MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  decpmatval0 Structured version   Visualization version   GIF version

Theorem decpmatval0 22640
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 22639 . . 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 5900 . . . . . 6 (𝑚 = 𝑀 → dom 𝑚 = dom 𝑀)
43adantr 480 . . . . 5 ((𝑚 = 𝑀𝑘 = 𝐾) → dom 𝑚 = dom 𝑀)
54dmeqd 5902 . . . 4 ((𝑚 = 𝑀𝑘 = 𝐾) → dom dom 𝑚 = dom dom 𝑀)
6 oveq 7420 . . . . . . 7 (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
76fveq2d 6895 . . . . . 6 (𝑚 = 𝑀 → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
87adantr 480 . . . . 5 ((𝑚 = 𝑀𝑘 = 𝐾) → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
9 simpr 484 . . . . 5 ((𝑚 = 𝑀𝑘 = 𝐾) → 𝑘 = 𝐾)
108, 9fveq12d 6898 . . . 4 ((𝑚 = 𝑀𝑘 = 𝐾) → ((coe1‘(𝑖𝑚𝑗))‘𝑘) = ((coe1‘(𝑖𝑀𝑗))‘𝐾))
115, 5, 10mpoeq123dv 7489 . . 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 3488 . . 3 (𝑀𝑉𝑀 ∈ V)
1413adantr 480 . 2 ((𝑀𝑉𝐾 ∈ ℕ0) → 𝑀 ∈ V)
15 simpr 484 . 2 ((𝑀𝑉𝐾 ∈ ℕ0) → 𝐾 ∈ ℕ0)
16 dmexg 7901 . . . . . 6 (𝑀𝑉 → dom 𝑀 ∈ V)
1716dmexd 7903 . . . . 5 (𝑀𝑉 → dom dom 𝑀 ∈ V)
1817, 17jca 511 . . . 4 (𝑀𝑉 → (dom dom 𝑀 ∈ V ∧ dom dom 𝑀 ∈ V))
1918adantr 480 . . 3 ((𝑀𝑉𝐾 ∈ ℕ0) → (dom dom 𝑀 ∈ V ∧ dom dom 𝑀 ∈ V))
20 mpoexga 8074 . . 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 7565 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 395   = wceq 1534  wcel 2099  Vcvv 3469  dom cdm 5672  cfv 6542  (class class class)co 7414  cmpo 7416  0cn0 12488  coe1cco1 22071   decompPMat cdecpmat 22638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7985  df-2nd 7986  df-decpmat 22639
This theorem is referenced by:  decpmatval  22641
  Copyright terms: Public domain W3C validator