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Theorem fveere 26001
Description: The function value of a point is a real. (Contributed by Scott Fenton, 10-Jun-2013.)
Assertion
Ref Expression
fveere ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐼 ∈ (1...𝑁)) → (𝐴𝐼) ∈ ℝ)

Proof of Theorem fveere
StepHypRef Expression
1 eleei 25997 . 2 (𝐴 ∈ (𝔼‘𝑁) → 𝐴:(1...𝑁)⟶ℝ)
21ffvelrnda 6523 1 ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐼 ∈ (1...𝑁)) → (𝐴𝐼) ∈ ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2139  cfv 6049  (class class class)co 6814  cr 10147  1c1 10149  ...cfz 12539  𝔼cee 25988
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-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205
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-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  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-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-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-map 8027  df-ee 25991
This theorem is referenced by:  fveecn  26002  eqeelen  26004  brbtwn2  26005  colinearalglem4  26009  colinearalg  26010  eleesub  26011  eleesubd  26012  axcgrid  26016  axsegconlem1  26017  axsegconlem2  26018  axsegconlem3  26019  axsegconlem8  26024  axsegconlem9  26025  axsegconlem10  26026  ax5seglem3a  26030  ax5seg  26038  axpasch  26041  axeuclidlem  26062  axcontlem2  26065
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