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Theorem coe1fval 19790
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 3364 . 2 (𝐹𝑉𝐹 ∈ V)
2 coe1fval.a . . 3 𝐴 = (coe1𝐹)
3 fveq1 6331 . . . . 5 (𝑓 = 𝐹 → (𝑓‘(1𝑜 × {𝑛})) = (𝐹‘(1𝑜 × {𝑛})))
43mpteq2dv 4879 . . . 4 (𝑓 = 𝐹 → (𝑛 ∈ ℕ0 ↦ (𝑓‘(1𝑜 × {𝑛}))) = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1𝑜 × {𝑛}))))
5 df-coe1 19768 . . . 4 coe1 = (𝑓 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (𝑓‘(1𝑜 × {𝑛}))))
6 nn0ex 11500 . . . . 5 0 ∈ V
76mptex 6630 . . . 4 (𝑛 ∈ ℕ0 ↦ (𝐹‘(1𝑜 × {𝑛}))) ∈ V
84, 5, 7fvmpt 6424 . . 3 (𝐹 ∈ V → (coe1𝐹) = (𝑛 ∈ ℕ0 ↦ (𝐹‘(1𝑜 × {𝑛}))))
92, 8syl5eq 2817 . 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 1631  wcel 2145  Vcvv 3351  {csn 4316  cmpt 4863   × cxp 5247  cfv 6031  1𝑜c1o 7706  0cn0 11494  coe1cco1 19763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-i2m1 10206  ax-1ne0 10207  ax-rrecex 10210  ax-cnre 10211
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-nn 11223  df-n0 11495  df-coe1 19768
This theorem is referenced by:  coe1fv  19791  coe1fval3  19793
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