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

Theorem eqeefv 25982
Description: Two points are equal iff they agree in all dimensions. (Contributed by Scott Fenton, 10-Jun-2013.)
Assertion
Ref Expression
eqeefv ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (𝐴 = 𝐵 ↔ ∀𝑖 ∈ (1...𝑁)(𝐴𝑖) = (𝐵𝑖)))
Distinct variable groups:   𝐴,𝑖   𝐵,𝑖   𝑖,𝑁

Proof of Theorem eqeefv
StepHypRef Expression
1 eleei 25976 . . 3 (𝐴 ∈ (𝔼‘𝑁) → 𝐴:(1...𝑁)⟶ℝ)
2 ffn 6206 . . 3 (𝐴:(1...𝑁)⟶ℝ → 𝐴 Fn (1...𝑁))
31, 2syl 17 . 2 (𝐴 ∈ (𝔼‘𝑁) → 𝐴 Fn (1...𝑁))
4 eleei 25976 . . 3 (𝐵 ∈ (𝔼‘𝑁) → 𝐵:(1...𝑁)⟶ℝ)
5 ffn 6206 . . 3 (𝐵:(1...𝑁)⟶ℝ → 𝐵 Fn (1...𝑁))
64, 5syl 17 . 2 (𝐵 ∈ (𝔼‘𝑁) → 𝐵 Fn (1...𝑁))
7 eqfnfv 6474 . 2 ((𝐴 Fn (1...𝑁) ∧ 𝐵 Fn (1...𝑁)) → (𝐴 = 𝐵 ↔ ∀𝑖 ∈ (1...𝑁)(𝐴𝑖) = (𝐵𝑖)))
83, 6, 7syl2an 495 1 ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (𝐴 = 𝐵 ↔ ∀𝑖 ∈ (1...𝑁)(𝐴𝑖) = (𝐵𝑖)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6813  cr 10127  1c1 10129  ...cfz 12519  𝔼cee 25967
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 7114  ax-cnex 10184  ax-resscn 10185
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-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-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-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-map 8025  df-ee 25970
This theorem is referenced by:  eqeelen  25983  brbtwn2  25984  colinearalg  25989  axcgrid  25995  ax5seglem4  26011  ax5seglem5  26012  axbtwnid  26018  axeuclid  26042  axcontlem2  26044  axcontlem4  26046  axcontlem7  26049
  Copyright terms: Public domain W3C validator