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Theorem s8eqd 14226
Description: Equality theorem for a length 8 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)
s5eqd.5 (𝜑𝐸 = 𝑅)
s6eqd.6 (𝜑𝐹 = 𝑆)
s7eqd.6 (𝜑𝐺 = 𝑇)
s8eqd.6 (𝜑𝐻 = 𝑈)
Assertion
Ref Expression
s8eqd (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇𝑈”⟩)

Proof of Theorem s8eqd
StepHypRef Expression
1 s2eqd.1 . . . 4 (𝜑𝐴 = 𝑁)
2 s2eqd.2 . . . 4 (𝜑𝐵 = 𝑂)
3 s3eqd.3 . . . 4 (𝜑𝐶 = 𝑃)
4 s4eqd.4 . . . 4 (𝜑𝐷 = 𝑄)
5 s5eqd.5 . . . 4 (𝜑𝐸 = 𝑅)
6 s6eqd.6 . . . 4 (𝜑𝐹 = 𝑆)
7 s7eqd.6 . . . 4 (𝜑𝐺 = 𝑇)
81, 2, 3, 4, 5, 6, 7s7eqd 14225 . . 3 (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇”⟩)
9 s8eqd.6 . . . 4 (𝜑𝐻 = 𝑈)
109s1eqd 13950 . . 3 (𝜑 → ⟨“𝐻”⟩ = ⟨“𝑈”⟩)
118, 10oveq12d 7157 . 2 (𝜑 → (⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ ++ ⟨“𝐻”⟩) = (⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇”⟩ ++ ⟨“𝑈”⟩))
12 df-s8 14211 . 2 ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ ++ ⟨“𝐻”⟩)
13 df-s8 14211 . 2 ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇𝑈”⟩ = (⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇”⟩ ++ ⟨“𝑈”⟩)
1411, 12, 133eqtr4g 2861 1 (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇𝑈”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  (class class class)co 7139   ++ cconcat 13917  ⟨“cs1 13944  ⟨“cs7 14203  ⟨“cs8 14204
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 2114  ax-9 2122  ax-ext 2773
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 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-iota 6287  df-fv 6336  df-ov 7142  df-s1 13945  df-s2 14205  df-s3 14206  df-s4 14207  df-s5 14208  df-s6 14209  df-s7 14210  df-s8 14211
This theorem is referenced by: (None)
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