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Theorem coe1fval 22207
Description: Value of the univariate polynomial coefficient function. (Contributed by Stefan O'Rear, 21-Mar-2015.)
Hypothesis
Ref Expression
coe1fval.a 𝐴 = (coe1𝐹)
Assertion
Ref Expression
coe1fval (𝐹𝑉𝐴 = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))))
Distinct variable group:   𝑛,𝐹
Allowed substitution hints:   𝐴(𝑛)   𝑉(𝑛)

Proof of Theorem coe1fval
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 elex 3501 . 2 (𝐹𝑉𝐹 ∈ V)
2 coe1fval.a . . 3 𝐴 = (coe1𝐹)
3 fveq1 6905 . . . . 5 (𝑓 = 𝐹 → (𝑓‘(1o × {𝑛})) = (𝐹‘(1o × {𝑛})))
43mpteq2dv 5244 . . . 4 (𝑓 = 𝐹 → (𝑛 ∈ ℕ0 ↦ (𝑓‘(1o × {𝑛}))) = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))))
5 df-coe1 22184 . . . 4 coe1 = (𝑓 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (𝑓‘(1o × {𝑛}))))
6 nn0ex 12532 . . . . 5 0 ∈ V
76mptex 7243 . . . 4 (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))) ∈ V
84, 5, 7fvmpt 7016 . . 3 (𝐹 ∈ V → (coe1𝐹) = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))))
92, 8eqtrid 2789 . 2 (𝐹 ∈ V → 𝐴 = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))))
101, 9syl 17 1 (𝐹𝑉𝐴 = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1o × {𝑛}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  {csn 4626  cmpt 5225   × cxp 5683  cfv 6561  1oc1o 8499  0cn0 12526  coe1cco1 22179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-1cn 11213  ax-addcl 11215
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-nn 12267  df-n0 12527  df-coe1 22184
This theorem is referenced by:  coe1fv  22208  coe1fval3  22210
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