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Theorem decpmatval0 20791
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 20790 . . 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 5479 . . . . . 6 (𝑚 = 𝑀 → dom 𝑚 = dom 𝑀)
43adantr 472 . . . . 5 ((𝑚 = 𝑀𝑘 = 𝐾) → dom 𝑚 = dom 𝑀)
54dmeqd 5481 . . . 4 ((𝑚 = 𝑀𝑘 = 𝐾) → dom dom 𝑚 = dom dom 𝑀)
6 oveq 6820 . . . . . . 7 (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
76fveq2d 6357 . . . . . 6 (𝑚 = 𝑀 → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
87adantr 472 . . . . 5 ((𝑚 = 𝑀𝑘 = 𝐾) → (coe1‘(𝑖𝑚𝑗)) = (coe1‘(𝑖𝑀𝑗)))
9 simpr 479 . . . . 5 ((𝑚 = 𝑀𝑘 = 𝐾) → 𝑘 = 𝐾)
108, 9fveq12d 6359 . . . 4 ((𝑚 = 𝑀𝑘 = 𝐾) → ((coe1‘(𝑖𝑚𝑗))‘𝑘) = ((coe1‘(𝑖𝑀𝑗))‘𝐾))
115, 5, 10mpt2eq123dv 6883 . . 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 473 . 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 3352 . . 3 (𝑀𝑉𝑀 ∈ V)
1413adantr 472 . 2 ((𝑀𝑉𝐾 ∈ ℕ0) → 𝑀 ∈ V)
15 simpr 479 . 2 ((𝑀𝑉𝐾 ∈ ℕ0) → 𝐾 ∈ ℕ0)
16 dmexg 7263 . . . . . 6 (𝑀𝑉 → dom 𝑀 ∈ V)
17 dmexg 7263 . . . . . 6 (dom 𝑀 ∈ V → dom dom 𝑀 ∈ V)
1816, 17syl 17 . . . . 5 (𝑀𝑉 → dom dom 𝑀 ∈ V)
1918, 18jca 555 . . . 4 (𝑀𝑉 → (dom dom 𝑀 ∈ V ∧ dom dom 𝑀 ∈ V))
2019adantr 472 . . 3 ((𝑀𝑉𝐾 ∈ ℕ0) → (dom dom 𝑀 ∈ V ∧ dom dom 𝑀 ∈ V))
21 mpt2exga 7415 . . 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 6954 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 1632  wcel 2139  Vcvv 3340  dom cdm 5266  cfv 6049  (class class class)co 6814  cmpt2 6816  0cn0 11504  coe1cco1 19770   decompPMat cdecpmat 20789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-decpmat 20790
This theorem is referenced by:  decpmatval  20792
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