ((𝑚 = 𝑀𝑘 = 𝐾) → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
9 simpr 477 . . . . 5 ((𝑚 = 𝑀𝑘 = 𝐾) → 𝑘 = 𝐾)
108, 9fveq12d 6156 . . . 4 ((𝑚 = 𝑀𝑘 = 𝐾) → ((coe1‘(𝑖𝑚𝑗))‘𝑘) = ((coe1‘(𝑖𝑀𝑗))‘𝐾))
115, 5, 10mpt2eq123dv 6671 . . 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 3203 . . 3 (𝑀𝑉𝑀 ∈ V)
1413adantr 481 . 2 ((𝑀𝑉𝐾 ∈ ℕ0) → 𝑀 ∈ V)
15 simpr 477 . 2 ((𝑀𝑉𝐾 ∈ ℕ0) → 𝐾 ∈ ℕ0)
16 dmexg 7045 . . . . . 6 (𝑀𝑉 → dom 𝑀 ∈ V)
17 dmexg 7045 . . . . . 6 (dom 𝑀 ∈ V → dom dom 𝑀 ∈ V)
1816, 17syl 17 . . . . 5 (𝑀𝑉 → dom dom 𝑀 ∈ V)
1918, 18jca 554 . . . 4 (𝑀𝑉 → (dom dom 𝑀 ∈ V ∧ dom dom 𝑀 ∈ V))
2019adantr 481 . . 3 ((𝑀𝑉𝐾 ∈ ℕ0) → (dom dom 𝑀 ∈ V ∧ dom dom 𝑀 ∈ V))
21 mpt2exga 7192 . . 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 6742 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 384   = wceq 1480  wcel 1992  Vcvv 3191  dom cdm 5079  cfv 5850  (class class class)co 6605  cmpt2 6607  0cn0 11237  coe1cco1 19462   decompPMat cdecpmat 20481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-decpmat 20482
This theorem is referenced by:  decpmatval  20484
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