MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  s4eqd Structured version   Visualization version   GIF version

Theorem s4eqd 14218
Description: Equality theorem for a length 4 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
Hypotheses
Ref Expression
s2eqd.1 (𝜑𝐴 = 𝑁)
s2eqd.2 (𝜑𝐵 = 𝑂)
s3eqd.3 (𝜑𝐶 = 𝑃)
s4eqd.4 (𝜑𝐷 = 𝑄)
Assertion
Ref Expression
s4eqd (𝜑 → ⟨“𝐴𝐵𝐶𝐷”⟩ = ⟨“𝑁𝑂𝑃𝑄”⟩)

Proof of Theorem s4eqd
StepHypRef Expression
1 s2eqd.1 . . . 4 (𝜑𝐴 = 𝑁)
2 s2eqd.2 . . . 4 (𝜑𝐵 = 𝑂)
3 s3eqd.3 . . . 4 (𝜑𝐶 = 𝑃)
41, 2, 3s3eqd 14217 . . 3 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝑁𝑂𝑃”⟩)
5 s4eqd.4 . . . 4 (𝜑𝐷 = 𝑄)
65s1eqd 13946 . . 3 (𝜑 → ⟨“𝐷”⟩ = ⟨“𝑄”⟩)
74, 6oveq12d 7158 . 2 (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩) = (⟨“𝑁𝑂𝑃”⟩ ++ ⟨“𝑄”⟩))
8 df-s4 14203 . 2 ⟨“𝐴𝐵𝐶𝐷”⟩ = (⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)
9 df-s4 14203 . 2 ⟨“𝑁𝑂𝑃𝑄”⟩ = (⟨“𝑁𝑂𝑃”⟩ ++ ⟨“𝑄”⟩)
107, 8, 93eqtr4g 2882 1 (𝜑 → ⟨“𝐴𝐵𝐶𝐷”⟩ = ⟨“𝑁𝑂𝑃𝑄”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  (class class class)co 7140   ++ cconcat 13913  ⟨“cs1 13940  ⟨“cs3 14195  ⟨“cs4 14196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-v 3471  df-un 3913  df-in 3915  df-ss 3925  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-iota 6293  df-fv 6342  df-ov 7143  df-s1 13941  df-s2 14201  df-s3 14202  df-s4 14203
This theorem is referenced by:  s5eqd  14219
  Copyright terms: Public domain W3C validator